Curvature
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 -smooth functions. In some cases the concepts are defined in terms of integrals, and they remain valid without the
-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.
Contents
The curvature of a curve.
Let be a regular curve in the
-dimensional Euclidean space, parametrized in terms of its natural parameter
. Let
and
be the angle between the tangents to
at the points
and
of
and the length of the arc of the curve between
and
, respectively. Then the limit
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is called the curvature of the curve at
. The curvature of the curve is equal to the absolute value of the vector
, and the direction of this vector is just the direction of the principal normal to the curve. For the curve
to coincide with some segment of a straight line or with an entire line it is necessary and sufficient that its curvature
vanishes identically.
The curvature of a surface.
Let be a regular surface in the three-dimensional Euclidean space. Let
be a point of
,
the tangent plane to
at
,
the normal to
at
, and
the plane through
and some unit vector
in
. The intersection
of the plane
and the surface
is a curve, called the normal section of the surface
at the point
in the direction
. The number
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where is the natural parameter on
, is called the normal curvature of
in the direction
. The normal curvature is equal to the curvature of the curve
up to the sign.
The tangent plane contains two perpendicular directions
and
such that the normal curvature in any direction can be expressed by Euler's formula:
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where is the angle between
and
. The numbers
and
are called the principal curvatures, and the directions
and
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
, the equation
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where is the radius vector, defines a certain curve of the second order in the tangent plane
, 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 is principal if and only if
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where is the radius vector of the surface and
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 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
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is called the mean curvature of the surface. The quantity
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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:
![]() | (1) |
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where 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 . A surface is locally isometric to a plane if and only if its Gaussian curvature vanishes identically.
The curvature of a Riemannian space.
Let be a regular
-dimensional Riemannian space and let
be the space of regular vector fields on
. The curvature of
is usually characterized by the Riemann (curvature) tensor (cf. Riemann tensor), i.e. by the multilinear mapping
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defined by
![]() | (2) |
where is the Levi-Civita connection on
and
denotes the Lie bracket. If one puts
,
in some local coordinate system
, one can rewrite (2) as follows:
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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 may be expressed in terms of the Christoffel symbols and the coefficients of the metric tensor, as follows:
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where is the Riemann tensor with fourth covariant index, or — in a coordinate-free notation — the mapping
(where
denotes the scalar product).
The Riemann tensor possesses the following symmetry properties:
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which may be written in local coordinates in the form:
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The Riemann tensor has algebraically independent components. The covariant derivatives of the Riemann tensor satisfy the (second) Bianchi identity:
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where is the covariant derivative of
with respect to
. In local coordinates, this identity is
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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 . Let
be a two-dimensional linear space in the tangent space
to
at a point
. Then the sectional curvature of
at
in the direction
is defined as
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where and
are vectors defining
. The same area element
may be defined by different vectors
and
, but
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:
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where
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Weaker characteristics of the curvature of a Riemannian space are also used — the Ricci tensor, or Ricci curvature:
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and the scalar curvature:
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The Ricci tensor is symmetric: .
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
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which is used in investigating the Schwarzschild gravity field.
For a two-dimensional space, the Riemann tensor is
![]() | (3) |
where is the Gaussian curvature. In this case the scalar curvature is equal to
. For a three-dimensional space the Riemann tensor has the form
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where is the metric tensor,
is the Ricci tensor and
is the scalar curvature.
If the sectional curvatures are independent both of the point and of the two-dimensional direction, the space is known as a space of constant curvature; the Riemann tensor of such a space has the form (3) (the constant
is then called the curvature of the space
). When
it turns out that, if in all points the curvature is independent of the direction, then
is a space of constant curvature (Schur's theorem).
The curvature of submanifolds.
Let be a regular surface in
, let
be a curve on
and let
be the tangent plane to
at a point
on
. Suppose that a small neighbourhood of
is projected onto the plane
and let
be the projection of the curve
on
. The geodesic curvature
of the curve
at
is defined as the number equal in absolute value to the curvature of the curve
at
. The geodesic curvature is considered positive if the rotation of the tangent to
as one passes through
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
. It can be evaluated from the formula
![]() | (4) |
where is the natural equation of the curve
in local coordinates
on
,
are the components of the metric tensor of
in these coordinates,
are the Christoffel symbols, and
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 be a two-dimensional submanifold of a three-dimensional Riemannian space
. There are two approaches to the definition of the curvature for
. On the one hand, one can consider
as a Riemannian space whose metric is induced by that of
, 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:
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where is the curvature of
in the direction of the tangent plane to
, and
and
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, . Other examples of this type are the conformal curvature tensor and the projective curvature tensor. The conformal curvature tensor (Weyl tensor) is
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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
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where is the Kronecker symbol and
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 at its vertices:
![]() | (5) |
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 , 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 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
even predetermine the topological structure of the space as a whole. For example:
The sphere theorem. Let be a complete simply-connected Riemannian space of dimension
and let
. Then
is homeomorphic to the sphere
.
The Hadamard–Cartan and Gromoll–Meyer theorems. Let be a complete Riemannian space of dimension
. If
everywhere and
is simply connected, or if
everywhere and
is not compact, then
is homeomorphic to the Euclidean space
.
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:
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The first Bianchi identity is the usual name given to the fourth symmetry relation for the Riemann tensor, i.e. to . The second Bianchi identity is the relation
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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 , where
is assumed to be described by its arc length parameter (natural parameter) and
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 at
in the direction of the tangent plane
is also called the Riemannian curvature.
Let denote the Ricci tensor and let
be the quadratic form on
given by
at
. Then the value
for a unit vector
is the mean of
for all plane directions
in
containing
, and is called the Ricci curvature or mean curvature of the direction
at
. The mean
of all the
is the scalar curvature at
, cf. also Ricci tensor and Ricci curvature. If
is a Kähler manifold and
is restricted to a complex plane (i.e. a plane invariant under the almost-complex structure), then
is called the holomorphic sectional curvature.
For a simply-closed space curve of length
the integral
is called the total curvature of
; generally
, and
if and only if
is a closed curve lying in a plane (W. Fenchel). Fix an origin 0 in
and consider the unit sphere
around 0. For each point
of
let
be the point on
such that
is the (displaced) unit tangent vector to
at
. As
runs over
the
trace out a curve on
, the spherical indicatrix
of
. The correspondence
is called the spherical representation. The total curvature of
is equal to the length of
. Instead of the tangents to
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) |
Curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Curvature&oldid=12026