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''of a surface''
 
''of a surface''
  
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If
 
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/g/g043/g043590/g0435901.png" /></td> </tr></table>
+
$$
 +
\textrm{ I }  = ds  ^ {2}  = \
 +
E  du  ^ {2} + 2 F  du  dv + G  dv  ^ {2}
 +
$$
  
 
is the [[First fundamental form|first fundamental form]] of the surface and
 
is the [[First fundamental form|first fundamental form]] of the surface and
  
<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/g/g043/g043590/g0435902.png" /></td> </tr></table>
+
$$
 +
\textrm{ II }  = L  du  ^ {2} + 2 M  du  dv + N  dv  ^ {2}
 +
$$
  
 
is the [[Second fundamental form|second fundamental form]] of the surface, then the Gaussian curvature can be computed by the formula
 
is the [[Second fundamental form|second fundamental form]] of the surface, then the Gaussian curvature can be computed by the formula
  
<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/g/g043/g043590/g0435903.png" /></td> </tr></table>
+
$$
 +
=
 +
\frac{LN - M  ^ {2} }{EG - F  ^ {2} }
 +
.
 +
$$
  
 
The Gaussian curvature is identical with the Jacobi determinant of the [[Spherical map|spherical map]]:
 
The Gaussian curvature is identical with the Jacobi determinant of the [[Spherical map|spherical map]]:
  
<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/g/g043/g043590/g0435904.png" /></td> </tr></table>
+
$$
 +
| K | _ {P _ {0}  }  = \
 +
\lim\limits _ {d( s) \rightarrow 0 } \
 +
{
 +
\frac{S}{s}
 +
} ,
 +
$$
 +
 
 +
where  $  P _ {0} $
 +
is a point on the surface,  $  s $
 +
is the area of a domain  $  U $
 +
which contains  $  P _ {0} $,
 +
$  S $
 +
is the area of the spherical image of  $  U $,
 +
and  $  d $
 +
is the diameter of the domain. The Gaussian curvature is positive at an [[Elliptic point|elliptic point]], negative at a [[Hyperbolic point|hyperbolic point]], and is zero at a [[Parabolic point|parabolic point]] or a [[Flat point|flat point]]. It may be expressed in terms of the coefficients of the first fundamental form and their derivatives alone (the [[Gauss theorem|Gauss theorem]]), viz.
 +
 
 +
$$
 +
K  = 
 +
\frac{1}{4W  ^ {4} }
 +
\left |
 +
 
 +
\begin{array}{ccc}
 +
E  &E _ {u}  &E _ {v}  \\
 +
F  &F _ {u}  &F _ {v}  \\
 +
G  &G _ {u}  &G _ {v}  \\
 +
\end{array}
 +
\
 +
\right | +
 +
\frac{1}{2W}
 +
 
 +
\left \{
 +
\frac \partial {\partial  u }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g0435905.png" /> is a point on the surface, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g0435906.png" /> is the area of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g0435907.png" /> which contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g0435908.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g0435909.png" /> is the area of the spherical image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359011.png" /> is the diameter of the domain. The Gaussian curvature is positive at an [[Elliptic point|elliptic point]], negative at a [[Hyperbolic point|hyperbolic point]], and is zero at a [[Parabolic point|parabolic point]] or a [[Flat point|flat point]]. It may be expressed in terms of the coefficients of the first fundamental form and their derivatives alone (the [[Gauss theorem|Gauss theorem]]), viz.
+
\frac{F _ {v} - G _ {u} }{W }
 +
+
  
<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/g/g043/g043590/g04359012.png" /></td> </tr></table>
+
\frac \partial {\partial  v }
 +
 +
\frac{F _ {u} - E _ {v} }{W }
 +
\right \} ,
 +
$$
  
 
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/g/g043/g043590/g04359013.png" /></td> </tr></table>
+
$$
 +
W  ^ {2}  = EG - F  ^ {2} .
 +
$$
  
 
Since the Gaussian curvature depends on the metric only, i.e. on the coefficients of the first fundamental form, the Gaussian curvature is invariant under isometric deformation (cf. [[Deformation, isometric|Deformation, isometric]]). The Gaussian curvature plays a special role in the theory of surfaces, and many formulas are available for its computation, [[#References|[2]]].
 
Since the Gaussian curvature depends on the metric only, i.e. on the coefficients of the first fundamental form, the Gaussian curvature is invariant under isometric deformation (cf. [[Deformation, isometric|Deformation, isometric]]). The Gaussian curvature plays a special role in the theory of surfaces, and many formulas are available for its computation, [[#References|[2]]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.F. Gauss,  "Allgemeine Flächentheorie" , W. Engelmann , Leipzig  (1900)  (Translated from Latin)  {{MR|}}  {{ZBL|31.0599.11}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Reichardt,  "Einführung in die Differentialgeometrie" , Springer  (1960)  {{MR|0116267}} {{ZBL|0091.34001}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.F. Gauss,  "Allgemeine Flächentheorie" , W. Engelmann , Leipzig  (1900)  (Translated from Latin)  {{MR|}}  {{ZBL|31.0599.11}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Reichardt,  "Einführung in die Differentialgeometrie" , Springer  (1960)  {{MR|0116267}} {{ZBL|0091.34001}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
The total Gaussian curvature (often also abbreviated to total curvature) is the quantity
 
The total Gaussian curvature (often also abbreviated to total curvature) is 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/g/g043/g043590/g04359014.png" /></td> </tr></table>
+
$$
 +
\int\limits \int\limits K  d \sigma .
 +
$$
  
 
(See also [[Gauss–Bonnet theorem|Gauss–Bonnet theorem]].)
 
(See also [[Gauss–Bonnet theorem|Gauss–Bonnet theorem]].)
  
For a smooth space curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359015.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359016.png" />, the total curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359018.png" /> is defined as the length of the spherical image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359019.png" /> (cf. also [[Spherical indicatrix|Spherical indicatrix]]) and can be expressed using the [[Frénet formulas|Frénet formulas]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359022.png" /> for a Frénet frame (cf. [[Frénet trihedron|Frénet trihedron]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359023.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359024.png" /> by
+
For a smooth space curve $  C $
 +
given by $  x = x ( s) $,  
 +
the total curvature $  K $
 +
of $  C $
 +
is defined as the length of the spherical image of $  C $(
 +
cf. also [[Spherical indicatrix|Spherical indicatrix]]) and can be expressed using the [[Frénet formulas|Frénet formulas]] $  e _ {1}  ^  \prime  = \kappa _ {1} e _ {2} $,  
 +
$  e _ {2}  ^  \prime  = - \kappa _ {1} e _ {1} + \kappa _ {2} e _ {3} $,  
 +
$  e _ {3}  ^  \prime  = - \kappa _ {2} e _ {2} $
 +
for a Frénet frame (cf. [[Frénet trihedron|Frénet trihedron]]) $  ( x, e _ {1} , e _ {2} , e _ {3} ) $
 +
along $  C $
 +
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/g/g043/g043590/g04359025.png" /></td> </tr></table>
+
$$
 +
= \int\limits \kappa _ {1}  ds.
 +
$$

Latest revision as of 19:41, 5 June 2020


of a surface

The product of the principal curvatures (cf. Principal curvature) of a regular surface at a given point.

If

$$ \textrm{ I } = ds ^ {2} = \ E du ^ {2} + 2 F du dv + G dv ^ {2} $$

is the first fundamental form of the surface and

$$ \textrm{ II } = L du ^ {2} + 2 M du dv + N dv ^ {2} $$

is the second fundamental form of the surface, then the Gaussian curvature can be computed by the formula

$$ K = \frac{LN - M ^ {2} }{EG - F ^ {2} } . $$

The Gaussian curvature is identical with the Jacobi determinant of the spherical map:

$$ | K | _ {P _ {0} } = \ \lim\limits _ {d( s) \rightarrow 0 } \ { \frac{S}{s} } , $$

where $ P _ {0} $ is a point on the surface, $ s $ is the area of a domain $ U $ which contains $ P _ {0} $, $ S $ is the area of the spherical image of $ U $, and $ d $ is the diameter of the domain. The Gaussian curvature is positive at an elliptic point, negative at a hyperbolic point, and is zero at a parabolic point or a flat point. It may be expressed in terms of the coefficients of the first fundamental form and their derivatives alone (the Gauss theorem), viz.

$$ K = \frac{1}{4W ^ {4} } \left | \begin{array}{ccc} E &E _ {u} &E _ {v} \\ F &F _ {u} &F _ {v} \\ G &G _ {u} &G _ {v} \\ \end{array} \ \right | + \frac{1}{2W} \left \{ \frac \partial {\partial u } \frac{F _ {v} - G _ {u} }{W } + \frac \partial {\partial v } \frac{F _ {u} - E _ {v} }{W } \right \} , $$

where

$$ W ^ {2} = EG - F ^ {2} . $$

Since the Gaussian curvature depends on the metric only, i.e. on the coefficients of the first fundamental form, the Gaussian curvature is invariant under isometric deformation (cf. Deformation, isometric). The Gaussian curvature plays a special role in the theory of surfaces, and many formulas are available for its computation, [2].

The concept was introduced by C.F. Gauss [1], and was named after him.

References

[1] C.F. Gauss, "Allgemeine Flächentheorie" , W. Engelmann , Leipzig (1900) (Translated from Latin) Zbl 31.0599.11
[2] H. Reichardt, "Einführung in die Differentialgeometrie" , Springer (1960) MR0116267 Zbl 0091.34001

Comments

The total Gaussian curvature (often also abbreviated to total curvature) is the quantity

$$ \int\limits \int\limits K d \sigma . $$

(See also Gauss–Bonnet theorem.)

For a smooth space curve $ C $ given by $ x = x ( s) $, the total curvature $ K $ of $ C $ is defined as the length of the spherical image of $ C $( cf. also Spherical indicatrix) and can be expressed using the Frénet formulas $ e _ {1} ^ \prime = \kappa _ {1} e _ {2} $, $ e _ {2} ^ \prime = - \kappa _ {1} e _ {1} + \kappa _ {2} e _ {3} $, $ e _ {3} ^ \prime = - \kappa _ {2} e _ {2} $ for a Frénet frame (cf. Frénet trihedron) $ ( x, e _ {1} , e _ {2} , e _ {3} ) $ along $ C $ by

$$ K = \int\limits \kappa _ {1} ds. $$

How to Cite This Entry:
Gaussian curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gaussian_curvature&oldid=28198
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article