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A collective term for a series of quantitative characteristics (in terms of numbers, vectors, tensors) describing the degree to which some object (a curve, a surface, a Riemannian space, etc.) deviates in its properties from certain other objects (a straight line, a plane, a Euclidean space, etc.) which are considered to be flat. The concepts of curvature are usually defined locally, i.e. at each point. These concepts of curvature are connected with the examination of deviations which are small to the second order; hence the object in question is assumed to be specified by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c0273201.png" />-smooth functions. In some cases the concepts are defined in terms of integrals, and they remain valid without the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c0273202.png" />-smoothness condition. As a rule, if the curvature vanishes at all points, the object in question is identical (in small sections, not in the large) with the corresponding  "flat"  object.
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A collective term for a series of quantitative characteristics (in terms of numbers, vectors, tensors) describing the degree to which some object (a curve, a surface, a Riemannian space, etc.) deviates in its properties from certain other objects (a straight line, a plane, a Euclidean space, etc.) which are considered to be flat. The concepts of curvature are usually defined locally, i.e. at each point. These concepts of curvature are connected with the examination of deviations which are small to the second order; hence the object in question is assumed to be specified by $  C  ^ {2} $-
 +
smooth functions. In some cases the concepts are defined in terms of integrals, and they remain valid without the $  C  ^ {2} $-
 +
smoothness condition. As a rule, if the curvature vanishes at all points, the object in question is identical (in small sections, not in the large) with the corresponding  "flat"  object.
  
 
==The curvature of a curve.==
 
==The curvature of a curve.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c0273203.png" /> be a regular curve in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c0273204.png" />-dimensional Euclidean space, parametrized in terms of its natural parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c0273205.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c0273206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c0273207.png" /> be the angle between the tangents to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c0273208.png" /> at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c0273209.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732011.png" /> and the length of the arc of the curve between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732013.png" />, respectively. Then the limit
+
Let $  \gamma $
 +
be a regular curve in the $  n $-
 +
dimensional Euclidean space, parametrized in terms of its natural parameter $  t $.  
 +
Let $  \alpha ( P, P _ {1} ) $
 +
and $  s ( P, P _ {1} ) $
 +
be the angle between the tangents to $  \gamma $
 +
at the points $  P $
 +
and $  P _ {1} $
 +
of $  \gamma $
 +
and the length of the arc of the curve between $  P $
 +
and $  P _ {1} $,  
 +
respectively. Then the limit
 +
 
 +
$$
 +
k  =  \lim\limits _ {P _ {1} \rightarrow P } \
 +
 
 +
\frac{\alpha ( P, P _ {1} ) }{s ( P, P _ {1} ) }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732014.png" /></td> </tr></table>
+
$$
  
is called the curvature of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732015.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732016.png" />. The curvature of the curve is equal to the absolute value of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732017.png" />, and the direction of this vector is just the direction of the principal normal to the curve. For the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732018.png" /> to coincide with some segment of a straight line or with an entire line it is necessary and sufficient that its curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732019.png" /> vanishes identically.
+
is called the curvature of the curve $  \gamma $
 +
at $  P $.  
 +
The curvature of the curve is equal to the absolute value of the vector $  d  ^ {2} \gamma ( t)/dt  ^ {2} $,  
 +
and the direction of this vector is just the direction of the principal normal to the curve. For the curve $  \gamma $
 +
to coincide with some segment of a straight line or with an entire line it is necessary and sufficient that its curvature $  k $
 +
vanishes identically.
  
 
==The curvature of a surface.==
 
==The curvature of a surface.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732020.png" /> be a regular surface in the three-dimensional Euclidean space. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732021.png" /> be a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732023.png" /> the tangent plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732024.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732026.png" /> the normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732027.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732028.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732029.png" /> the plane through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732030.png" /> and some unit vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732032.png" />. The intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732033.png" /> of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732034.png" /> and the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732035.png" /> is a curve, called the normal section of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732036.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732037.png" /> in the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732038.png" />. The number
+
Let $  \Phi $
 +
be a regular surface in the three-dimensional Euclidean space. Let $  P $
 +
be a point of $  \Phi $,  
 +
$  T _ {p} $
 +
the tangent plane to $  \Phi $
 +
at $  P $,  
 +
$  \mathbf n $
 +
the normal to $  \Phi $
 +
at $  P $,  
 +
and $  \alpha $
 +
the plane through $  \mathbf n $
 +
and some unit vector $  \mathbf l $
 +
in $  T _ {p} $.  
 +
The intersection $  \gamma _ {\mathbf l} $
 +
of the plane $  \alpha $
 +
and the surface $  \Phi $
 +
is a curve, called the normal section of the surface $  \Phi $
 +
at the point $  P $
 +
in the direction $  \mathbf l $.  
 +
The number
 +
 
 +
$$
 +
k _ {\mathbf l}  = \
 +
\left (
 +
 
 +
\frac{d  ^ {2} \gamma _ {\mathbf l} }{dt  ^ {2} }
 +
, \mathbf n \
 +
\right ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732039.png" /></td> </tr></table>
+
where  $  t $
 +
is the natural parameter on  $  \gamma $,
 +
is called the normal curvature of  $  \Phi $
 +
in the direction  $  \mathbf l $.  
 +
The normal curvature is equal to the curvature of the curve  $  \gamma _ {\mathbf l} $
 +
up to the sign.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732040.png" /> is the natural parameter on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732041.png" />, is called the normal curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732042.png" /> in the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732043.png" />. The normal curvature is equal to the curvature of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732044.png" /> up to the sign.
+
The tangent plane  $  T _ {p} $
 +
contains two perpendicular directions  $  \mathbf l _ {1} $
 +
and  $  \mathbf l _ {2} $
 +
such that the normal curvature in any direction can be expressed by Euler's formula:
  
The tangent plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732045.png" /> contains two perpendicular directions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732047.png" /> such that the normal curvature in any direction can be expressed by Euler's formula:
+
$$
 +
k _ {\mathbf l}  = \
 +
k _ {1}  \cos  ^ {2}  \theta +
 +
k _ {2}  \sin  ^ {2}  \theta ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732048.png" /></td> </tr></table>
+
where  $  \theta $
 +
is the angle between  $  \mathbf l _ {1} $
 +
and  $  \mathbf l $.
 +
The numbers  $  k _ {1} $
 +
and  $  k _ {2} $
 +
are called the principal curvatures, and the directions  $  \mathbf l _ {1} $
 +
and  $  \mathbf l _ {2} $
 +
are known as the principal directions of the surface. The principal curvatures are extremal values of the normal curvature. The construction of the normal curvature at a given point of the surface may be represented graphically as follows. When  $  k _ {\mathbf l} \neq 0 $,
 +
the equation
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732049.png" /> is the angle between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732051.png" />. The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732053.png" /> are called the principal curvatures, and the directions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732055.png" /> are known as the principal directions of the surface. The principal curvatures are extremal values of the normal curvature. The construction of the normal curvature at a given point of the surface may be represented graphically as follows. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732056.png" />, the equation
+
$$
 +
\mathbf r ( \mathbf l )  = \
 +
\mathbf l \left |
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732057.png" /></td> </tr></table>
+
\frac{1}{k _ {\mathbf l} }
 +
\
 +
\right |  ^ {1/2} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732058.png" /> is the radius vector, defines a certain curve of the second order in the tangent plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732059.png" />, known as the Dupin indicatrix. The Dupin indicatrix can only be one of the following three curves: an ellipse, a hyperbola or a pair of parallel lines. The points of the surface are accordingly classified as elliptic, hyperbolic or parabolic. At an elliptic point, the second fundamental form of the surface is of fixed sign; at a hyperbolic point the form is of variable sign; and at a parabolic point it is degenerate. If all normal curvatures at a point are zero, the point is said to be flat. If the Dupin indicatrix is a circle it is called an umbilical (or spherical) point.
+
where $  \mathbf r ( \mathbf l ) $
 +
is the radius vector, defines a certain curve of the second order in the tangent plane $  T _ {p} $,  
 +
known as the Dupin indicatrix. The Dupin indicatrix can only be one of the following three curves: an ellipse, a hyperbola or a pair of parallel lines. The points of the surface are accordingly classified as elliptic, hyperbolic or parabolic. At an elliptic point, the second fundamental form of the surface is of fixed sign; at a hyperbolic point the form is of variable sign; and at a parabolic point it is degenerate. If all normal curvatures at a point are zero, the point is said to be flat. If the Dupin indicatrix is a circle it is called an umbilical (or spherical) point.
  
The principal directions are uniquely determined (up to the order), unless the point in question is an umbilical point or a flat point. In these cases every direction is principal. In this connection one has the following theorem of Rodrigues: A direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732060.png" /> is principal if and only if
+
The principal directions are uniquely determined (up to the order), unless the point in question is an umbilical point or a flat point. In these cases every direction is principal. In this connection one has the following theorem of Rodrigues: A direction $  \mathbf l $
 +
is principal if and only if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732061.png" /></td> </tr></table>
+
$$
 +
d \mathbf n  = \
 +
- \lambda  d \mathbf r \ \
 +
\mathop{\rm in}  \textrm{ the }  \textrm{ direction }  \mathbf l ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732062.png" /> is the radius vector of the surface and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732063.png" /> the unit normal vector.
+
where $  \mathbf r $
 +
is the radius vector of the surface and $  \mathbf n $
 +
the unit normal vector.
  
A curve on a surface is called a curvature line if its direction at every point is principal. In a neighbourhood of every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732064.png" /> on a surface, other than an umbilical point or a flat point, the surface may be so parametrized that its coordinate curves are curvature lines.
+
A curve on a surface is called a curvature line if its direction at every point is principal. In a neighbourhood of every point $  P $
 +
on a surface, other than an umbilical point or a flat point, the surface may be so parametrized that its coordinate curves are curvature lines.
  
 
The quantity
 
The quantity
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732065.png" /></td> </tr></table>
+
$$
 +
= \
 +
{
 +
\frac{1}{2}
 +
}
 +
( k _ {1} + k _ {2} )
 +
$$
  
 
is called the mean curvature of the surface. The quantity
 
is called the mean curvature of the surface. The quantity
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732066.png" /></td> </tr></table>
+
$$
 +
= k _ {1} k _ {2}  $$
  
 
is called the Gaussian (or total) curvature of the surface. The Gaussian curvature is an object of the intrinsic geometry of the surface, i.e. it can be expressed in terms of the first fundamental form:
 
is called the Gaussian (or total) curvature of the surface. The Gaussian curvature is an object of the intrinsic geometry of the surface, i.e. it can be expressed in terms of the first fundamental form:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732067.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
= \
 +
 
 +
\frac{1}{( EG - F  ^ {2} )  ^ {2} }
 +
 
 +
\left |
 +
 
 +
\begin{array}{lll}
 +
E  &E _ {u}  &E _ {v}  \\
 +
F  &F _ {u}  &F _ {v}  \\
 +
G  &G _ {u}  &G _ {v}  \\
 +
\end{array}
 +
\
 +
\right | -
 +
$$
 +
 
 +
$$
 +
-  
 +
 
 +
\frac{1}{2 \sqrt {EG - F  ^ {2} } }
 +
\left \{ \left (  
 +
\frac{E _ {v} - F _ {u} }{\sqrt {EG - F  ^ {2} } }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732068.png" /></td> </tr></table>
+
\right ) _ {v} - \left (
 +
\frac{F _ {v} - G _ {u} }{\sqrt {EG - F  ^ {2} } }
 +
\right ) _ {u} \right \} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732069.png" /> are the coefficients of the first fundamental form of the surface.
+
where $  E, F, G $
 +
are the coefficients of the first fundamental form of the surface.
  
Using formula (1), one defines the Gaussian curvature for an abstract two-dimensional Riemannian manifold with line element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732070.png" />. A surface is locally isometric to a plane if and only if its Gaussian curvature vanishes identically.
+
Using formula (1), one defines the Gaussian curvature for an abstract two-dimensional Riemannian manifold with line element $  ds  ^ {2} $.  
 +
A surface is locally isometric to a plane if and only if its Gaussian curvature vanishes identically.
  
 
==The curvature of a Riemannian space.==
 
==The curvature of a Riemannian space.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732071.png" /> be a regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732072.png" />-dimensional [[Riemannian space|Riemannian space]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732073.png" /> be the space of regular vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732074.png" />. The curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732075.png" /> is usually characterized by the Riemann (curvature) tensor (cf. [[Riemann tensor|Riemann tensor]]), i.e. by the multilinear mapping
+
Let $  M  ^ {n} $
 +
be a regular $  n $-
 +
dimensional [[Riemannian space|Riemannian space]] and let $  BM  ^ {n} $
 +
be the space of regular vector fields on $  M  ^ {n} $.  
 +
The curvature of $  M  ^ {n} $
 +
is usually characterized by the Riemann (curvature) tensor (cf. [[Riemann tensor|Riemann tensor]]), i.e. by the multilinear mapping
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732076.png" /></td> </tr></table>
+
$$
 +
R: BM  ^ {n} \times BM  ^ {n} \times BM  ^ {n}  \rightarrow  BM  ^ {n} ,
 +
$$
  
 
defined by
 
defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732077.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
R ( X, Y) Z  = \
 +
\nabla _ {X} \nabla _ {Y} Z -
 +
\nabla _ {Y} \nabla _ {X} Z -
 +
\nabla _ {[ X, Y] }  Z,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732078.png" /> is the Levi-Civita connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732080.png" /> denotes the Lie bracket. If one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732082.png" /> in some local coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732083.png" />, one can rewrite (2) as follows:
+
where $  \nabla $
 +
is the Levi-Civita connection on $  M  ^ {n} $
 +
and $  [  , ] $
 +
denotes the Lie bracket. If one puts $  X = \partial  / \partial  x  ^ {k} $,  
 +
$  Y = \partial  / \partial  x  ^ {l} $
 +
in some local coordinate system $  x  ^ {i} $,  
 +
one can rewrite (2) as follows:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732084.png" /></td> </tr></table>
+
$$
 +
Z _ {; k; l }  ^ { i } -
 +
Z _ {; l; k }  ^ { i }  = \
 +
Z  ^ {m} R _ {mkl} ^ { i } ,
 +
$$
  
 
where; is the symbol for covariant differentiation.
 
where; is the symbol for covariant differentiation.
Line 68: Line 221:
 
Thus, the Riemann tensor is a quantitative characteristic of the non-commutativity of the second covariant derivatives in a Riemannian space. It also yields a quantitative description of certain other properties of Riemannian spaces — properties that distinguish them from Euclidean spaces.
 
Thus, the Riemann tensor is a quantitative characteristic of the non-commutativity of the second covariant derivatives in a Riemannian space. It also yields a quantitative description of certain other properties of Riemannian spaces — properties that distinguish them from Euclidean spaces.
  
The coefficients of the Riemann tensor in the local coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732085.png" /> may be expressed in terms of the Christoffel symbols and the coefficients of the metric tensor, as follows:
+
The coefficients of the Riemann tensor in the local coordinate system $  x  ^ {i} $
 +
may be expressed in terms of the Christoffel symbols and the coefficients of the metric tensor, as follows:
 +
 
 +
$$
 +
R _ {ijk} ^ { l }  = \
 +
 
 +
\frac{\partial  \Gamma _ {jk}  ^ {l} }{\partial  x  ^ {i} }
 +
-
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732086.png" /></td> </tr></table>
+
\frac{\partial  \Gamma _ {ik}  ^ {l} }{\partial  x  ^ {j} }
 +
+
 +
\Gamma _ {ir}  ^ {l} \Gamma _ {jk}  ^ {r} -
 +
\Gamma _ {jr}  ^ {l} \Gamma _ {ik}  ^ {r} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732087.png" /></td> </tr></table>
+
$$
 +
R _ {iklm}  = {
 +
\frac{1}{2}
 +
} \left (
 +
\frac{\partial  ^ {2} g _ {il} }{\partial  x  ^ {k} \partial  x  ^ {m} }
 +
+
 +
\frac{\partial  ^ {2} g _ {km} }{\partial  x  ^ {i} \partial  x  ^ {l} }
 +
-
 +
\frac{\partial
 +
^ {2} g _ {im} }{\partial  x  ^ {k} \partial  x
 +
^ {l} }
 +
-  
 +
\frac{\partial  ^ {2} g _ {kl} }{\partial  x  ^ {i} \partial  x  ^ {m} }
 +
\right ) +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732088.png" /></td> </tr></table>
+
$$
 +
+
 +
g _ {np} \left ( \Gamma _ {km}  ^ {p} \Gamma _ {il}  ^ {n} - \Gamma _ {im}  ^ {p} \Gamma _ {kl}  ^ {n} \right ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732089.png" /> is the Riemann tensor with fourth covariant index, or — in a coordinate-free notation — the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732090.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732091.png" /> denotes the scalar product).
+
where $  R _ {iklm} $
 +
is the Riemann tensor with fourth covariant index, or — in a coordinate-free notation — the mapping $  \langle  R ( X, Y) U, Z \rangle $(
 +
where $  \langle  \cdot , \cdot \rangle $
 +
denotes the scalar product).
  
 
The Riemann tensor possesses the following symmetry properties:
 
The Riemann tensor possesses the following symmetry properties:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732092.png" /></td> </tr></table>
+
$$
 +
R ( X, Y) Z  = \
 +
- R ( Y, X) Z,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732093.png" /></td> </tr></table>
+
$$
 +
\langle  R ( X, Y) Z, U \rangle  = - \langle  R ( X, Y) U, Z \rangle ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732094.png" /></td> </tr></table>
+
$$
 +
\langle  R ( X, Y) Z, U \rangle  = \langle  R ( Z, U) X, Y \rangle ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732095.png" /></td> </tr></table>
+
$$
 +
R ( X, Y) Z + R ( Y, Z) X + R ( Z, X) Y  = 0,
 +
$$
  
 
which may be written in local coordinates in the form:
 
which may be written in local coordinates in the form:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732096.png" /></td> </tr></table>
+
$$
 +
R _ {iklm}  = \
 +
- R _ {kilm}  = \
 +
- R _ {ikml} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732097.png" /></td> </tr></table>
+
$$
 +
R _ {iklm}  = R _ {lmik} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732098.png" /></td> </tr></table>
+
$$
 +
R _ {iklm} + R _ {imkl} + R _ {ilmk}  = 0.
 +
$$
  
The Riemann tensor has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c02732099.png" /> algebraically independent components. The covariant derivatives of the Riemann tensor satisfy the (second) Bianchi identity:
+
The Riemann tensor has $  n  ^ {2} ( n  ^ {2} - 1)/12 $
 +
algebraically independent components. The covariant derivatives of the Riemann tensor satisfy the (second) Bianchi identity:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320100.png" /></td> </tr></table>
+
$$
 +
( \nabla _ {X} R) ( Y, Z, U) +
 +
( \nabla _ {Y} R) ( Z, X, U) +
 +
( \nabla _ {Z} R) ( X, Y, U)  = 0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320101.png" /> is the covariant derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320102.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320103.png" />. In local coordinates, this identity is
+
where $  ( \nabla _ {X} R) ( Y, Z, U) $
 +
is the covariant derivative of $  R ( Y, Z) U $
 +
with respect to $  X $.  
 +
In local coordinates, this identity is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320104.png" /></td> </tr></table>
+
$$
 +
R _ {ikl;m} ^ { n } +
 +
R _ {imk;l} ^ { n } +
 +
R _ {ilm;k} ^ { n }  = 0.
 +
$$
  
 
The Riemann tensor is sometimes defined with the opposite sign.
 
The Riemann tensor is sometimes defined with the opposite sign.
Line 108: Line 321:
 
A Riemannian space is locally isometric to a Euclidean space if and only if its Riemann tensor vanishes identically.
 
A Riemannian space is locally isometric to a Euclidean space if and only if its Riemann tensor vanishes identically.
  
Another, equivalent, approach is sometimes adopted with regard to describing the curvature of a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320105.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320106.png" /> be a two-dimensional linear space in the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320107.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320108.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320109.png" />. Then the sectional curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320110.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320111.png" /> in the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320112.png" /> is defined as
+
Another, equivalent, approach is sometimes adopted with regard to describing the curvature of a Riemannian space $  M  ^ {n} $.  
 +
Let $  \sigma $
 +
be a two-dimensional linear space in the tangent space $  TM  ^ {n} $
 +
to $  M  ^ {n} $
 +
at a point $  P $.  
 +
Then the sectional curvature of $  M  ^ {n} $
 +
at $  P $
 +
in the direction $  \sigma $
 +
is defined as
 +
 
 +
$$
 +
K _  \sigma  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320113.png" /></td> </tr></table>
+
\frac{\langle  R ( V, W) W, V \rangle }{\langle  V, V > < W, W \rangle - \langle  V, W \rangle  ^ {2} }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320114.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320115.png" /> are vectors defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320116.png" />. The same area element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320117.png" /> may be defined by different vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320118.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320119.png" />, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320120.png" /> is independent of the specific vectors chosen. For a two-dimensional Riemannian space, the sectional curvature coincides with the Gaussian curvature. The Riemann tensor can be expressed in terms of the sectional curvatures:
+
where $  V $
 +
and $  W $
 +
are vectors defining $  \sigma $.  
 +
The same area element $  \sigma $
 +
may be defined by different vectors $  V $
 +
and $  W $,  
 +
but $  K _  \sigma  $
 +
is independent of the specific vectors chosen. For a two-dimensional Riemannian space, the sectional curvature coincides with the Gaussian curvature. The Riemann tensor can be expressed in terms of the sectional curvatures:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320121.png" /></td> </tr></table>
+
$$
 +
\langle  R ( X, Y) Z, U \rangle  = \
 +
{
 +
\frac{1}{6}
 +
} \{
 +
k ( X + U, Y + Z) -
 +
k ( X + U, Y) -
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320122.png" /></td> </tr></table>
+
$$
 +
-  
 +
k ( X + U, Z) - k ( X, Y + Z) - k ( U, Y + Z) + k ( X, Z) +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320123.png" /></td> </tr></table>
+
$$
 +
+
 +
k ( U, Y) - k ( Y + U, X + Z) + k ( Y + U, X) +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320124.png" /></td> </tr></table>
+
$$
 +
+
 +
k ( Y + U, Z) + k ( Y, Z + X) +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320125.png" /></td> </tr></table>
+
$$
 +
+
 +
{} k ( U, Z + X) - k ( Y, Z) - k ( U, X) \} ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320126.png" /></td> </tr></table>
+
$$
 +
k ( V, W)  = \
 +
K _  \sigma  ( \langle  V, V > < W, W \rangle - \langle  V, W \rangle  ^ {2} ).
 +
$$
  
 
Weaker characteristics of the curvature of a Riemannian space are also used — the Ricci tensor, or Ricci curvature:
 
Weaker characteristics of the curvature of a Riemannian space are also used — the Ricci tensor, or Ricci curvature:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320127.png" /></td> </tr></table>
+
$$
 +
R _ {ik}  = \
 +
R _ {ilk} ^ { l } ,
 +
$$
  
 
and the scalar curvature:
 
and the scalar curvature:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320128.png" /></td> </tr></table>
+
$$
 +
= g  ^ {ik} R _ {ik} .
 +
$$
  
The Ricci tensor is symmetric: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320129.png" />.
+
The Ricci tensor is symmetric: $  R _ {ik} = R _ {ki} $.
  
 
The curvature is sometimes characterized in terms of more complicated constructions — particularly quadratic ones — based on the Riemann tensor. One of the most common invariants of this type is
 
The curvature is sometimes characterized in terms of more complicated constructions — particularly quadratic ones — based on the Riemann tensor. One of the most common invariants of this type is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320130.png" /></td> </tr></table>
+
$$
 +
= \
 +
R _ {klm} ^ { i } R _ {i} ^ { klm } ,
 +
$$
  
 
which is used in investigating the Schwarzschild gravity field.
 
which is used in investigating the Schwarzschild gravity field.
Line 146: Line 409:
 
For a two-dimensional space, the Riemann tensor is
 
For a two-dimensional space, the Riemann tensor is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320131.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
R ( X, Y) Z  = \
 +
K ( \langle  Y, Z \rangle X - \langle  X, Z \rangle Y),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320132.png" /> is the Gaussian curvature. In this case the scalar curvature is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320133.png" />. For a three-dimensional space the Riemann tensor has the form
+
where $  K $
 +
is the Gaussian curvature. In this case the scalar curvature is equal to $  K $.  
 +
For a three-dimensional space the Riemann tensor has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320134.png" /></td> </tr></table>
+
$$
 +
R _ {iklm}  = \
 +
R _ {il} g _ {km} -
 +
R _ {im} g _ {kl} +
 +
R _ {km} g _ {il} -
 +
R _ {kl} g _ {im} +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320135.png" /></td> </tr></table>
+
$$
 +
+
 +
{
 +
\frac{R}{2}
 +
} ( g _ {im} g _ {kl} - g _ {il} g _ {km} ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320136.png" /> is the metric tensor, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320137.png" /> is the Ricci tensor and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320138.png" /> is the scalar curvature.
+
where $  g _ {ij} $
 +
is the metric tensor, $  R _ {ij} $
 +
is the Ricci tensor and $  R $
 +
is the scalar curvature.
  
If the sectional curvatures are independent both of the point and of the two-dimensional direction, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320139.png" /> is known as a space of constant curvature; the Riemann tensor of such a space has the form (3) (the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320140.png" /> is then called the curvature of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320141.png" />). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320142.png" /> it turns out that, if in all points the curvature is independent of the direction, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320143.png" /> is a space of constant curvature (Schur's theorem).
+
If the sectional curvatures are independent both of the point and of the two-dimensional direction, the space $  M  ^ {n} $
 +
is known as a space of constant curvature; the Riemann tensor of such a space has the form (3) (the constant $  K $
 +
is then called the curvature of the space $  M  ^ {n} $).  
 +
When $  n > 2 $
 +
it turns out that, if in all points the curvature is independent of the direction, then $  M  ^ {n} $
 +
is a space of constant curvature (Schur's theorem).
  
 
==The curvature of submanifolds.==
 
==The curvature of submanifolds.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320144.png" /> be a regular surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320145.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320146.png" /> be a curve on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320147.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320148.png" /> be the tangent plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320149.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320150.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320151.png" />. Suppose that a small neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320152.png" /> is projected onto the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320153.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320154.png" /> be the projection of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320155.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320156.png" />. The [[Geodesic curvature|geodesic curvature]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320157.png" /> of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320158.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320159.png" /> is defined as the number equal in absolute value to the curvature of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320160.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320161.png" />. The geodesic curvature is considered positive if the rotation of the tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320162.png" /> as one passes through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320163.png" /> forms a right-handed screw with the direction of the normal to the surface. The geodesic curvature is an object of the intrinsic geometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320164.png" />. It can be evaluated from the formula
+
Let $  \Phi $
 +
be a regular surface in $  E  ^ {3} $,  
 +
let $  \gamma $
 +
be a curve on $  \Phi $
 +
and let $  \alpha _ {P} $
 +
be the tangent plane to $  \Phi $
 +
at a point $  P $
 +
on $  \gamma $.  
 +
Suppose that a small neighbourhood of $  P $
 +
is projected onto the plane $  \alpha _ {P} $
 +
and let $  \overline \gamma \; $
 +
be the projection of the curve $  \gamma $
 +
on $  \alpha _ {P} $.  
 +
The [[Geodesic curvature|geodesic curvature]] $  \kappa $
 +
of the curve $  \gamma $
 +
at $  P $
 +
is defined as the number equal in absolute value to the curvature of the curve $  \overline \gamma \; $
 +
at $  P $.  
 +
The geodesic curvature is considered positive if the rotation of the tangent to $  \overline \gamma \; $
 +
as one passes through $  P $
 +
forms a right-handed screw with the direction of the normal to the surface. The geodesic curvature is an object of the intrinsic geometry of $  \Phi $.  
 +
It can be evaluated from the formula
 +
 
 +
$$ \tag{4 }
 +
\kappa  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320165.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
\frac{e _ {ij} \left (
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320166.png" /> is the [[Natural equation|natural equation]] of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320167.png" /> in local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320168.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320169.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320170.png" /> are the components of the metric tensor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320171.png" /> in these coordinates, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320172.png" /> are the Christoffel symbols, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320173.png" /> is the totally [[Discriminant tensor|discriminant tensor]]. Using formula (4) one can define the geodesic curvature for curves on an abstract two-dimensional Riemannian space. A curve on a Riemannian manifold coincides with a geodesic or with part of a geodesic if and only if its geodesic curvature vanishes identically.
+
\frac{d  ^ {2} x  ^ {i} }{ds  ^ {2} }
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320174.png" /> be a two-dimensional submanifold of a three-dimensional Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320175.png" />. There are two approaches to the definition of the curvature for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320176.png" />. On the one hand, one can consider <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320177.png" /> as a Riemannian space whose metric is induced by that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320178.png" />, and then use formula (1) to define its curvature. This yields what is called the internal curvature. On the other hand, one can carry out the same construction that gives the definition of the curvature for surfaces in a Euclidean space and apply it to submanifolds in a Riemannian space. The result is a different concept of the curvature, known as the external curvature. One has the following relationship:
+
\frac{d  ^ {2} x  ^ {j} }{ds  ^ {2} }
 +
+
 +
\Gamma _ {kl}  ^ {i}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320179.png" /></td> </tr></table>
+
\frac{dx  ^ {k} }{ds}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320180.png" /> is the curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320181.png" /> in the direction of the tangent plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320182.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320183.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320184.png" /> are the internal and external curvatures, respectively.
+
\frac{dx  ^ {l} }{ds}
 +
 
 +
\frac{dx  ^ {j} }{ds}
 +
 
 +
\right ) }{\left ( g _ {ij}
 +
\frac{dx  ^ {i} }{ds}
 +
 
 +
\frac{dx  ^ {j} }{ds}
 +
 
 +
\right )  ^ {3/2} }
 +
,
 +
$$
 +
 
 +
where $  x  ^ {i} ( s) $
 +
is the [[Natural equation|natural equation]] of the curve  $  \gamma $
 +
in local coordinates  $  x  ^ {i} $
 +
on  $  \Phi $,
 +
$  g _ {ij} $
 +
are the components of the metric tensor of  $  \Phi $
 +
in these coordinates,  $  \Gamma _ {kl}  ^ {i} $
 +
are the Christoffel symbols, and  $  e _ {ij} $
 +
is the totally [[Discriminant tensor|discriminant tensor]]. Using formula (4) one can define the geodesic curvature for curves on an abstract two-dimensional Riemannian space. A curve on a Riemannian manifold coincides with a geodesic or with part of a geodesic if and only if its geodesic curvature vanishes identically.
 +
 
 +
Let  $  \Phi $
 +
be a two-dimensional submanifold of a three-dimensional Riemannian space  $  M $.  
 +
There are two approaches to the definition of the curvature for  $  \Phi $.
 +
On the one hand, one can consider  $  \Phi $
 +
as a Riemannian space whose metric is induced by that of  $  M $,
 +
and then use formula (1) to define its curvature. This yields what is called the internal curvature. On the other hand, one can carry out the same construction that gives the definition of the curvature for surfaces in a Euclidean space and apply it to submanifolds in a Riemannian space. The result is a different concept of the curvature, known as the external curvature. One has the following relationship:
 +
 
 +
$$
 +
K _ {i}  = \
 +
K _ {e} + K _  \sigma  ,
 +
$$
 +
 
 +
where  $  K _  \sigma  $
 +
is the curvature of  $  M $
 +
in the direction of the tangent plane to $  \Phi $,  
 +
and $  K _ {i} $
 +
and $  K _ {e} $
 +
are the internal and external curvatures, respectively.
  
 
The concepts of normal, internal and external curvatures can be generalized with respect to the dimension and codimension of the submanifold in question.
 
The concepts of normal, internal and external curvatures can be generalized with respect to the dimension and codimension of the submanifold in question.
  
The concept of the Riemann tensor may be generalized to various spaces with a weaker structure than Riemannian spaces. For example, the Riemann and Ricci tensors depend only on the affine structure of the space and may also be defined in spaces with an affine connection, although in that case they do not possess all the symmetry properties as above. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320185.png" />. Other examples of this type are the conformal curvature tensor and the projective curvature tensor. The conformal curvature tensor (Weyl tensor) is
+
The concept of the Riemann tensor may be generalized to various spaces with a weaker structure than Riemannian spaces. For example, the Riemann and Ricci tensors depend only on the affine structure of the space and may also be defined in spaces with an affine connection, although in that case they do not possess all the symmetry properties as above. For example, $  R _ {ik} \neq R _ {ki} $.  
 +
Other examples of this type are the conformal curvature tensor and the projective curvature tensor. The conformal curvature tensor (Weyl tensor) is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320186.png" /></td> </tr></table>
+
$$
 +
C _ {iklm}  = \
 +
R _ {iklm} -
 +
R _ {l[i }  g _ {k]m }  +
 +
R _ {m[i }  g _ {k]l }  +
 +
{
 +
\frac{1}{3}
 +
}
 +
R _ {l[i }  g _ {k]m }  ,
 +
$$
  
 
where the brackets denote alternation with respect to the relevant indices. Vanishing of the conformal curvature tensor is a necessary and sufficient condition for the space to coincide locally with a conformal Euclidean space. The projective curvature tensor is
 
where the brackets denote alternation with respect to the relevant indices. Vanishing of the conformal curvature tensor is a necessary and sufficient condition for the space to coincide locally with a conformal Euclidean space. The projective curvature tensor is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320187.png" /></td> </tr></table>
+
$$
 +
P _ {lki} ^ { q }  = \
 +
R _ {lki} ^ { q } +
 +
\delta _ {k}  ^ {q}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320188.png" /> is the Kronecker symbol and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320189.png" /> is the dimension of the space. Vanishing of the projective curvature tensor is a necessary and sufficient condition for the space to coincide locally with a projective Euclidean space.
+
\frac{R _ {li} + R _ {il} }{n  ^ {2} - 1 }
 +
+
 +
\delta _ {l}  ^ {q}
  
The concept of curvature generalizes to the case of non-regular objects, in particular, to the case of the theory of two-dimensional manifolds of bounded curvature. Here the curvature in a space is defined not at a point, but in a domain, and one is concerned with the total or integral curvature of a domain. In the regular case the total curvature is equal to the integral of the Gaussian curvature. The total curvature of a geodesic triangle may be expressed in terms of the angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320190.png" /> at its vertices:
+
\frac{R _ {ki} + R _ {ik} }{n  ^ {2} - 1 }
 +
+
 +
\delta _ {i}  ^ {q}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320191.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
\frac{R _ {lk} - R _ {kl} }{n + 1 }
 +
,
 +
$$
 +
 
 +
where  $  \delta _ {i}  ^ {k} $
 +
is the Kronecker symbol and  $  n $
 +
is the dimension of the space. Vanishing of the projective curvature tensor is a necessary and sufficient condition for the space to coincide locally with a projective Euclidean space.
 +
 
 +
The concept of curvature generalizes to the case of non-regular objects, in particular, to the case of the theory of two-dimensional manifolds of bounded curvature. Here the curvature in a space is defined not at a point, but in a domain, and one is concerned with the total or integral curvature of a domain. In the regular case the total curvature is equal to the integral of the Gaussian curvature. The total curvature of a geodesic triangle may be expressed in terms of the angles  $  \beta _ {i} $
 +
at its vertices:
 +
 
 +
$$ \tag{5 }
 +
K  = \
 +
\sum \beta _ {i} - \pi ,
 +
$$
  
 
this relationship is a special case of the [[Gauss–Bonnet theorem|Gauss–Bonnet theorem]]. Formula (5) has been used as a basis for the definition of the total curvature in manifolds of bounded curvature.
 
this relationship is a special case of the [[Gauss–Bonnet theorem|Gauss–Bonnet theorem]]. Formula (5) has been used as a basis for the definition of the total curvature in manifolds of bounded curvature.
  
The curvature is one of the fundamental concepts in modern differential geometry. Restrictions on the curvature usually yield meaningful information about an object. For example, in the theory of surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320192.png" />, the sign of the Gaussian curvature defines the type of a point (elliptic, hyperbolic or parabolic). Surfaces with an everywhere non-negative Gaussian curvature share a whole spectrum of properties, by virtue of which they can be grouped together in one natural class (see [[#References|[4]]], [[#References|[6]]]). Surfaces with zero mean curvature (see [[Minimal surface|Minimal surface]]) have many specific properties. The theory of non-regular surfaces especially studies classes of surfaces of bounded integral absolute Gaussian or mean curvature.
+
The curvature is one of the fundamental concepts in modern differential geometry. Restrictions on the curvature usually yield meaningful information about an object. For example, in the theory of surfaces in $  E  ^ {3} $,  
 +
the sign of the Gaussian curvature defines the type of a point (elliptic, hyperbolic or parabolic). Surfaces with an everywhere non-negative Gaussian curvature share a whole spectrum of properties, by virtue of which they can be grouped together in one natural class (see [[#References|[4]]], [[#References|[6]]]). Surfaces with zero mean curvature (see [[Minimal surface|Minimal surface]]) have many specific properties. The theory of non-regular surfaces especially studies classes of surfaces of bounded integral absolute Gaussian or mean curvature.
  
In Riemannian spaces, a uniform bound on the sectional curvatures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320193.png" /> at any point and in any two-dimensional direction makes it possible to use comparison theorems. The latter enable one to compare the rate of deviation of the geodesics and the volumes of domains in a given space with the characteristics of the corresponding curves and domains in a space of constant curvature. Some of the restrictions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320194.png" /> even predetermine the topological structure of the space as a whole. For example:
+
In Riemannian spaces, a uniform bound on the sectional curvatures $  K _  \sigma  $
 +
at any point and in any two-dimensional direction makes it possible to use comparison theorems. The latter enable one to compare the rate of deviation of the geodesics and the volumes of domains in a given space with the characteristics of the corresponding curves and domains in a space of constant curvature. Some of the restrictions on $  K _  \sigma  $
 +
even predetermine the topological structure of the space as a whole. For example:
  
The sphere theorem. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320195.png" /> be a complete simply-connected Riemannian space of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320196.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320197.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320198.png" /> is homeomorphic to the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320199.png" />.
+
The sphere theorem. Let $  M $
 +
be a complete simply-connected Riemannian space of dimension $  n \geq  2 $
 +
and let $  1/4 < \delta \leq  K _  \sigma  \leq  1 $.  
 +
Then $  M $
 +
is homeomorphic to the sphere $  S  ^ {n} $.
  
The Hadamard–Cartan and Gromoll–Meyer theorems. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320200.png" /> be a complete Riemannian space of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320201.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320202.png" /> everywhere and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320203.png" /> is simply connected, or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320204.png" /> everywhere and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320205.png" /> is not compact, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320206.png" /> is homeomorphic to the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320207.png" />.
+
The Hadamard–Cartan and Gromoll–Meyer theorems. Let $  M $
 +
be a complete Riemannian space of dimension $  n \geq  2 $.  
 +
If $  K _  \sigma  \leq  0 $
 +
everywhere and $  M $
 +
is simply connected, or if $  K _  \sigma  > 0 $
 +
everywhere and $  M $
 +
is not compact, then $  M $
 +
is homeomorphic to the Euclidean space $  E  ^ {n} $.
  
 
The concepts of curvature are utilized in various natural sciences. Thus, when a body is moving along a trajectory, there is a relationship between the curvature of the trajectory and the centrifugal force. The Gaussian curvature first appeared in Gauss' work on cartography. The mean curvature of the surface of a liquid is related to the capillary effect. In [[Relativity theory|relativity theory]] there is a connection between the distribution of mass and energy (more precisely, between the energy-momentum tensor) and the curvature of [[Space-time|space-time]]. The conformal curvature tensor is used in the theory of formation of particles in a gravitational field.
 
The concepts of curvature are utilized in various natural sciences. Thus, when a body is moving along a trajectory, there is a relationship between the curvature of the trajectory and the centrifugal force. The Gaussian curvature first appeared in Gauss' work on cartography. The mean curvature of the surface of a liquid is related to the capillary effect. In [[Relativity theory|relativity theory]] there is a connection between the distribution of mass and energy (more precisely, between the energy-momentum tensor) and the curvature of [[Space-time|space-time]]. The conformal curvature tensor is used in the theory of formation of particles in a gravitational field.
Line 202: Line 600:
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,  "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Pogorelov,  "Differential geometry" , Noordhoff  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Blaschke,  "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Elementare Differentialgeometrie" , '''1''' , Springer  (1921)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.D. Aleksandrov,  "Die innere Geometrie der konvexen Flächen" , Akademie Verlag  (1955)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D. Gromoll,  W. Klingenberg,  W. Meyer,  "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.V. Pogorelov,  "Extrinsic geometry of convex surfaces" , Amer. Math. Soc.  (1972)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,  "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Pogorelov,  "Differential geometry" , Noordhoff  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Blaschke,  "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Elementare Differentialgeometrie" , '''1''' , Springer  (1921)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.D. Aleksandrov,  "Die innere Geometrie der konvexen Flächen" , Akademie Verlag  (1955)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D. Gromoll,  W. Klingenberg,  W. Meyer,  "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.V. Pogorelov,  "Extrinsic geometry of convex surfaces" , Amer. Math. Soc.  (1972)  (Translated from Russian)</TD></TR></table>
  
 +
====Comments====
 +
Formula (1) can be expressed in various ways, e.g., in [[#References|[2]]] it reads:
  
 +
$$
 +
K  =  {
 +
\frac{1}{( EG - F  ^ {2} )  ^ {2} }
 +
} \times
 +
$$
  
====Comments====
+
$$
Formula (1) can be expressed in various ways, e.g., in [[#References|[2]]] it reads:
+
\times
 +
\left \{  \left |  
 +
\begin{array}{ccc}
 +
-
 +
\frac{G _ {uu}  }{2}
 +
+ F _ {uv} -
 +
\frac{E _ {vv} }{2}
 +
  &
 +
\frac{E _ {u} }{2}
 +
  &F _ {u} -
 +
\frac{E _ {v} }{2}
 +
  \\
 +
F _ {v} -
 +
\frac{G _ {u} }{2}
 +
  & E  & F  \\
 +
 
 +
\frac{G _ {v} }{2}
 +
  & F  & G  \\
 +
\end{array}
 +
\right | \right .  -
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320208.png" /></td> </tr></table>
+
$$
 +
- \
 +
\left . \left |
 +
\begin{array}{ccc}
 +
0 &
 +
\frac{E _ {v} }{2}
 +
  &
 +
\frac{G _ {u} }{2}
 +
  \\
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320209.png" /></td> </tr></table>
+
\frac{E _ {v} }{2}
 +
  & E  & F  \\
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320210.png" /></td> </tr></table>
+
\frac{G _ {u} }{2}
 +
  & F  & G  \\
 +
\end{array}
 +
\right |  \right \} .
 +
$$
  
The first Bianchi identity is the usual name given to the fourth symmetry relation for the Riemann tensor, i.e. to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320211.png" />. The second Bianchi identity is the relation
+
The first Bianchi identity is the usual name given to the fourth symmetry relation for the Riemann tensor, i.e. to $  R ( X, Y) Z + R ( Y, Z) X + R ( Z, X) Y = 0 $.  
 +
The second Bianchi identity is the relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320212.png" /></td> </tr></table>
+
$$
 +
\nabla _ {X} R ( Y, Z, U) + \nabla _ {Y} R ( Z, X, U) +
 +
\nabla _ {Z} R ( X, Y, U) = 0 ,
 +
$$
  
 
called the Bianchi identity above.
 
called the Bianchi identity above.
  
Such concepts as mean curvature, conformal curvature tensors, geodesic curvature, and projective curvature tensor are also defined in higher dimensional settings (than surfaces), cf. e.g. [[#References|[a2]]] (mean curvature), [[#References|[a3]]], [[#References|[1]]] (conformal and projective curvature tensors). (Cf. also [[Conformal Euclidean space|Conformal Euclidean space]].) The absolute value of the geodesic curvature of a curve on a surface is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320213.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320214.png" /> is assumed to be described by its arc length parameter (natural parameter) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320215.png" /> is the Levi-Civita connection on the surface. For the concepts of natural parameter and natural equation of a curve, cf. [[Natural equation|Natural equation]]. The various fundamental (quadratic) forms of a surface are discussed in [[Fundamental forms of a surface|Fundamental forms of a surface]]; [[Geometry of immersed manifolds|Geometry of immersed manifolds]] and [[Second fundamental form|Second fundamental form]].
+
Such concepts as mean curvature, conformal curvature tensors, geodesic curvature, and projective curvature tensor are also defined in higher dimensional settings (than surfaces), cf. e.g. [[#References|[a2]]] (mean curvature), [[#References|[a3]]], [[#References|[1]]] (conformal and projective curvature tensors). (Cf. also [[Conformal Euclidean space|Conformal Euclidean space]].) The absolute value of the geodesic curvature of a curve on a surface is $  | \nabla _ {\dot \gamma  }  \dot \gamma  | $,  
 +
where $  \gamma $
 +
is assumed to be described by its arc length parameter (natural parameter) and $  \nabla $
 +
is the Levi-Civita connection on the surface. For the concepts of natural parameter and natural equation of a curve, cf. [[Natural equation|Natural equation]]. The various fundamental (quadratic) forms of a surface are discussed in [[Fundamental forms of a surface|Fundamental forms of a surface]]; [[Geometry of immersed manifolds|Geometry of immersed manifolds]] and [[Second fundamental form|Second fundamental form]].
  
The sectional curvature of a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320216.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320217.png" /> in the direction of the tangent plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320218.png" /> is also called the [[Riemannian curvature|Riemannian curvature]].
+
The sectional curvature of a Riemannian space $  M  ^ {n} $
 +
at $  P $
 +
in the direction of the tangent plane $  \sigma $
 +
is also called the [[Riemannian curvature|Riemannian curvature]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320219.png" /> denote the Ricci tensor and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320220.png" /> be the quadratic form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320221.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320222.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320223.png" />. Then the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320224.png" /> for a unit vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320225.png" /> is the mean of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320226.png" /> for all plane directions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320227.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320228.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320229.png" />, and is called the Ricci curvature or mean curvature of the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320230.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320231.png" />. The mean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320232.png" /> of all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320233.png" /> is the scalar curvature at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320234.png" />, cf. also [[Ricci tensor|Ricci tensor]] and [[Ricci curvature|Ricci curvature]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320235.png" /> is a [[Kähler manifold|Kähler manifold]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320236.png" /> is restricted to a complex plane (i.e. a plane invariant under the almost-complex structure), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320237.png" /> is called the holomorphic sectional curvature.
+
Let $  R _ {ij} $
 +
denote the Ricci tensor and let $  Q $
 +
be the quadratic form on $  T _ {P} M  ^ {n} $
 +
given by $  R _ {ij} $
 +
at $  P \in M  ^ {n} $.  
 +
Then the value $  Q ( \xi ) $
 +
for a unit vector $  \xi \in T _ {p} M  ^ {n} $
 +
is the mean of $  K _  \sigma  $
 +
for all plane directions $  \sigma $
 +
in $  T _ {P} M  ^ {n} $
 +
containing $  \xi $,  
 +
and is called the Ricci curvature or mean curvature of the direction $  \xi $
 +
at $  P $.  
 +
The mean $  R $
 +
of all the $  Q ( \xi ) $
 +
is the scalar curvature at $  P $,  
 +
cf. also [[Ricci tensor|Ricci tensor]] and [[Ricci curvature|Ricci curvature]]. If $  M $
 +
is a [[Kähler manifold|Kähler manifold]] and $  \sigma $
 +
is restricted to a complex plane (i.e. a plane invariant under the almost-complex structure), then $  K _  \sigma  $
 +
is called the holomorphic sectional curvature.
  
For a simply-closed space curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320238.png" /> of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320239.png" /> the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320240.png" /> is called the total curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320241.png" />; generally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320242.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320243.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320244.png" /> is a closed curve lying in a plane (W. Fenchel). Fix an origin 0 in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320245.png" /> and consider the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320246.png" /> around 0. For each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320247.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320248.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320249.png" /> be the point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320250.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320251.png" /> is the (displaced) unit tangent vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320252.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320253.png" />. As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320254.png" /> runs over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320255.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320256.png" /> trace out a curve on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320257.png" />, the spherical indicatrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320258.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320259.png" />. The correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320260.png" /> is called the spherical representation. The total curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320261.png" /> is equal to the length of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320262.png" />. Instead of the tangents to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027320/c027320263.png" /> one can also use the principal normal and binormal vectors and perform a similar construction yielding other spherical indicatrices, cf. [[Spherical indicatrix|Spherical indicatrix]].
+
For a simply-closed space curve $  C $
 +
of length $  L $
 +
the integral $  K = \int _ {0}  ^ {L} k ( s)  ds $
 +
is called the total curvature of $  C $;  
 +
generally $  K \leq  2 \pi $,  
 +
and $  K = 2 \pi $
 +
if and only if $  C $
 +
is a closed curve lying in a plane (W. Fenchel). Fix an origin 0 in $  E  ^ {3} $
 +
and consider the unit sphere $  S  ^ {2} $
 +
around 0. For each point $  P $
 +
of $  C $
 +
let $  \overline{P}\; $
 +
be the point on $  S  ^ {2} $
 +
such that 0 \overline{P}\; $
 +
is the (displaced) unit tangent vector to $  C $
 +
at $  P $.  
 +
As $  P $
 +
runs over $  C $
 +
the $  \overline{P}\; $
 +
trace out a curve on $  S  ^ {2} $,  
 +
the spherical indicatrix $  \overline{C}\; $
 +
of $  C $.  
 +
The correspondence $  C \mapsto \overline{C}\; $
 +
is called the spherical representation. The total curvature of $  C $
 +
is equal to the length of $  \overline{C}\; $.  
 +
Instead of the tangents to $  C $
 +
one can also use the principal normal and binormal vectors and perform a similar construction yielding other spherical indicatrices, cf. [[Spherical indicatrix|Spherical indicatrix]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.C. Hsiung,  "A first course in differential geometry" , Wiley  (1981)  pp. Chapt. 3, Sect. 4</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''2''' , Interscience  (1969)  pp. 33</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.A. Schouten,  "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer  (1954)  pp. Chapt. VI  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.C. Hsiung,  "A first course in differential geometry" , Wiley  (1981)  pp. Chapt. 3, Sect. 4</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''2''' , Interscience  (1969)  pp. 33</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.A. Schouten,  "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer  (1954)  pp. Chapt. VI  (Translated from German)</TD></TR></table>

Latest revision as of 17:31, 5 June 2020


A collective term for a series of quantitative characteristics (in terms of numbers, vectors, tensors) describing the degree to which some object (a curve, a surface, a Riemannian space, etc.) deviates in its properties from certain other objects (a straight line, a plane, a Euclidean space, etc.) which are considered to be flat. The concepts of curvature are usually defined locally, i.e. at each point. These concepts of curvature are connected with the examination of deviations which are small to the second order; hence the object in question is assumed to be specified by $ C ^ {2} $- smooth functions. In some cases the concepts are defined in terms of integrals, and they remain valid without the $ C ^ {2} $- smoothness condition. As a rule, if the curvature vanishes at all points, the object in question is identical (in small sections, not in the large) with the corresponding "flat" object.

The curvature of a curve.

Let $ \gamma $ be a regular curve in the $ n $- dimensional Euclidean space, parametrized in terms of its natural parameter $ t $. Let $ \alpha ( P, P _ {1} ) $ and $ s ( P, P _ {1} ) $ be the angle between the tangents to $ \gamma $ at the points $ P $ and $ P _ {1} $ of $ \gamma $ and the length of the arc of the curve between $ P $ and $ P _ {1} $, respectively. Then the limit

$$ k = \lim\limits _ {P _ {1} \rightarrow P } \ \frac{\alpha ( P, P _ {1} ) }{s ( P, P _ {1} ) } $$

is called the curvature of the curve $ \gamma $ at $ P $. The curvature of the curve is equal to the absolute value of the vector $ d ^ {2} \gamma ( t)/dt ^ {2} $, and the direction of this vector is just the direction of the principal normal to the curve. For the curve $ \gamma $ to coincide with some segment of a straight line or with an entire line it is necessary and sufficient that its curvature $ k $ vanishes identically.

The curvature of a surface.

Let $ \Phi $ be a regular surface in the three-dimensional Euclidean space. Let $ P $ be a point of $ \Phi $, $ T _ {p} $ the tangent plane to $ \Phi $ at $ P $, $ \mathbf n $ the normal to $ \Phi $ at $ P $, and $ \alpha $ the plane through $ \mathbf n $ and some unit vector $ \mathbf l $ in $ T _ {p} $. The intersection $ \gamma _ {\mathbf l} $ of the plane $ \alpha $ and the surface $ \Phi $ is a curve, called the normal section of the surface $ \Phi $ at the point $ P $ in the direction $ \mathbf l $. The number

$$ k _ {\mathbf l} = \ \left ( \frac{d ^ {2} \gamma _ {\mathbf l} }{dt ^ {2} } , \mathbf n \ \right ) , $$

where $ t $ is the natural parameter on $ \gamma $, is called the normal curvature of $ \Phi $ in the direction $ \mathbf l $. The normal curvature is equal to the curvature of the curve $ \gamma _ {\mathbf l} $ up to the sign.

The tangent plane $ T _ {p} $ contains two perpendicular directions $ \mathbf l _ {1} $ and $ \mathbf l _ {2} $ such that the normal curvature in any direction can be expressed by Euler's formula:

$$ k _ {\mathbf l} = \ k _ {1} \cos ^ {2} \theta + k _ {2} \sin ^ {2} \theta , $$

where $ \theta $ is the angle between $ \mathbf l _ {1} $ and $ \mathbf l $. The numbers $ k _ {1} $ and $ k _ {2} $ are called the principal curvatures, and the directions $ \mathbf l _ {1} $ and $ \mathbf l _ {2} $ are known as the principal directions of the surface. The principal curvatures are extremal values of the normal curvature. The construction of the normal curvature at a given point of the surface may be represented graphically as follows. When $ k _ {\mathbf l} \neq 0 $, the equation

$$ \mathbf r ( \mathbf l ) = \ \mathbf l \left | \frac{1}{k _ {\mathbf l} } \ \right | ^ {1/2} , $$

where $ \mathbf r ( \mathbf l ) $ is the radius vector, defines a certain curve of the second order in the tangent plane $ T _ {p} $, known as the Dupin indicatrix. The Dupin indicatrix can only be one of the following three curves: an ellipse, a hyperbola or a pair of parallel lines. The points of the surface are accordingly classified as elliptic, hyperbolic or parabolic. At an elliptic point, the second fundamental form of the surface is of fixed sign; at a hyperbolic point the form is of variable sign; and at a parabolic point it is degenerate. If all normal curvatures at a point are zero, the point is said to be flat. If the Dupin indicatrix is a circle it is called an umbilical (or spherical) point.

The principal directions are uniquely determined (up to the order), unless the point in question is an umbilical point or a flat point. In these cases every direction is principal. In this connection one has the following theorem of Rodrigues: A direction $ \mathbf l $ is principal if and only if

$$ d \mathbf n = \ - \lambda d \mathbf r \ \ \mathop{\rm in} \textrm{ the } \textrm{ direction } \mathbf l , $$

where $ \mathbf r $ is the radius vector of the surface and $ \mathbf n $ the unit normal vector.

A curve on a surface is called a curvature line if its direction at every point is principal. In a neighbourhood of every point $ P $ on a surface, other than an umbilical point or a flat point, the surface may be so parametrized that its coordinate curves are curvature lines.

The quantity

$$ H = \ { \frac{1}{2} } ( k _ {1} + k _ {2} ) $$

is called the mean curvature of the surface. The quantity

$$ K = k _ {1} k _ {2} $$

is called the Gaussian (or total) curvature of the surface. The Gaussian curvature is an object of the intrinsic geometry of the surface, i.e. it can be expressed in terms of the first fundamental form:

$$ \tag{1 } K = \ \frac{1}{( EG - F ^ {2} ) ^ {2} } \left | \begin{array}{lll} E &E _ {u} &E _ {v} \\ F &F _ {u} &F _ {v} \\ G &G _ {u} &G _ {v} \\ \end{array} \ \right | - $$

$$ - \frac{1}{2 \sqrt {EG - F ^ {2} } } \left \{ \left ( \frac{E _ {v} - F _ {u} }{\sqrt {EG - F ^ {2} } } \right ) _ {v} - \left ( \frac{F _ {v} - G _ {u} }{\sqrt {EG - F ^ {2} } } \right ) _ {u} \right \} , $$

where $ E, F, G $ are the coefficients of the first fundamental form of the surface.

Using formula (1), one defines the Gaussian curvature for an abstract two-dimensional Riemannian manifold with line element $ ds ^ {2} $. A surface is locally isometric to a plane if and only if its Gaussian curvature vanishes identically.

The curvature of a Riemannian space.

Let $ M ^ {n} $ be a regular $ n $- dimensional Riemannian space and let $ BM ^ {n} $ be the space of regular vector fields on $ M ^ {n} $. The curvature of $ M ^ {n} $ is usually characterized by the Riemann (curvature) tensor (cf. Riemann tensor), i.e. by the multilinear mapping

$$ R: BM ^ {n} \times BM ^ {n} \times BM ^ {n} \rightarrow BM ^ {n} , $$

defined by

$$ \tag{2 } R ( X, Y) Z = \ \nabla _ {X} \nabla _ {Y} Z - \nabla _ {Y} \nabla _ {X} Z - \nabla _ {[ X, Y] } Z, $$

where $ \nabla $ is the Levi-Civita connection on $ M ^ {n} $ and $ [ , ] $ denotes the Lie bracket. If one puts $ X = \partial / \partial x ^ {k} $, $ Y = \partial / \partial x ^ {l} $ in some local coordinate system $ x ^ {i} $, one can rewrite (2) as follows:

$$ Z _ {; k; l } ^ { i } - Z _ {; l; k } ^ { i } = \ Z ^ {m} R _ {mkl} ^ { i } , $$

where; is the symbol for covariant differentiation.

Thus, the Riemann tensor is a quantitative characteristic of the non-commutativity of the second covariant derivatives in a Riemannian space. It also yields a quantitative description of certain other properties of Riemannian spaces — properties that distinguish them from Euclidean spaces.

The coefficients of the Riemann tensor in the local coordinate system $ x ^ {i} $ may be expressed in terms of the Christoffel symbols and the coefficients of the metric tensor, as follows:

$$ R _ {ijk} ^ { l } = \ \frac{\partial \Gamma _ {jk} ^ {l} }{\partial x ^ {i} } - \frac{\partial \Gamma _ {ik} ^ {l} }{\partial x ^ {j} } + \Gamma _ {ir} ^ {l} \Gamma _ {jk} ^ {r} - \Gamma _ {jr} ^ {l} \Gamma _ {ik} ^ {r} , $$

$$ R _ {iklm} = { \frac{1}{2} } \left ( \frac{\partial ^ {2} g _ {il} }{\partial x ^ {k} \partial x ^ {m} } + \frac{\partial ^ {2} g _ {km} }{\partial x ^ {i} \partial x ^ {l} } - \frac{\partial ^ {2} g _ {im} }{\partial x ^ {k} \partial x ^ {l} } - \frac{\partial ^ {2} g _ {kl} }{\partial x ^ {i} \partial x ^ {m} } \right ) + $$

$$ + g _ {np} \left ( \Gamma _ {km} ^ {p} \Gamma _ {il} ^ {n} - \Gamma _ {im} ^ {p} \Gamma _ {kl} ^ {n} \right ) , $$

where $ R _ {iklm} $ is the Riemann tensor with fourth covariant index, or — in a coordinate-free notation — the mapping $ \langle R ( X, Y) U, Z \rangle $( where $ \langle \cdot , \cdot \rangle $ denotes the scalar product).

The Riemann tensor possesses the following symmetry properties:

$$ R ( X, Y) Z = \ - R ( Y, X) Z, $$

$$ \langle R ( X, Y) Z, U \rangle = - \langle R ( X, Y) U, Z \rangle , $$

$$ \langle R ( X, Y) Z, U \rangle = \langle R ( Z, U) X, Y \rangle , $$

$$ R ( X, Y) Z + R ( Y, Z) X + R ( Z, X) Y = 0, $$

which may be written in local coordinates in the form:

$$ R _ {iklm} = \ - R _ {kilm} = \ - R _ {ikml} , $$

$$ R _ {iklm} = R _ {lmik} , $$

$$ R _ {iklm} + R _ {imkl} + R _ {ilmk} = 0. $$

The Riemann tensor has $ n ^ {2} ( n ^ {2} - 1)/12 $ algebraically independent components. The covariant derivatives of the Riemann tensor satisfy the (second) Bianchi identity:

$$ ( \nabla _ {X} R) ( Y, Z, U) + ( \nabla _ {Y} R) ( Z, X, U) + ( \nabla _ {Z} R) ( X, Y, U) = 0, $$

where $ ( \nabla _ {X} R) ( Y, Z, U) $ is the covariant derivative of $ R ( Y, Z) U $ with respect to $ X $. In local coordinates, this identity is

$$ R _ {ikl;m} ^ { n } + R _ {imk;l} ^ { n } + R _ {ilm;k} ^ { n } = 0. $$

The Riemann tensor is sometimes defined with the opposite sign.

A Riemannian space is locally isometric to a Euclidean space if and only if its Riemann tensor vanishes identically.

Another, equivalent, approach is sometimes adopted with regard to describing the curvature of a Riemannian space $ M ^ {n} $. Let $ \sigma $ be a two-dimensional linear space in the tangent space $ TM ^ {n} $ to $ M ^ {n} $ at a point $ P $. Then the sectional curvature of $ M ^ {n} $ at $ P $ in the direction $ \sigma $ is defined as

$$ K _ \sigma = \ \frac{\langle R ( V, W) W, V \rangle }{\langle V, V > < W, W \rangle - \langle V, W \rangle ^ {2} } , $$

where $ V $ and $ W $ are vectors defining $ \sigma $. The same area element $ \sigma $ may be defined by different vectors $ V $ and $ W $, but $ K _ \sigma $ is independent of the specific vectors chosen. For a two-dimensional Riemannian space, the sectional curvature coincides with the Gaussian curvature. The Riemann tensor can be expressed in terms of the sectional curvatures:

$$ \langle R ( X, Y) Z, U \rangle = \ { \frac{1}{6} } \{ k ( X + U, Y + Z) - k ( X + U, Y) - $$

$$ - k ( X + U, Z) - k ( X, Y + Z) - k ( U, Y + Z) + k ( X, Z) + $$

$$ + k ( U, Y) - k ( Y + U, X + Z) + k ( Y + U, X) + $$

$$ + k ( Y + U, Z) + k ( Y, Z + X) + $$

$$ + {} k ( U, Z + X) - k ( Y, Z) - k ( U, X) \} , $$

where

$$ k ( V, W) = \ K _ \sigma ( \langle V, V > < W, W \rangle - \langle V, W \rangle ^ {2} ). $$

Weaker characteristics of the curvature of a Riemannian space are also used — the Ricci tensor, or Ricci curvature:

$$ R _ {ik} = \ R _ {ilk} ^ { l } , $$

and the scalar curvature:

$$ R = g ^ {ik} R _ {ik} . $$

The Ricci tensor is symmetric: $ R _ {ik} = R _ {ki} $.

The curvature is sometimes characterized in terms of more complicated constructions — particularly quadratic ones — based on the Riemann tensor. One of the most common invariants of this type is

$$ C = \ R _ {klm} ^ { i } R _ {i} ^ { klm } , $$

which is used in investigating the Schwarzschild gravity field.

For a two-dimensional space, the Riemann tensor is

$$ \tag{3 } R ( X, Y) Z = \ K ( \langle Y, Z \rangle X - \langle X, Z \rangle Y), $$

where $ K $ is the Gaussian curvature. In this case the scalar curvature is equal to $ K $. For a three-dimensional space the Riemann tensor has the form

$$ R _ {iklm} = \ R _ {il} g _ {km} - R _ {im} g _ {kl} + R _ {km} g _ {il} - R _ {kl} g _ {im} + $$

$$ + { \frac{R}{2} } ( g _ {im} g _ {kl} - g _ {il} g _ {km} ), $$

where $ g _ {ij} $ is the metric tensor, $ R _ {ij} $ is the Ricci tensor and $ R $ is the scalar curvature.

If the sectional curvatures are independent both of the point and of the two-dimensional direction, the space $ M ^ {n} $ is known as a space of constant curvature; the Riemann tensor of such a space has the form (3) (the constant $ K $ is then called the curvature of the space $ M ^ {n} $). When $ n > 2 $ it turns out that, if in all points the curvature is independent of the direction, then $ M ^ {n} $ is a space of constant curvature (Schur's theorem).

The curvature of submanifolds.

Let $ \Phi $ be a regular surface in $ E ^ {3} $, let $ \gamma $ be a curve on $ \Phi $ and let $ \alpha _ {P} $ be the tangent plane to $ \Phi $ at a point $ P $ on $ \gamma $. Suppose that a small neighbourhood of $ P $ is projected onto the plane $ \alpha _ {P} $ and let $ \overline \gamma \; $ be the projection of the curve $ \gamma $ on $ \alpha _ {P} $. The geodesic curvature $ \kappa $ of the curve $ \gamma $ at $ P $ is defined as the number equal in absolute value to the curvature of the curve $ \overline \gamma \; $ at $ P $. The geodesic curvature is considered positive if the rotation of the tangent to $ \overline \gamma \; $ as one passes through $ P $ forms a right-handed screw with the direction of the normal to the surface. The geodesic curvature is an object of the intrinsic geometry of $ \Phi $. It can be evaluated from the formula

$$ \tag{4 } \kappa = \ \frac{e _ {ij} \left ( \frac{d ^ {2} x ^ {i} }{ds ^ {2} } \frac{d ^ {2} x ^ {j} }{ds ^ {2} } + \Gamma _ {kl} ^ {i} \frac{dx ^ {k} }{ds} \frac{dx ^ {l} }{ds} \frac{dx ^ {j} }{ds} \right ) }{\left ( g _ {ij} \frac{dx ^ {i} }{ds} \frac{dx ^ {j} }{ds} \right ) ^ {3/2} } , $$

where $ x ^ {i} ( s) $ is the natural equation of the curve $ \gamma $ in local coordinates $ x ^ {i} $ on $ \Phi $, $ g _ {ij} $ are the components of the metric tensor of $ \Phi $ in these coordinates, $ \Gamma _ {kl} ^ {i} $ are the Christoffel symbols, and $ e _ {ij} $ is the totally discriminant tensor. Using formula (4) one can define the geodesic curvature for curves on an abstract two-dimensional Riemannian space. A curve on a Riemannian manifold coincides with a geodesic or with part of a geodesic if and only if its geodesic curvature vanishes identically.

Let $ \Phi $ be a two-dimensional submanifold of a three-dimensional Riemannian space $ M $. There are two approaches to the definition of the curvature for $ \Phi $. On the one hand, one can consider $ \Phi $ as a Riemannian space whose metric is induced by that of $ M $, and then use formula (1) to define its curvature. This yields what is called the internal curvature. On the other hand, one can carry out the same construction that gives the definition of the curvature for surfaces in a Euclidean space and apply it to submanifolds in a Riemannian space. The result is a different concept of the curvature, known as the external curvature. One has the following relationship:

$$ K _ {i} = \ K _ {e} + K _ \sigma , $$

where $ K _ \sigma $ is the curvature of $ M $ in the direction of the tangent plane to $ \Phi $, and $ K _ {i} $ and $ K _ {e} $ are the internal and external curvatures, respectively.

The concepts of normal, internal and external curvatures can be generalized with respect to the dimension and codimension of the submanifold in question.

The concept of the Riemann tensor may be generalized to various spaces with a weaker structure than Riemannian spaces. For example, the Riemann and Ricci tensors depend only on the affine structure of the space and may also be defined in spaces with an affine connection, although in that case they do not possess all the symmetry properties as above. For example, $ R _ {ik} \neq R _ {ki} $. Other examples of this type are the conformal curvature tensor and the projective curvature tensor. The conformal curvature tensor (Weyl tensor) is

$$ C _ {iklm} = \ R _ {iklm} - R _ {l[i } g _ {k]m } + R _ {m[i } g _ {k]l } + { \frac{1}{3} } R _ {l[i } g _ {k]m } , $$

where the brackets denote alternation with respect to the relevant indices. Vanishing of the conformal curvature tensor is a necessary and sufficient condition for the space to coincide locally with a conformal Euclidean space. The projective curvature tensor is

$$ P _ {lki} ^ { q } = \ R _ {lki} ^ { q } + \delta _ {k} ^ {q} \frac{R _ {li} + R _ {il} }{n ^ {2} - 1 } + \delta _ {l} ^ {q} \frac{R _ {ki} + R _ {ik} }{n ^ {2} - 1 } + \delta _ {i} ^ {q} \frac{R _ {lk} - R _ {kl} }{n + 1 } , $$

where $ \delta _ {i} ^ {k} $ is the Kronecker symbol and $ n $ is the dimension of the space. Vanishing of the projective curvature tensor is a necessary and sufficient condition for the space to coincide locally with a projective Euclidean space.

The concept of curvature generalizes to the case of non-regular objects, in particular, to the case of the theory of two-dimensional manifolds of bounded curvature. Here the curvature in a space is defined not at a point, but in a domain, and one is concerned with the total or integral curvature of a domain. In the regular case the total curvature is equal to the integral of the Gaussian curvature. The total curvature of a geodesic triangle may be expressed in terms of the angles $ \beta _ {i} $ at its vertices:

$$ \tag{5 } K = \ \sum \beta _ {i} - \pi , $$

this relationship is a special case of the Gauss–Bonnet theorem. Formula (5) has been used as a basis for the definition of the total curvature in manifolds of bounded curvature.

The curvature is one of the fundamental concepts in modern differential geometry. Restrictions on the curvature usually yield meaningful information about an object. For example, in the theory of surfaces in $ E ^ {3} $, the sign of the Gaussian curvature defines the type of a point (elliptic, hyperbolic or parabolic). Surfaces with an everywhere non-negative Gaussian curvature share a whole spectrum of properties, by virtue of which they can be grouped together in one natural class (see [4], [6]). Surfaces with zero mean curvature (see Minimal surface) have many specific properties. The theory of non-regular surfaces especially studies classes of surfaces of bounded integral absolute Gaussian or mean curvature.

In Riemannian spaces, a uniform bound on the sectional curvatures $ K _ \sigma $ at any point and in any two-dimensional direction makes it possible to use comparison theorems. The latter enable one to compare the rate of deviation of the geodesics and the volumes of domains in a given space with the characteristics of the corresponding curves and domains in a space of constant curvature. Some of the restrictions on $ K _ \sigma $ even predetermine the topological structure of the space as a whole. For example:

The sphere theorem. Let $ M $ be a complete simply-connected Riemannian space of dimension $ n \geq 2 $ and let $ 1/4 < \delta \leq K _ \sigma \leq 1 $. Then $ M $ is homeomorphic to the sphere $ S ^ {n} $.

The Hadamard–Cartan and Gromoll–Meyer theorems. Let $ M $ be a complete Riemannian space of dimension $ n \geq 2 $. If $ K _ \sigma \leq 0 $ everywhere and $ M $ is simply connected, or if $ K _ \sigma > 0 $ everywhere and $ M $ is not compact, then $ M $ is homeomorphic to the Euclidean space $ E ^ {n} $.

The concepts of curvature are utilized in various natural sciences. Thus, when a body is moving along a trajectory, there is a relationship between the curvature of the trajectory and the centrifugal force. The Gaussian curvature first appeared in Gauss' work on cartography. The mean curvature of the surface of a liquid is related to the capillary effect. In relativity theory there is a connection between the distribution of mass and energy (more precisely, between the energy-momentum tensor) and the curvature of space-time. The conformal curvature tensor is used in the theory of formation of particles in a gravitational field.

References

[1] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[2] A.V. Pogorelov, "Differential geometry" , Noordhoff (1959) (Translated from Russian)
[3] W. Blaschke, "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Elementare Differentialgeometrie" , 1 , Springer (1921)
[4] A.D. Aleksandrov, "Die innere Geometrie der konvexen Flächen" , Akademie Verlag (1955) (Translated from Russian)
[5] D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[6] A.V. Pogorelov, "Extrinsic geometry of convex surfaces" , Amer. Math. Soc. (1972) (Translated from Russian)

Comments

Formula (1) can be expressed in various ways, e.g., in [2] it reads:

$$ K = { \frac{1}{( EG - F ^ {2} ) ^ {2} } } \times $$

$$ \times \left \{ \left | \begin{array}{ccc} - \frac{G _ {uu} }{2} + F _ {uv} - \frac{E _ {vv} }{2} & \frac{E _ {u} }{2} &F _ {u} - \frac{E _ {v} }{2} \\ F _ {v} - \frac{G _ {u} }{2} & E & F \\ \frac{G _ {v} }{2} & F & G \\ \end{array} \right | \right . - $$

$$ - \ \left . \left | \begin{array}{ccc} 0 & \frac{E _ {v} }{2} & \frac{G _ {u} }{2} \\ \frac{E _ {v} }{2} & E & F \\ \frac{G _ {u} }{2} & F & G \\ \end{array} \right | \right \} . $$

The first Bianchi identity is the usual name given to the fourth symmetry relation for the Riemann tensor, i.e. to $ R ( X, Y) Z + R ( Y, Z) X + R ( Z, X) Y = 0 $. The second Bianchi identity is the relation

$$ \nabla _ {X} R ( Y, Z, U) + \nabla _ {Y} R ( Z, X, U) + \nabla _ {Z} R ( X, Y, U) = 0 , $$

called the Bianchi identity above.

Such concepts as mean curvature, conformal curvature tensors, geodesic curvature, and projective curvature tensor are also defined in higher dimensional settings (than surfaces), cf. e.g. [a2] (mean curvature), [a3], [1] (conformal and projective curvature tensors). (Cf. also Conformal Euclidean space.) The absolute value of the geodesic curvature of a curve on a surface is $ | \nabla _ {\dot \gamma } \dot \gamma | $, where $ \gamma $ is assumed to be described by its arc length parameter (natural parameter) and $ \nabla $ is the Levi-Civita connection on the surface. For the concepts of natural parameter and natural equation of a curve, cf. Natural equation. The various fundamental (quadratic) forms of a surface are discussed in Fundamental forms of a surface; Geometry of immersed manifolds and Second fundamental form.

The sectional curvature of a Riemannian space $ M ^ {n} $ at $ P $ in the direction of the tangent plane $ \sigma $ is also called the Riemannian curvature.

Let $ R _ {ij} $ denote the Ricci tensor and let $ Q $ be the quadratic form on $ T _ {P} M ^ {n} $ given by $ R _ {ij} $ at $ P \in M ^ {n} $. Then the value $ Q ( \xi ) $ for a unit vector $ \xi \in T _ {p} M ^ {n} $ is the mean of $ K _ \sigma $ for all plane directions $ \sigma $ in $ T _ {P} M ^ {n} $ containing $ \xi $, and is called the Ricci curvature or mean curvature of the direction $ \xi $ at $ P $. The mean $ R $ of all the $ Q ( \xi ) $ is the scalar curvature at $ P $, cf. also Ricci tensor and Ricci curvature. If $ M $ is a Kähler manifold and $ \sigma $ is restricted to a complex plane (i.e. a plane invariant under the almost-complex structure), then $ K _ \sigma $ is called the holomorphic sectional curvature.

For a simply-closed space curve $ C $ of length $ L $ the integral $ K = \int _ {0} ^ {L} k ( s) ds $ is called the total curvature of $ C $; generally $ K \leq 2 \pi $, and $ K = 2 \pi $ if and only if $ C $ is a closed curve lying in a plane (W. Fenchel). Fix an origin 0 in $ E ^ {3} $ and consider the unit sphere $ S ^ {2} $ around 0. For each point $ P $ of $ C $ let $ \overline{P}\; $ be the point on $ S ^ {2} $ such that $ 0 \overline{P}\; $ is the (displaced) unit tangent vector to $ C $ at $ P $. As $ P $ runs over $ C $ the $ \overline{P}\; $ trace out a curve on $ S ^ {2} $, the spherical indicatrix $ \overline{C}\; $ of $ C $. The correspondence $ C \mapsto \overline{C}\; $ is called the spherical representation. The total curvature of $ C $ is equal to the length of $ \overline{C}\; $. Instead of the tangents to $ C $ one can also use the principal normal and binormal vectors and perform a similar construction yielding other spherical indicatrices, cf. Spherical indicatrix.

References

[a1] C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4
[a2] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 2 , Interscience (1969) pp. 33
[a3] J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) pp. Chapt. VI (Translated from German)
How to Cite This Entry:
Curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Curvature&oldid=12026
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article