Difference between revisions of "Riemannian space, generalized"
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200200.png" /> be a convex space with an internal metric; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200201.png" /> be the cone of shortests spanned over a closed rectifiable curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200202.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200203.png" /> from a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200204.png" />, and, moreover, let, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200205.png" />, the length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200206.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200207.png" /> be less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200208.png" />. Then in a space of constant curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200209.png" /> there exists a convex domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200210.png" /> majorizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200211.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200212.png" /> for the corresponding majorizing mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200213.png" />. This property is characteristic for spaces of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200214.png" />. The existence of a length-preserving non-stretching mapping of the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200215.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200216.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200217.png" /> is already sufficient (see [[#References|[8]]]). | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200200.png" /> be a convex space with an internal metric; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200201.png" /> be the cone of shortests spanned over a closed rectifiable curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200202.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200203.png" /> from a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200204.png" />, and, moreover, let, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200205.png" />, the length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200206.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200207.png" /> be less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200208.png" />. Then in a space of constant curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200209.png" /> there exists a convex domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200210.png" /> majorizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200211.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200212.png" /> for the corresponding majorizing mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200213.png" />. This property is characteristic for spaces of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200214.png" />. The existence of a length-preserving non-stretching mapping of the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200215.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200216.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200217.png" /> is already sufficient (see [[#References|[8]]]). | ||
− | A continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200218.png" /> from a disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200219.png" /> into a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200220.png" /> is called a surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200221.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200222.png" /> be a triangulated polygon, i.e. a complex of triangles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200223.png" /> inscribed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200224.png" />. To the triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200225.png" /> with vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200226.png" /> there corresponds the Euclidean triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200227.png" /> with sides equal to the distances between points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200228.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200229.png" /> be the sum of the areas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200230.png" /> of all triangles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200231.png" />; then the area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200232.png" /> of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200233.png" /> is defined (see [[#References|[3]]]) as the limes inferior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200234.png" /> under the condition that the vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200235.png" /> unboundedly contract in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200236.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200237.png" />. This definition is modified as follows (see [[#References|[6]]]). Instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200238.png" />, the vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200239.png" /> of the triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200240.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200241.png" /> are put into correspondence with points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200242.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200243.png" />, where, moreover, to vertices of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200244.png" /> correspond the same points if and only if the images of the vertices under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200245.png" /> coincide. For the area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200246.png" /> of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200247.png" /> one takes the limes inferior of the sums of the areas of the Euclidean triangles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200248.png" /> with sides equal to the distances between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200249.png" />, under the additional assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200250.png" /> tends to zero for all vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200251.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200252.png" />. One always has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200253.png" />. | + | A continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200218.png" /> from a disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200219.png" /> into a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200220.png" /> is called a surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200221.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200222.png" /> be a triangulated polygon, i.e. a complex of triangles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200223.png" /> inscribed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200224.png" />. To the triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200225.png" /> with vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200226.png" /> there corresponds the Euclidean triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200227.png" /> with sides equal to the distances between points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200228.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200229.png" /> be the sum of the areas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200230.png" /> of all triangles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200231.png" />; then the area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200232.png" /> of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200233.png" /> is defined (see [[#References|[3]]]) as the [[limes inferior]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200234.png" /> under the condition that the vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200235.png" /> unboundedly contract in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200236.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200237.png" />. This definition is modified as follows (see [[#References|[6]]]). Instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200238.png" />, the vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200239.png" /> of the triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200240.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200241.png" /> are put into correspondence with points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200242.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200243.png" />, where, moreover, to vertices of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200244.png" /> correspond the same points if and only if the images of the vertices under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200245.png" /> coincide. For the area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200246.png" /> of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200247.png" /> one takes the limes inferior of the sums of the areas of the Euclidean triangles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200248.png" /> with sides equal to the distances between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200249.png" />, under the additional assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200250.png" /> tends to zero for all vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200251.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200252.png" />. One always has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200253.png" />. |
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200254.png" />) If a sequence of surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200255.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200256.png" /> converges uniformly to a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200257.png" />, then | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200254.png" />) If a sequence of surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200255.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200256.png" /> converges uniformly to a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200257.png" />, then |
Revision as of 13:14, 8 May 2017
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:
![]() | (1) |
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,
![]() | (2) |
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,
![]() | (3) |
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).
References
[1] | A.D. Aleksandrov, "Die innere Geometrie der konvexen Flächen" , Akademie Verlag (1955) (Translated from Russian) |
[2] | A.D. Aleksandrov, "A theorem on triangles in metric space and certain applications" Trudy Mat. Inst. Steklov. , 38 (1951) pp. 5–23 (In Russian) |
[3] | A.D. [A.D. Aleksandrov ] Alexandroff, "Über eine Verallgemeinerung der Riemannschen Geometrie" Schrift. Inst. Math. Deutsch. Akad. Wiss. , 1 (1957) pp. 33–84 |
[4] | V.N. Berestovskii, "Introduction of a Riemann structure into certain metric spaces" Sib. Math. J. , 16 : 4 (1975) pp. 499–507 Sibirsk. Mat. Zh. , 16 : 4 (1975) pp. 651–662 |
[5] | I.G. Nikolaev, "Space of directions at a point in a space of curvature not greater than ![]() |
[6] | I.G. Nikolaev, "Solution of Plateau's problem in spaces of curvature not greater than ![]() |
[7] | Yu.G. Reshetnyak, "To the theory of spaces with curvature not greater than ![]() |
[8] | Yu.G. Reshetnyak, "Inextensible mappings in a space of curvature no greater than ![]() |
[9] | I.G. Nikolaev, "Smoothness of the metric of spaces with two-sided bounded A.D. Aleksandrov curvature" Sib. Math. J. , 24 : 2 (1983) pp. 247–263 Sibersk. Mat. Zh. , 24 : 2 (1983) pp. 114–132 |
[10] | V.N. Berestovskii, "Spaces with bounded curvature and distance geometry" Sib. Math. J. , 27 : 1 (1986) pp. 8–18 Sibersk. Mat. Zh. , 27 : 1 (1986) pp. 11–25 |
Comments
References
[a1] | W. Rinow, "Die innere Geometrie der metrischen Räume" , Springer (1961) |
[a2] | M. Gromov, "Structures métriques pour les variétés riemanniennes" , Cedec-Nathan (1981) (Translated from Russian) |
[a3] | "Elie Cartan et les mathématiques d'aujourd'hui" Astérisque (1985) |
[a4] | A.V. Pogorelov, "Intrinsic geometry of surfaces" , Amer. Math. Soc. (1973) (Translated from Russian) |
[a5] | H. Busemann, "The geometry of geodesics" , Acad. Press (1955) |
Riemannian space, generalized. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemannian_space,_generalized&oldid=41327