The theory of Riemannian spaces. A
Riemannian space
is an
-dimensional
connected
differentiable manifold
on which a differentiable tensor field
of rank 2 is given which is covariant, symmetric and positive definite. The tensor
is called a
metric tensor.
Riemannian geometry is a multi-dimensional generalization of the intrinsic geometry (cf.
Interior geometry)
of two-dimensional surfaces in the Euclidean space
.
The metric of a Riemannian space coincides with
the Euclidean metric of the domain under consideration up
to the first order of smallness. The difference between these
metrics is (locally) estimated by the Riemannian curvature —
a multi-dimensional generalization of the concept of the
Gaussian curvature
of a surface in
.
At the foundation of Riemannian geometry there are three ideas. The first
of these is the realization of the fact that a non-Euclidean geometry
exists — the geometry of
N.I. Lobachevskii.
The second is the concept
of the interior geometry of surfaces created by
C.F. Gauss.
The third is the concept of an
-dimensional
space, developed in the first half of the
19th century
by
B. Riemann,
who unified and generalized these ideas
in his lecture
"On the hypotheses underlying the foundations of geometry"
[1].
The concepts of Riemannian geometry played an important role in the
formulation of the general theory of relativity by
A. Einstein,
and, further,
its development was related to the creation of the apparatus
of tensor analysis. Riemannian geometry and its many generalizations have
been successfully developed, particularly in that part known as
Riemannian geometry in the large,
and find wide and profound application in mechanics and physics
[4].
The basic concepts of Riemannian geometry are the following.
Scalar product.
In each tangent space
,
,
the tensor
determines a scalar (inner) product
according to the formula
The converse is also true: If for any
in
a scalar product is defined which depends differentiably on
,
then it defines a tensor field
with the properties listed above. The degrees of smoothness of
and
vary, depending on the problem posed. In most cases it is sufficient to demand that
be three times continuously differentiable and that the field of
be twice continuously differentiable (below, the necessary degree
of smoothness will not be indicated). In local coordinates
with a local basis
,
,
the components of
take the form
so that
where
A Riemannian space as a metric space.
The length
of a smooth curve
is determined by the formula
where
is the tangent vector to
.
The length of a piecewise-smooth curve is equal to
the total length of its smooth parts. If
is the equation of
in local coordinates, then
In view of this formula, the metric in
is written in the conventional form
and
is called the element of length, the functions
being the coefficients of the metric (first fundamental) form. The
angle
between two curves at a point of intersection is defined
as the angle between the tangents to them. The volume
of a domain
which belongs to a coordinate neighbourhood is determined by the formula
where
.
The volume of an arbitrary domain is equal to the sum of
the volumes of its parts, each of them lying in a specific coordinate neighbourhood.
The
distance
between two points
is defined as the greatest lower bound of
the lengths of all piecewise-smooth curves that join
and
.
The metric
in an arbitrary connected domain
is defined in the same way. Two Riemannian spaces
and
are called
isometric
if there is a transformation
under which
or, which is the same,
,
where
is an arbitrary curve in
.
If
is an
isometry,
then for any point
there is a coordinate neighbourhood
and a coordinate neighbourhood
such that
,
,
.
An
isometric mapping
of
onto itself is called a
motion.
A curve with ends at two points
and
is called a
shortest curve
if its length is equal to
.
A stationary curve of the length functional
is called a
geodesic.
Each shortest curve in
is a geodesic and each sufficiently small arc of a geodesic is a shortest curve. A domain
is called
geodesically convex
if the shortest curves, determined from the metric
,
are geodesics of
.
If
,
,
are the equations of a geodesic in a local coordinate system
,
then the functions
satisfy a system of equations that, when
is a parameter proportional to the arc length, takes the form
where
are the Christoffel symbols (cf.
Christoffel symbol),
and
are the elements of the matrix inverse to
,
.
A Riemannian space is called
complete
(geodesically complete)
if it is complete as a metric space (if any arc of a
geodesic can be extended indefinitely on both sides). A Riemannian space is
complete if and only if it is geodesically complete. In
a complete Riemannian space any two points can be connected by a
shortest curve (that is not necessarily unique). The structure of
a complete Riemannian space may be introduced on any differentiable manifold.
A Riemannian space as a manifold with a connection.
A
covariant derivative
is called
symmetric and compatible with the metric
of the space
if the symmetry conditions
and compatibility conditions
are fulfilled, where
are vector fields and
is their Lie bracket. These conditions determine the derivative
uniquely in terms of the field of the metric tensor
.
In local coordinates
the components of the connection
take the form
,
coincide with the Christoffel symbol of the first kind, and
The
covariant derivative
of any tensor is determined by an analogous formula.
A vector field
along a curve
is called
parallel
if
.
Analytically a parallel field
is determined by a solution to the system
where
are the equations of the curve
.
The solutions to this system under different
initial conditions determine a transformation of
into
;
it turns out to be an isometry and is called the
Levi-Civita parallel displacement.
The result of the transfer depends, as a rule, not only on the end-points
and
,
but also on the arc
itself. A curve for which
is a geodesic, and this property of a geodesic can be taken as its definition.
Submanifolds of a Riemannian space.
If
,
,
is a differentiable
submanifold of a Riemannian space
,
then in each tangent space to
a scalar product is induced, and thus there arises on
the structure of a Riemannian space with a metric tensor
,
the components of which are calculated by the formulas
where
are the equations of
in local coordinates. The extrinsic geometry of
,
,
is described by the second fundamental forms
,
which are determined for each unit normal
to
by the formula
where
and
are tangent vector fields on
and
is an arbitrary field of unit normals containing
.
For any form
the normal curvatures, the principal directions and curvatures, the
mean curvature,
the
complete curvature,
etc., are defined, and the Gauss–Codazzi–Ricci
equations can be derived, connecting the coefficients of the
first and second fundamental forms. Important classes of submanifolds
are characterized by properties of the second fundamental
form, i.e. minimal, totally geodesic, convex, etc. For
(a smooth curve) a theory has been constructed similar to that of curves in
,
the first, second, etc., curvatures have been determined, and equations similar to the
Frénet formulas
have been derived. The first curvature of a curve,
,
is generally called the
geodesic curvature,
and is calculated from the formula
if
is the arc length parameter; in local coordinates
where
and
are the equations of
.
Many problems of Riemannian geometry are connected with
isometric immersion
of one Riemannian space into another, and with the study of
the properties of such immersions. These problems are difficult and little
research has been done on them (in the
two-dimensional case they were studied in more detail).
The
exponential mapping
is determined by the condition
where
is the end of the arc of the geodesic that starts at
with direction
and length
.
If a coordinate system is introduced into a neighbourhood of a point
by associating with the point
the Cartesian coordinates of the point
,
then it turns out that
that is,
these are the so-called (Riemannian)
normal coordinates.
Curvature.
If normal coordinates are introduced in a neighbourhood of a point
,
then the components of the metric tensor take the form
where
as
,
.
From this, an important property of a Riemannian metric can be derived: For any point
the exponential mapping
possesses the property
where
as
.
In general it is impossible to obtain a higher-order coincidence of the metrics of
and
by a more successful choice of the mapping
.
Therefore the coefficients
characterize the deviation between the metric of
and the Euclidean metric of
.
These coefficients are components of the so-called
curvature tensor
or
Riemann–Christoffel tensor
(at the point
).
In local coordinates
they are expressed in terms of the coefficients of the
metric tensor and its first and second derivatives by the formula
A series of other concepts is associated with the curvature tensor. These
also (from various sides) characterize the extent of the deviation of the metric of
from the Euclidean metric. Thus, in terms of the curvature tensor one can define the
Ricci tensor
and the
Einstein tensor
where
is known as the
scalar curvature
of
.
The trilinear transformation that assigns the field
to the three vector fields
,
is called the
curvature transformation.
Its properties are:
1)
;
2)
(the
first Bianchi identity);
3)
;
4)
.
In addition, the
second Bianchi identity
holds. The curvature transformation may be connected by some construction with
the parallel transfer. The algebraic properties of the curvature tensor
are derived from the properties of the curvature transformation, since in terms
of it, and in particular in terms of the
"bi-quadratic form"
,
the curvature tensor (or, more precisely, its value on the vectors
)
can be uniquely (algebraically) expressed (see
Curvature).
Sectional curvature.
Let
be a two-dimensional surface in
passing through
,
let
.
Let
be a simply-closed curve in
passing through
,
let
be the area of the domain in
bounded by the curve
,
let
,
let
be the vector obtained from
by parallel transfer along
,
and let
be the angle between
and the tangential component of
.
Then when
is contracted to the point
,
the limit
exists and is called the Riemannian
sectional curvature
of
at
in the given two-dimensional direction
(
does not depend on the surface
but only on
).
The sectional curvature shows the extent of
"bending"
of
at a given point and in a given two-dimensional direction.
In general, this bending varies with different two-dimensional
directions; if, however, at each point the curvature
does not depend on the choice of
,
then it does not change from point to point
(Schur's theorem).
The identical vanishing of the sectional curvature
is a necessary and sufficient condition for
to be locally isometric to
(in the large it can be different from
).
The sectional curvature of
may be connected with other objects of Riemannian geometry
as well, such as the defect (excess) of a geodesic triangle (see
Gauss–Bonnet theorem).
Riemann defined the sectional curvature as the
Gaussian curvature of the two-dimensional surface
calculated by Gauss' formula at
.
The metric of
is uniquely defined by the sectional curvature in the
following sense: If the sectional curvatures of two manifolds
and
are constant and equal to the same number
,
then
and
are locally isometric, and if they are also both simply connected, then
they are simply isometric. A simply-connected
Riemannian space of constant sectional curvature
is isometric to: the
-dimensional
Lobachevskii space
when
;
the
-dimensional
Euclidean space
when
;
and the
-dimensional
sphere
in
of radius
when
.
In general, the following result is known: If
is an analytic Riemannian space of non-constant sectional
curvature and if there is a diffeomorphism
such that
,
then if
the transformation
is an isometry (cf.
Isometric mapping);
if
this statement has been proved under certain additional assumptions, while if
the theorem is not true. However, it is not known
(1983),
apart from the two-dimensional case, what the function
must be in order for it to have a metric
for which it is the sectional curvature. Only some negative
results have so far been obtained in this direction.
The sectional curvature is connected with the curvature transformation by the formula
and in terms of the components of the curvature tensor it is expressed as:
where
is determined by the vectors
The value of the Ricci tensor
at a vector
is connected with the sectional curvature in the following way: Let the vectors
form an orthonormal basis in
;
then
where
is the two-dimensional direction of the vectors
and
.
Special classes of Riemannian spaces.
In addition to general (arbitrary) Riemannian spaces there are Riemannian
spaces onto which additional structures may be introduced. These structures
arise when some kind of geometric or algebraic conditions are directly
imposed on the metrics. This is how some important classes of
Riemannian spaces are defined: a manifold of constant sectional curvature (see
Space forms),
a
homogeneous space,
a
symmetric space,
Hermitian and Kähler manifolds (cf.
Kähler manifold),
Einstein spaces, etc.
Generalizations.
The development of the ideas of Riemannian geometry and geometry in the large
has led to a series of generalizations of the concept of Riemannian geometry.
Pseudo-Riemannian geometry
is the theory of a
pseudo-Riemannian space.
This is a differentiable manifold on which
a non-degenerate symmetric tensor field is given.
Finsler geometry
is the theory of a differentiable manifold in the tangent bundle of which a function
is given that is homogeneous of the first degree in
.
The length
of a curve
is calculated as follows:
Spaces of bounded curvature
refers to the theory of two-dimensional metric manifolds with an
internal metric
(without any assumption of smoothness) on which the
total curvature of any bounded Borel set is defined. Related
to this theory is the intrinsic geometry of convex surfaces
[5].
This class of metric spaces may be obtained by
adjoining to two-dimensional Riemannian spaces two-dimensional metric manifolds whose metric
in a neighbourhood of each point admits a
uniform approximation by Riemannian metrics with integrals of the
absolute Gaussian curvature that are bounded in aggregate.
Spaces of curvature not exceeding
refers to the theory of complete metric manifolds with
internal metrics, in which the sum of the interior angles of
triangles which are made up of shortest curves does not exceed the sum
of the angles of a triangle in a plane of constant curvature
with sides of the same length (furthermore, it is assumed that any
two points may be connected by a single shortest curve). See also
Geodesic geometry;
Conformal geometry;
Riemannian space, generalized;
Riemannian geometry in the large.