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:
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
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
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
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:
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.