# 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 $C ^ {2}$- 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 , ). 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.

How to Cite This Entry:
Curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Curvature&oldid=46565
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article