A space with an
internal metric
subject to certain restrictions on the curvature. Spaces
of
"bounded curvature from above"
and others belong to this class (see
[3]).
Generalized Riemannian spaces differ from Riemannian spaces (cf.
Riemannian space)
not only by greater generality but also by the fact that
they are defined and studied on the basis of their metric
alone, without coordinates. Under a certain combination of conditions
concerning the curvature and the behaviour of shortests curves (i.e. curves whose lengths
are equal to the distances between the end points), a generalized Riemannian
space turns out to be Riemannian, which gives
a purely metric definition of a Riemannian space.
Definitions of generalized Riemannian spaces are based on the classical
relation between the curvature and the excess of a
geodesic triangle
(excess
sum of the angles minus
).
These concepts are carried over to a space with
an internal metric, such that each point of it has
a neighbourhood in which any two points can be connected by a
shortest curve. This condition is assumed hereafter without further stipulation. A
triangle
is triplet of shortests curves
— the
sides of the triangle — connecting in pairs three different points
— the vertices of the triangle. The angle between curves can
be defined in any metric space: Let
be curves starting at the same point
in a space with metric
.
One chooses points
,
and constructs the Euclidean triangle with sides
,
,
and angle
opposite to the side
.
One defines the
upper angle
between
and
as:
The upper angles of the triangle are the upper angles
between its sides at the vertices
and the excess of the triangle is
.
A generalized Riemannian space of bounded curvature
(
and
)
is defined by the following condition:
)
for any sequence of triangles
contracting to a point,
where
is the area of the Euclidean triangle with the same sides as
(if
,
then
).
Such a space turns out to be Riemannian under two natural additional conditions:
1)
local compactness of the space
(in a space with an internal metric this already
ensures the condition of local existence of shortests);
2)
local extendibility of shortests,
i.e. each point has a neighbourhood
such that any shortest
,
where
,
can be extended beyond its end points. Under
all these conditions the space is Riemannian (see
[4]);
moreover, in a neighbourhood of each point one can introduce coordinates
so that the metric will be given by a line element
with coefficients
,
.
Thus, a
parallel displacement
is given (with continuous
)
and, almost everywhere, a
curvature tensor
(see
[9]).
Moreover, it has been proved
[9]
that the coordinates
can be taken harmonic, i.e. satisfying the equalities
.
Harmonic coordinate systems
form an atlas of class
for any
,
.
A generalized Riemannian
space of bounded curvature
with
and satisfying conditions 1) and 2) is a Riemannian space of constant
Riemannian curvature
(see
[3]).
Any Riemannian space of Riemannian curvature contained in between
and
(
)
is a generalized Riemannian space of curvature
and
and satisfies conditions 1) and 2).
A
"space of curvature ≤ K"
is defined by the left inequality in 2), i.e. by the condition:
)
for any sequence of triangles
contracting to a point,
Another, equivalent, definition and a starting point for the study of
generalized Riemannian spaces are based on the comparison between an arbitrary triangle
and a triangle
with sides of the same lengths in a space of constant curvature
.
Let
be the angles of such a triangle; the
relative upper excess
of the triangle
is defined as
.
Condition
)
in the definition of a space of curvature
can be replaced by the following condition:
)
any point has a neighbourhood
in which
for any triangle
.
An even stronger property of concavity of the metric also holds. Namely, let
and
be shortests starting at the same point
and let
be the angle in the triangle
with sides
,
,
,
,
,
in a space of constant curvature
,
opposite to the side
.
In
(locally) the angle
turns out to be a non-decreasing function
(
for
,
,
a
-concave metric).
Hence one obtains the following local properties:
I) between any two shortests starting at the same point there
exists an angle and even an
"angle in the strong sense"
(so that, in particular, if
,
);
II) for the angles
of a triangle in
and the corresponding triangle
,
III) in
,
if
,
,
then the shortests
(thus, a shortest with given end points is unique in
).
Dual to spaces of curvature
are the spaces of curvature
subject to the condition dual to
-concavity:
)
each point has a neighbourhood
in which the angle
for two shortests
is a non-increasing function (a
-concave metric,
cf. also
Convex metric).
Similarly to spaces of curvature
,
for spaces of curvature
the following (local) properties analogous to I) and II) are valid:
Between two shortests there exists an angle in the strong sense;
,
,
for any triangle in
.
Instead of III) the condition of non-overlapping of shortests or,
which is the same, uniqueness of their extension holds: If
and
in
,
then either
or
.
Thus, a space of bounded curvature is obtained by combining the
conditions determining both classes of spaces — with curvature bounded from above
and from below (moreover, on the left-hand side of
inequality
(3)
there is no need to take
).
Condition
)
can be replaced, similar to
),
by the condition:
)
each point has a neighbourhood
,
where
,
for any triangle
.
The above turns out to be equivalent to the following:
)
for any quadruple of points in
there exists a quadruple of points with the same
pairwise distances in a space of constant curvature
,
where
and
depends, in general, on the chosen quadruple of points in
(see
[10]).
An example of a generalized Riemannian space of curvature
is a domain of a Riemannian space such that the
Riemannian curvatures of all two-dimensional surface elements at all
points of this domain are bounded from above by
(from below by
).
A set
in a space with an internal metric is called
convex
if any two points
can be connected by a shortest
and if every such shortest lies in
.
The following result
[7]
has been established: If a space
with an internal metric is obtained by glueing together of two spaces
of curvatures
along convex sets
and
,
then
itself is a space of curvature
.
The glueing condition is that
,
and the metrics of
are induced by that of the space
.
By definition, two curves
,
starting at a point
have the
same direction
at
if the upper angle between them is equal to zero (if
,
is said to have a definite oriented direction at
).
A
direction
at the point
is defined as a class of curves having the same direction at
.
The directions at the point
form a metric space in which the distance
between two directions is determined by the upper angle between any
two representatives of them. Such a space is called a
space of directions
at
.
The following has been proved
[5]:
If the point
lies in a neighbourhood of a space of curvature
homeomorphic to
,
then the space of directions at the point
has curvature
.
In the general case it is not homeomorphic to the
-dimensional
sphere.
In the two-dimensional case, the theory of manifolds of curvature
is included as a special case in the theory of manifolds of bounded curvature (see
Two-dimensional manifold of bounded curvature).
An example of a two-dimensional manifold of curvature
is a ruled surface in
provided with an internal metric, i.e. the surface formed by
the interior parts of shortests whose ends cut out two rectifiable curves
.
If the curve
degenerates to a point
,
the surface is called the
cone of shortests
spanned from the point
over the curve
.
If
is a triangle
,
then such a cone is called a
surface triangle
(see
[3]).
A mapping
of metric spaces is called
non-stretching
if
for any
.
A mapping
of a closed curve
in
onto a closed curve
in
is called
length-preserving
if the lengths of corresponding arcs of
and
coincide under
.
Let
be a convex domain in a space of constant curvature
and
be the boundary contour of
.
The domain
is said to
majorize
a closed curve
in a metric space
if there exists a non-stretching mapping from
into
that is length-preserving from
to
.
The mapping itself is called
majorizing.
Let
be a convex space with an internal metric; let
be the cone of shortests spanned over a closed rectifiable curve
in
from a point
,
and, moreover, let, if
,
the length
of
be less than
.
Then in a space of constant curvature
there exists a convex domain
majorizing
and such that
for the corresponding majorizing mapping
.
This property is characteristic for spaces of curvature
.
The existence of a length-preserving non-stretching mapping of the contour
of
onto
is already sufficient (see
[8]).
A continuous mapping
from a disc
into a metric space
is called a
surface
in
.
Let
be a triangulated polygon, i.e. a complex of triangles
inscribed in
.
To the triangle
with vertices
there corresponds the Euclidean triangle
with sides equal to the distances between points
.
Let
be the sum of the areas
of all triangles
;
then the
area
of the surface
is defined (see
[3])
as the limes inferior of
under the condition that the vertices of
unboundedly contract in
:
.
This definition is modified as follows (see
[6]).
Instead of
,
the vertices
of the triangle
of the complex
are put into correspondence with points
in
,
where, moreover, to vertices of the complex
correspond the same points if and only if the images of the vertices under
coincide. For the area
of the surface
one takes the limes inferior of the sums of the areas of the Euclidean triangles
with sides equal to the distances between
,
under the additional assumption that
tends to zero for all vertices
of the complex
.
One always has
.
)
If a sequence of surfaces
in
converges uniformly to a surface
,
then
)
If
is a non-stretching mapping from
into
and
is a surface in
,
then
)
The area
of a surface triangle
in
is not larger than the area of the corresponding triangle
and is equal to it if and only if
is isometric to
(the local property).
)
Under the conditions of the existence theorem for
a majorizing mapping (see above), the area
is not larger than the area of the disc of perimeter 1 in a space of constant curvature
(the isoperimetric inequality) (see
[3],
[6]).
In
[6]
the
Plateau problem
on the existence of a surface of minimal area spanned over a closed curve
in
is solved. The following has been proved. Let
be a metrically-complete space of curvature
(for
,
the diameter
)
and let
be a closed Jordan curve in
.
Then there exists a surface
of minimal area
spanned over the curve
.
Let
,
be closed Jordan curves in such a space and let
,
be the minimal areas of the surfaces spanned over
and
,
respectively. If the
converge under some parametrizations uniformly to
,
then
.
Two-dimensional manifolds with an indefinite metric of bounded curvature have
been studied. The problem of a coordinate-free definition of
multi-dimensional spaces with an indefinite metric of bounded curvature, and, in particular,
of spaces in the general theory of relativity, has not yet been solved
(1990).