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.

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

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

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:

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

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

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

is called the mean curvature of the surface. The quantity

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)

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

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:

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:

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:

which may be written in local coordinates in the form:

The Riemann tensor has algebraically independent components. The covariant derivatives of the Riemann tensor satisfy the (second) Bianchi identity:

where is the covariant derivative of with respect to . In local coordinates, this identity is

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

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:

where

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

and the scalar curvature:

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

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

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:

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

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 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:

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

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)