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A space with an [[Internal metric|internal metric]] subject to certain restrictions on the curvature. Spaces of  "bounded curvature from above"  and others belong to this class (see [[#References|[3]]]). Generalized Riemannian spaces differ from Riemannian spaces (cf. [[Riemannian space|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.
 
A space with an [[Internal metric|internal metric]] subject to certain restrictions on the curvature. Spaces of  "bounded curvature from above"  and others belong to this class (see [[#References|[3]]]). Generalized Riemannian spaces differ from Riemannian spaces (cf. [[Riemannian space|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|geodesic triangle]] (excess <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r0822001.png" /> sum of the angles minus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r0822002.png" />). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r0822003.png" /> is triplet of shortests curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r0822004.png" /> — the sides of the triangle — connecting in pairs three different points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r0822005.png" /> — the vertices of the triangle. The angle between curves can be defined in any metric space: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r0822006.png" /> be curves starting at the same point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r0822007.png" /> in a space with metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r0822008.png" />. One chooses points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r0822009.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220010.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220011.png" /> and constructs the Euclidean triangle with sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220014.png" /> and angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220015.png" /> opposite to the side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220016.png" />. One defines the upper angle between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220018.png" /> as:
+
Definitions of generalized Riemannian spaces are based on the classical relation between the curvature and the excess of a [[Geodesic triangle|geodesic triangle]] (excess = $
 +
sum of the angles minus $  \pi $).  
 +
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 $  T = ABC $
 +
is triplet of shortests curves $  AB, BC, CA $—  
 +
the sides of the triangle — connecting in pairs three different points $  A, B, C $—  
 +
the vertices of the triangle. The angle between curves can be defined in any metric space: Let $  L, M $
 +
be curves starting at the same point $  O $
 +
in a space with metric $  \rho $.  
 +
One chooses points $  X \in L $,
 +
$  Y \in M $
 +
$  ( X, Y \neq O) $
 +
and constructs the Euclidean triangle with sides $  x = \rho ( O, X) $,
 +
$  y = \rho ( O, Y) $,  
 +
$  z = \rho ( X, Y) $
 +
and angle $  \gamma ( x, y) $
 +
opposite to the side $  z $.  
 +
One defines the upper angle between $  L $
 +
and $  M $
 +
as:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\overline \alpha \; = \overline{\lim\limits}\; _ {x,y \rightarrow 0 }  \gamma ( x, y).
 +
$$
  
The upper angles of the triangle are the upper angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220020.png" /> between its sides at the vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220021.png" /> and the excess of the triangle is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220022.png" />.
+
The upper angles of the triangle are the upper angles $  \widetilde \alpha  , \widetilde \beta  , \widetilde \gamma  $
 +
between its sides at the vertices $  A, B, C $
 +
and the excess of the triangle is $  \overline \delta \; ( T) = \widetilde \alpha  + \widetilde \beta  + \widetilde \gamma  - \pi $.
  
A generalized Riemannian space of bounded curvature (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220024.png" />) is defined by the following condition:
+
A generalized Riemannian space of bounded curvature ( $  \leq  K $
 +
and $  \geq  K  ^  \prime  $)  
 +
is defined by the following condition:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220025.png" />) for any sequence of triangles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220026.png" /> contracting to a point,
+
$  A $)  
 +
for any sequence of triangles $  T _ {n} $
 +
contracting to a point,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
K  \geq  \overline{\lim\limits}\;
 +
\frac{\overline \delta \; ( T _ {n} ) }{\sigma ( T _ {n}  ^ {0} ) }
 +
  \geq
 +
fnnme \underline{lim} 
 +
\frac{\overline \delta \; ( T _ {n} ) }{\sigma ( T _ {n}  ^ {0} ) }
 +
  \geq  K  ^  \prime  ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220028.png" /> is the area of the Euclidean triangle with the same sides as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220029.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220030.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220031.png" />). Such a space turns out to be Riemannian under two natural additional conditions:
+
where $  \sigma ( T _ {n}  ^ {0} ) $
 +
is the area of the Euclidean triangle with the same sides as $  T _ {n} $(
 +
if $  \sigma ( T _ {n}  ^ {0} ) = 0 $,  
 +
then $  \overline \delta \; ( T _ {n} ) = 0 $).  
 +
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);
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220032.png" /> such that any shortest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220033.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220034.png" />, can be extended beyond its end points. Under all these conditions the space is Riemannian (see [[#References|[4]]]); moreover, in a neighbourhood of each point one can introduce coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220035.png" /> so that the metric will be given by a line element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220036.png" /> with coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220038.png" />. Thus, a [[Parallel displacement(2)|parallel displacement]] is given (with continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220039.png" />) and, almost everywhere, a [[Curvature tensor|curvature tensor]] (see [[#References|[9]]]).
+
2) local extendibility of shortests, i.e. each point has a neighbourhood $  U $
 +
such that any shortest $  XY $,  
 +
where $  X, Y \in U $,  
 +
can be extended beyond its end points. Under all these conditions the space is Riemannian (see [[#References|[4]]]); moreover, in a neighbourhood of each point one can introduce coordinates $  x  ^ {i} $
 +
so that the metric will be given by a line element $  ds  ^ {2} = g _ {ij}  dx  ^ {i}  dx  ^ {j} $
 +
with coefficients $  g _ {ij} \in W _ {q}  ^ {2} \cap C ^ {1, \alpha } $,
 +
$  0 < \alpha < 1 $.  
 +
Thus, a [[Parallel displacement(2)|parallel displacement]] is given (with continuous $  \Gamma _ {jk}  ^ {i} $)  
 +
and, almost everywhere, a [[Curvature tensor|curvature tensor]] (see [[#References|[9]]]).
  
Moreover, it has been proved [[#References|[9]]] that the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220040.png" /> can be taken harmonic, i.e. satisfying the equalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220041.png" />. Harmonic coordinate systems form an atlas of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220042.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220044.png" />.
+
Moreover, it has been proved [[#References|[9]]] that the coordinates $  x  ^ {i} $
 +
can be taken harmonic, i.e. satisfying the equalities $  \sum _ {ij} g  ^ {ij} \Gamma _ {ij}  ^ {l} = 0 $.  
 +
Harmonic coordinate systems form an atlas of class $  C ^ {3, \alpha } $
 +
for any $  \alpha $,
 +
$  0 < \alpha < 1 $.
  
A generalized Riemannian space of bounded curvature with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220045.png" /> and satisfying conditions 1) and 2) is a Riemannian space of constant [[Riemannian curvature|Riemannian curvature]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220046.png" /> (see [[#References|[3]]]).
+
A generalized Riemannian space of bounded curvature with $  K = K  ^  \prime  $
 +
and satisfying conditions 1) and 2) is a Riemannian space of constant [[Riemannian curvature|Riemannian curvature]] $  K $(
 +
see [[#References|[3]]]).
  
Any Riemannian space of Riemannian curvature contained in between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220048.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220049.png" />) is a generalized Riemannian space of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220051.png" /> and satisfies conditions 1) and 2).
+
Any Riemannian space of Riemannian curvature contained in between $  K $
 +
and $  K  ^  \prime  $(
 +
$  K  ^  \prime  \leq  K $)  
 +
is a generalized Riemannian space of curvature $  \leq  K $
 +
and $  \geq  K  ^  \prime  $
 +
and satisfies conditions 1) and 2).
  
 
A  "space of curvature ≤ K"  is defined by the left inequality in 2), i.e. by the condition:
 
A  "space of curvature ≤ K"  is defined by the left inequality in 2), i.e. by the condition:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220052.png" />) for any sequence of triangles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220053.png" /> contracting to a point,
+
$  A  ^ {-} $)  
 +
for any sequence of triangles $  T _ {n} $
 +
contracting to a point,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220054.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\overline{\lim\limits}\;
 +
\frac{\overline \delta \; ( T _ {n} ) }{\sigma ( T _ {n}  ^ {0} ) }
 +
  \leq  K.
 +
$$
  
Another, equivalent, definition and a starting point for the study of generalized Riemannian spaces are based on the comparison between an arbitrary triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220055.png" /> and a triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220056.png" /> with sides of the same lengths in a space of constant curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220057.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220058.png" /> be the angles of such a triangle; the relative upper excess of the triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220059.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220060.png" />. Condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220061.png" />) in the definition of a space of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220062.png" /> can be replaced by the following condition:
+
Another, equivalent, definition and a starting point for the study of generalized Riemannian spaces are based on the comparison between an arbitrary triangle $  T = ABC $
 +
and a triangle $  T  ^ {k} $
 +
with sides of the same lengths in a space of constant curvature $  K $.  
 +
Let $  \alpha _ {k} , \beta _ {k} , \gamma _ {k} $
 +
be the angles of such a triangle; the relative upper excess of the triangle $  T $
 +
is defined as $  \overline \delta \; _ {k} ( T) = ( \widetilde \alpha  + \widetilde \beta  + \widetilde \gamma  ) - ( \alpha _ {k} + \beta _ {k} + \gamma _ {k} ) $.  
 +
Condition $  A  ^ {-} $)  
 +
in the definition of a space of curvature $  \leq  K $
 +
can be replaced by the following condition:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220063.png" />) any point has a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220064.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220065.png" /> for any triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220066.png" />.
+
$  A _ {1}  ^ {-} $)  
 +
any point has a neighbourhood $  R _ {k}  ^ {-} $
 +
in which $  \overline \delta \; _ {k} ( T) \leq  0 $
 +
for any triangle $  T $.
  
An even stronger property of concavity of the metric also holds. Namely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220068.png" /> be shortests starting at the same point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220069.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220070.png" /> be the angle in the triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220071.png" /> with sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220076.png" />, in a space of constant curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220077.png" />, opposite to the side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220078.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220079.png" /> (locally) the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220080.png" /> turns out to be a non-decreasing function (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220081.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220083.png" />, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220085.png" />-concave metric). Hence one obtains the following local properties:
+
An even stronger property of concavity of the metric also holds. Namely, let $  L $
 +
and $  M $
 +
be shortests starting at the same point $  O $
 +
and let $  \gamma _ {k} ( x, y) $
 +
be the angle in the triangle $  T  ^ {k} $
 +
with sides $  x = \rho ( O, X) $,
 +
$  y = \rho ( O, Y) $,  
 +
$  z = \rho ( X, Y) $,  
 +
$  X \in L $,  
 +
$  Y \in M $,  
 +
in a space of constant curvature $  K $,  
 +
opposite to the side $  z $.  
 +
In $  R _ {k}  ^ {-} $(
 +
locally) the angle $  \gamma _ {k} ( x, y) $
 +
turns out to be a non-decreasing function ( $  \gamma _ {k} ( x _ {1} , y _ {1} ) \leq  \gamma _ {k} ( x _ {2} , y _ {2} ) $
 +
for $  x _ {1} \leq  x _ {2} $,  
 +
$  y _ {1} \leq  y _ {2} $,  
 +
a $  k $-
 +
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"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220086.png" /> (so that, in particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220088.png" />);
+
I) between any two shortests starting at the same point there exists an angle and even an  "angle in the strong sense"   $ \alpha _ {C} = \lim\limits _ {x,y \rightarrow 0 }  \gamma _ {k} ( x, y) $(
 +
so that, in particular, if $  y = \textrm{ const } $,
 +
$  \lim\limits _ {x\rightarrow} 0 ( y- z)/x = \cos  \alpha _ {C} $);
  
II) for the angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220089.png" /> of a triangle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220090.png" /> and the corresponding triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220091.png" />,
+
II) for the angles $  \alpha , \beta , \gamma $
 +
of a triangle in $  R _ {k}  ^ {-} $
 +
and the corresponding triangle $  T  ^ {k} $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220092.png" /></td> </tr></table>
+
$$
 +
\alpha  \leq  \alpha _ {k} ,\  \beta  \leq  \beta _ {k} ,\  \gamma  \leq  \gamma _ {k} ;
 +
$$
  
III) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220093.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220094.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220095.png" />, then the shortests <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220096.png" /> (thus, a shortest with given end points is unique in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220097.png" />).
+
III) in $  R _ {k}  ^ {-} $,  
 +
if $  A _ {n} \rightarrow A $,  
 +
$  B _ {n} \rightarrow B $,  
 +
then the shortests $  A _ {n} B _ {n} \rightarrow AB $(
 +
thus, a shortest with given end points is unique in $  R _ {k} $).
  
Dual to spaces of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220098.png" /> are the spaces of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r08220099.png" /> subject to the condition dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200100.png" />-concavity:
+
Dual to spaces of curvature $  \leq  K $
 +
are the spaces of curvature $  \geq  K $
 +
subject to the condition dual to $  K $-
 +
concavity:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200101.png" />) each point has a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200102.png" /> in which the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200103.png" /> for two shortests <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200104.png" /> is a non-increasing function (a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200106.png" />-concave metric, cf. also [[Convex metric|Convex metric]]).
+
$  A _ {1}  ^ {+} $)  
 +
each point has a neighbourhood $  R _ {k}  ^ {+} $
 +
in which the angle $  \gamma _ {k} ( x, y) $
 +
for two shortests $  L, M $
 +
is a non-increasing function (a $  K $-
 +
concave metric, cf. also [[Convex metric|Convex metric]]).
  
Similarly to spaces of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200107.png" />, for spaces of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200108.png" /> the following (local) properties analogous to I) and II) are valid: Between two shortests there exists an angle in the strong sense; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200111.png" /> for any triangle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200112.png" />. Instead of III) the condition of non-overlapping of shortests or, which is the same, uniqueness of their extension holds: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200113.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200114.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200115.png" />, then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200116.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200117.png" />.
+
Similarly to spaces of curvature $  \leq  K $,  
 +
for spaces of curvature $  \geq  K $
 +
the following (local) properties analogous to I) and II) are valid: Between two shortests there exists an angle in the strong sense; $  \alpha \geq  \alpha _ {k} $,  
 +
$  \beta \geq  \beta _ {k} $,  
 +
$  \gamma \geq  \gamma _ {k} $
 +
for any triangle in $  R _ {k}  ^ {+} $.  
 +
Instead of III) the condition of non-overlapping of shortests or, which is the same, uniqueness of their extension holds: If $  AC \supset AB $
 +
and $  AC _ {1} \supset AB $
 +
in $  R _ {k}  ^ {+} $,  
 +
then either $  AC \supset AC _ {1} $
 +
or $  AC _ {1} \supset AC $.
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200118.png" />). Condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200119.png" />) can be replaced, similar to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200120.png" />), by the condition:
+
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 $  \underline \delta  $).  
 +
Condition $  A $)  
 +
can be replaced, similar to $  A _ {1}  ^ {-} $),  
 +
by the condition:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200121.png" />) each point has a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200122.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200124.png" /> for any triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200125.png" />.
+
$  A _ {1} $)  
 +
each point has a neighbourhood $  R _ {kk  ^  \prime  } $,  
 +
where $  \delta _ {k} ( T) \leq  0 $,  
 +
$  \delta _ {k  ^  \prime  } ( T) \geq  0 $
 +
for any triangle $  T $.
  
 
The above turns out to be equivalent to the following:
 
The above turns out to be equivalent to the following:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200126.png" />) for any quadruple of points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200127.png" /> there exists a quadruple of points with the same pairwise distances in a space of constant curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200128.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200129.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200130.png" /> depends, in general, on the chosen quadruple of points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200131.png" /> (see [[#References|[10]]]).
+
$  A _ {2} $)  
 +
for any quadruple of points in $  R _ {kk  ^  \prime  } $
 +
there exists a quadruple of points with the same pairwise distances in a space of constant curvature $  k $,  
 +
where $  K  ^  \prime  \leq  k \leq  K $
 +
and $  k $
 +
depends, in general, on the chosen quadruple of points in $  R _ {kk  ^  \prime  } $(
 +
see [[#References|[10]]]).
  
An example of a generalized Riemannian space of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200132.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200133.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200134.png" /> (from below by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200135.png" />).
+
An example of a generalized Riemannian space of curvature $  \leq  K $
 +
$  (\geq  K  ^  \prime  ) $
 +
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 $  K $(
 +
from below by $  K  ^  \prime  $).
  
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200136.png" /> in a space with an internal metric is called convex if any two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200137.png" /> can be connected by a shortest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200138.png" /> and if every such shortest lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200139.png" />.
+
A set $  V $
 +
in a space with an internal metric is called convex if any two points $  X, Y \in V $
 +
can be connected by a shortest $  XY $
 +
and if every such shortest lies in $  V $.
  
The following result [[#References|[7]]] has been established: If a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200140.png" /> with an internal metric is obtained by glueing together of two spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200141.png" /> of curvatures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200142.png" /> along convex sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200143.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200144.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200145.png" /> itself is a space of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200146.png" />. The glueing condition is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200147.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200148.png" /> and the metrics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200149.png" /> are induced by that of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200150.png" />.
+
The following result [[#References|[7]]] has been established: If a space $  R $
 +
with an internal metric is obtained by glueing together of two spaces $  R  ^  \prime  , R  ^ {\prime\prime} $
 +
of curvatures $  \leq  K $
 +
along convex sets $  V  ^  \prime  \subset  R  ^  \prime  $
 +
and $  V  ^ {\prime\prime} \subset  R  ^ {\prime\prime} $,  
 +
then $  R $
 +
itself is a space of curvature $  \leq  K $.  
 +
The glueing condition is that $  R = R  ^  \prime  \cup R  ^ {\prime\prime} $,  
 +
$  V  ^  \prime  = V  ^ {\prime\prime} = R  ^  \prime  \cup R  ^ {\prime\prime} $
 +
and the metrics of $  R  ^  \prime  , R  ^ {\prime\prime} $
 +
are induced by that of the space $  R $.
  
By definition, two curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200151.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200152.png" /> starting at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200153.png" /> have the same direction at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200154.png" /> if the upper angle between them is equal to zero (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200155.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200156.png" /> is said to have a definite oriented direction at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200157.png" />). A direction at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200158.png" /> is defined as a class of curves having the same direction at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200159.png" />. The directions at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200160.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200161.png" />.
+
By definition, two curves $  L $,  
 +
$  M $
 +
starting at a point $  O $
 +
have the same direction at $  O $
 +
if the upper angle between them is equal to zero (if $  L = M $,  
 +
$  L $
 +
is said to have a definite oriented direction at $  O $).  
 +
A direction at the point $  O $
 +
is defined as a class of curves having the same direction at $  O $.  
 +
The directions at the point $  O $
 +
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 $  O $.
  
The following has been proved [[#References|[5]]]: If the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200162.png" /> lies in a neighbourhood of a space of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200163.png" /> homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200164.png" />, then the space of directions at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200165.png" /> has curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200166.png" />. In the general case it is not homeomorphic to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200167.png" />-dimensional sphere.
+
The following has been proved [[#References|[5]]]: If the point $  O $
 +
lies in a neighbourhood of a space of curvature $  \leq  K $
 +
homeomorphic to $  E  ^ {n} $,  
 +
then the space of directions at the point $  O $
 +
has curvature $  \leq  1 $.  
 +
In the general case it is not homeomorphic to the $  ( n- 1) $-
 +
dimensional sphere.
  
In the two-dimensional case, the theory of manifolds of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200168.png" /> is included as a special case in the theory of manifolds of bounded curvature (see [[Two-dimensional manifold of bounded curvature|Two-dimensional manifold of bounded curvature]]). An example of a two-dimensional manifold of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200169.png" /> is a ruled surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200170.png" /> provided with an internal metric, i.e. the surface formed by the interior parts of shortests whose ends cut out two rectifiable curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200171.png" />. If the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200172.png" /> degenerates to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200173.png" />, the surface is called the cone of shortests spanned from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200174.png" /> over the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200175.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200176.png" /> is a triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200177.png" />, then such a cone is called a surface triangle (see [[#References|[3]]]).
+
In the two-dimensional case, the theory of manifolds of curvature $  \leq  K $
 +
is included as a special case in the theory of manifolds of bounded curvature (see [[Two-dimensional manifold of bounded curvature|Two-dimensional manifold of bounded curvature]]). An example of a two-dimensional manifold of curvature $  \leq  K $
 +
is a ruled surface in $  R _ {k} $
 +
provided with an internal metric, i.e. the surface formed by the interior parts of shortests whose ends cut out two rectifiable curves $  L _ {1} , L _ {2} $.  
 +
If the curve $  L _ {2} $
 +
degenerates to a point $  O $,  
 +
the surface is called the cone of shortests spanned from the point $  O $
 +
over the curve $  L _ {1} $.  
 +
If $  L _ {1} $
 +
is a triangle $  OAB $,  
 +
then such a cone is called a surface triangle (see [[#References|[3]]]).
  
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200178.png" /> of metric spaces is called non-stretching if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200179.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200180.png" />. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200181.png" /> of a closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200182.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200183.png" /> onto a closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200184.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200185.png" /> is called length-preserving if the lengths of corresponding arcs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200186.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200187.png" /> coincide under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200188.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200189.png" /> be a convex domain in a space of constant curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200190.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200191.png" /> be the boundary contour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200192.png" />. The domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200193.png" /> is said to majorize a closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200194.png" /> in a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200195.png" /> if there exists a non-stretching mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200196.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200197.png" /> that is length-preserving from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200198.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200199.png" />. The mapping itself is called majorizing.
+
A mapping $  \phi : M _ {1} \rightarrow M _ {2} $
 +
of metric spaces is called non-stretching if $  \rho _ {M _ {1}  } ( X, Y) \geq  \rho _ {M _ {2}  } ( \phi ( X), \phi ( Y)) $
 +
for any $  X, Y \in M _ {1} $.  
 +
A mapping $  \phi : \Gamma _ {1} \rightarrow \Gamma _ {2} $
 +
of a closed curve $  \Gamma _ {1} $
 +
in $  M _ {1} $
 +
onto a closed curve $  \Gamma _ {2} $
 +
in $  M _ {2} $
 +
is called length-preserving if the lengths of corresponding arcs of $  \Gamma _ {1} $
 +
and $  \Gamma _ {2} $
 +
coincide under $  \phi $.  
 +
Let $  V $
 +
be a convex domain in a space of constant curvature $  K $
 +
and $  L $
 +
be the boundary contour of $  V $.  
 +
The domain $  V $
 +
is said to majorize a closed curve $  \Gamma $
 +
in a metric space $  M $
 +
if there exists a non-stretching mapping from $  V $
 +
into $  M $
 +
that is length-preserving from $  L $
 +
to $  \Gamma $.  
 +
The mapping itself is called majorizing.
  
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 $  R _ {k} $
 +
be a convex space with an internal metric; let $  C $
 +
be the cone of shortests spanned over a closed rectifiable curve $  \Gamma $
 +
in $  R _ {k} $
 +
from a point $  O \in \Gamma $,  
 +
and, moreover, let, if $  K > 0 $,  
 +
the length $  l $
 +
of $  \Gamma $
 +
be less than $  2 \pi / \sqrt K $.  
 +
Then in a space of constant curvature $  K $
 +
there exists a convex domain $  V $
 +
majorizing $  \Gamma $
 +
and such that $  \phi ( V) = C $
 +
for the corresponding majorizing mapping $  \phi $.  
 +
This property is characteristic for spaces of curvature $  \leq  K $.  
 +
The existence of a length-preserving non-stretching mapping of the contour $  L $
 +
of $  V $
 +
onto $  \Gamma $
 +
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 $  f $
 +
from a disc $  B $
 +
into a metric space $  M $
 +
is called a surface in $  M $.  
 +
Let $  P $
 +
be a triangulated polygon, i.e. a complex of triangles $  T _ {i} $
 +
inscribed in $  B $.  
 +
To the triangle $  T _ {i} $
 +
with vertices $  X, Y, Z $
 +
there corresponds the Euclidean triangle $  T _ {i}  ^ {0} $
 +
with sides equal to the distances between points $  f( X), f( Y), f( Z) $.  
 +
Let $  S _ {0} ( P) $
 +
be the sum of the areas $  S( T _ {i}  ^ {0} ) $
 +
of all triangles $  T _ {i}  ^ {0} $;  
 +
then the area $  S( f  ) $
 +
of the surface $  f $
 +
is defined (see [[#References|[3]]]) as the [[limes inferior]] of $  S _ {0} ( f  ) $
 +
under the condition that the vertices of $  P $
 +
unboundedly contract in $  B $:  
 +
$  S( f  ) = \lim\limits  S _ {0} ( P) $.  
 +
This definition is modified as follows (see [[#References|[6]]]). Instead of $  f( X), f( Y), f( Z) $,  
 +
the vertices $  X, Y, Z $
 +
of the triangle $  T _ {i} $
 +
of the complex $  P $
 +
are put into correspondence with points $  X  ^ {P} , Y  ^ {P} , Z  ^ {P} $
 +
in $  M $,  
 +
where, moreover, to vertices of the complex $  P $
 +
correspond the same points if and only if the images of the vertices under $  f $
 +
coincide. For the area $  L( f  ) $
 +
of the surface $  f $
 +
one takes the limes inferior of the sums of the areas of the Euclidean triangles $  T _ {i}  ^ {0} $
 +
with sides equal to the distances between $  X  ^ {P} , Y  ^ {P} , Z  ^ {P} $,  
 +
under the additional assumption that $  \rho ( f( X _ {k} ), X _ {k}  ^ {P} ) $
 +
tends to zero for all vertices $  X _ {k} $
 +
of the complex $  P $.  
 +
One always has $  L( f  ) \leq  S( f  ) $.
  
<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
+
$  \alpha $)  
 +
If a sequence of surfaces $  f _ {n} $
 +
in $  R _ {k} $
 +
converges uniformly to a surface $  f $,  
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200258.png" /></td> </tr></table>
+
$$
 +
L( f  )  \leq  \lim\limits  L( f _ {n} ) \  ( \textrm{ semi"\AAh"continuity  } ).
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200259.png" />) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200260.png" /> is a non-stretching mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200261.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200262.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200263.png" /> is a surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200264.png" />, then
+
$  \beta $)  
 +
If $  p $
 +
is a non-stretching mapping from $  R _ {k} $
 +
into $  R _ {k} $
 +
and $  f $
 +
is a surface in $  R _ {k} $,  
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200265.png" /></td> </tr></table>
+
$$
 +
L( p \circ f  )  \leq  L( f  ) \  ( \textrm{ Kolmogorov\prime s  principle  } ).
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200266.png" />) The area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200267.png" /> of a surface triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200268.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200269.png" /> is not larger than the area of the corresponding triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200270.png" /> and is equal to it if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200271.png" /> is isometric to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200272.png" /> (the local property).
+
$  \delta $)  
 +
The area $  S( f  ) $
 +
of a surface triangle $  T $
 +
in $  R _ {k} $
 +
is not larger than the area of the corresponding triangle $  T  ^ {k} $
 +
and is equal to it if and only if $  T $
 +
is isometric to $  T  ^ {k} $(
 +
the local property).
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200273.png" />) Under the conditions of the existence theorem for a majorizing mapping (see above), the area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200274.png" /> is not larger than the area of the disc of perimeter 1 in a space of constant curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200275.png" /> (the isoperimetric inequality) (see [[#References|[3]]], [[#References|[6]]]).
+
$  \gamma $)  
 +
Under the conditions of the existence theorem for a majorizing mapping (see above), the area $  S( G) $
 +
is not larger than the area of the disc of perimeter 1 in a space of constant curvature $  K $(
 +
the isoperimetric inequality) (see [[#References|[3]]], [[#References|[6]]]).
  
In [[#References|[6]]] the [[Plateau problem|Plateau problem]] on the existence of a surface of minimal area spanned over a closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200276.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200277.png" /> is solved. The following has been proved. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200278.png" /> be a metrically-complete space of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200279.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200280.png" />, the diameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200281.png" />) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200282.png" /> be a closed Jordan curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200283.png" />. Then there exists a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200284.png" /> of minimal area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200285.png" /> spanned over the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200286.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200287.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200288.png" /> be closed Jordan curves in such a space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200289.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200290.png" /> be the minimal areas of the surfaces spanned over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200291.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200292.png" />, respectively. If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200293.png" /> converge under some parametrizations uniformly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200294.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200295.png" />.
+
In [[#References|[6]]] the [[Plateau problem|Plateau problem]] on the existence of a surface of minimal area spanned over a closed curve $  \Gamma $
 +
in $  R _ {k} $
 +
is solved. The following has been proved. Let $  R _ {k} $
 +
be a metrically-complete space of curvature $  \leq  K $(
 +
for $  K > 0 $,  
 +
the diameter $  d( R _ {k} ) < \pi /2 \sqrt K $)  
 +
and let $  \Gamma $
 +
be a closed Jordan curve in $  R _ {k} $.  
 +
Then there exists a surface $  f $
 +
of minimal area $  L( f  ) $
 +
spanned over the curve $  \Gamma $.  
 +
Let $  \Gamma , \Gamma _ {n} $,
 +
$  n = 1, 2 \dots $
 +
be closed Jordan curves in such a space and let $  a( \Gamma ) $,  
 +
$  a( \Gamma _ {n} ) $
 +
be the minimal areas of the surfaces spanned over $  \Gamma $
 +
and $  \Gamma _ {n} $,  
 +
respectively. If the $  \Gamma _ {n} $
 +
converge under some parametrizations uniformly to $  \Gamma $,  
 +
then $  a( \Gamma ) \geq  \lim\limits  a( \Gamma _ {n} ) $.
  
 
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).
 
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).
Line 95: Line 399:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.D. Aleksandrov,  "Die innere Geometrie der konvexen Flächen" , Akademie Verlag  (1955)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.D. Aleksandrov,  "A theorem on triangles in metric space and certain applications"  ''Trudy Mat. Inst. Steklov.'' , '''38'''  (1951)  pp. 5–23  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.D. [A.D. Aleksandrov ] Alexandroff,  "Über eine Verallgemeinerung der Riemannschen Geometrie"  ''Schrift. Inst. Math. Deutsch. Akad. Wiss.'' , '''1'''  (1957)  pp. 33–84</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.G. Nikolaev,  "Space of directions at a point in a space of curvature not greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200296.png" />"  ''Sib. Math. J.'' , '''19''' :  6  (1978)  pp. 944–948  ''Sibirsk. Math. Zh.'' , '''19''' :  6  (1978)  pp. 1341–1348</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I.G. Nikolaev,  "Solution of Plateau's problem in spaces of curvature not greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200297.png" />"  ''Sib. Math. J.'' , '''20''' :  2  (1979)  pp. 246–251  ''Sibirsk. Mat. Zh.'' , '''20''' :  2  (1979)  pp. 345–353</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  Yu.G. Reshetnyak,  "To the theory of spaces with curvature not greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200298.png" />"  ''Mat. Sb.'' , '''52''' :  3  (1960)  pp. 789–798  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  Yu.G. Reshetnyak,  "Inextensible mappings in a space of curvature no greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200299.png" />"  ''Sib. Math. J.'' , '''9''' :  4  (1968)  pp. 683–689  ''Sibirsk. Mat. Zh.'' , '''9''' :  4  (1968)  pp. 918–927</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  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</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.D. Aleksandrov,  "Die innere Geometrie der konvexen Flächen" , Akademie Verlag  (1955)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.D. Aleksandrov,  "A theorem on triangles in metric space and certain applications"  ''Trudy Mat. Inst. Steklov.'' , '''38'''  (1951)  pp. 5–23  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.D. [A.D. Aleksandrov ] Alexandroff,  "Über eine Verallgemeinerung der Riemannschen Geometrie"  ''Schrift. Inst. Math. Deutsch. Akad. Wiss.'' , '''1'''  (1957)  pp. 33–84</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.G. Nikolaev,  "Space of directions at a point in a space of curvature not greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200296.png" />"  ''Sib. Math. J.'' , '''19''' :  6  (1978)  pp. 944–948  ''Sibirsk. Math. Zh.'' , '''19''' :  6  (1978)  pp. 1341–1348</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I.G. Nikolaev,  "Solution of Plateau's problem in spaces of curvature not greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200297.png" />"  ''Sib. Math. J.'' , '''20''' :  2  (1979)  pp. 246–251  ''Sibirsk. Mat. Zh.'' , '''20''' :  2  (1979)  pp. 345–353</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  Yu.G. Reshetnyak,  "To the theory of spaces with curvature not greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200298.png" />"  ''Mat. Sb.'' , '''52''' :  3  (1960)  pp. 789–798  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  Yu.G. Reshetnyak,  "Inextensible mappings in a space of curvature no greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082200/r082200299.png" />"  ''Sib. Math. J.'' , '''9''' :  4  (1968)  pp. 683–689  ''Sibirsk. Mat. Zh.'' , '''9''' :  4  (1968)  pp. 918–927</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  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</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rinow,  "Die innere Geometrie der metrischen Räume" , Springer  (1961)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Gromov,  "Structures métriques pour les variétés riemanniennes" , Cedec-Nathan  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  "Elie Cartan et les mathématiques d'aujourd'hui"  ''Astérisque''  (1985)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.V. Pogorelov,  "Intrinsic geometry of surfaces" , Amer. Math. Soc.  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Busemann,  "The geometry of geodesics" , Acad. Press  (1955)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rinow,  "Die innere Geometrie der metrischen Räume" , Springer  (1961)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Gromov,  "Structures métriques pour les variétés riemanniennes" , Cedec-Nathan  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  "Elie Cartan et les mathématiques d'aujourd'hui"  ''Astérisque''  (1985)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.V. Pogorelov,  "Intrinsic geometry of surfaces" , Amer. Math. Soc.  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Busemann,  "The geometry of geodesics" , Acad. Press  (1955)</TD></TR></table>

Latest revision as of 08:11, 6 June 2020


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 $ \pi $). 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 $ T = ABC $ is triplet of shortests curves $ AB, BC, CA $— the sides of the triangle — connecting in pairs three different points $ A, B, C $— the vertices of the triangle. The angle between curves can be defined in any metric space: Let $ L, M $ be curves starting at the same point $ O $ in a space with metric $ \rho $. One chooses points $ X \in L $, $ Y \in M $ $ ( X, Y \neq O) $ and constructs the Euclidean triangle with sides $ x = \rho ( O, X) $, $ y = \rho ( O, Y) $, $ z = \rho ( X, Y) $ and angle $ \gamma ( x, y) $ opposite to the side $ z $. One defines the upper angle between $ L $ and $ M $ as:

$$ \tag{1 } \overline \alpha \; = \overline{\lim\limits}\; _ {x,y \rightarrow 0 } \gamma ( x, y). $$

The upper angles of the triangle are the upper angles $ \widetilde \alpha , \widetilde \beta , \widetilde \gamma $ between its sides at the vertices $ A, B, C $ and the excess of the triangle is $ \overline \delta \; ( T) = \widetilde \alpha + \widetilde \beta + \widetilde \gamma - \pi $.

A generalized Riemannian space of bounded curvature ( $ \leq K $ and $ \geq K ^ \prime $) is defined by the following condition:

$ A $) for any sequence of triangles $ T _ {n} $ contracting to a point,

$$ \tag{2 } K \geq \overline{\lim\limits}\; \frac{\overline \delta \; ( T _ {n} ) }{\sigma ( T _ {n} ^ {0} ) } \geq fnnme \underline{lim} \frac{\overline \delta \; ( T _ {n} ) }{\sigma ( T _ {n} ^ {0} ) } \geq K ^ \prime , $$

where $ \sigma ( T _ {n} ^ {0} ) $ is the area of the Euclidean triangle with the same sides as $ T _ {n} $( if $ \sigma ( T _ {n} ^ {0} ) = 0 $, then $ \overline \delta \; ( T _ {n} ) = 0 $). 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 $ U $ such that any shortest $ XY $, where $ X, Y \in U $, 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 $ x ^ {i} $ so that the metric will be given by a line element $ ds ^ {2} = g _ {ij} dx ^ {i} dx ^ {j} $ with coefficients $ g _ {ij} \in W _ {q} ^ {2} \cap C ^ {1, \alpha } $, $ 0 < \alpha < 1 $. Thus, a parallel displacement is given (with continuous $ \Gamma _ {jk} ^ {i} $) and, almost everywhere, a curvature tensor (see [9]).

Moreover, it has been proved [9] that the coordinates $ x ^ {i} $ can be taken harmonic, i.e. satisfying the equalities $ \sum _ {ij} g ^ {ij} \Gamma _ {ij} ^ {l} = 0 $. Harmonic coordinate systems form an atlas of class $ C ^ {3, \alpha } $ for any $ \alpha $, $ 0 < \alpha < 1 $.

A generalized Riemannian space of bounded curvature with $ K = K ^ \prime $ and satisfying conditions 1) and 2) is a Riemannian space of constant Riemannian curvature $ K $( see [3]).

Any Riemannian space of Riemannian curvature contained in between $ K $ and $ K ^ \prime $( $ K ^ \prime \leq K $) is a generalized Riemannian space of curvature $ \leq K $ and $ \geq K ^ \prime $ and satisfies conditions 1) and 2).

A "space of curvature ≤ K" is defined by the left inequality in 2), i.e. by the condition:

$ A ^ {-} $) for any sequence of triangles $ T _ {n} $ contracting to a point,

$$ \tag{3 } \overline{\lim\limits}\; \frac{\overline \delta \; ( T _ {n} ) }{\sigma ( T _ {n} ^ {0} ) } \leq K. $$

Another, equivalent, definition and a starting point for the study of generalized Riemannian spaces are based on the comparison between an arbitrary triangle $ T = ABC $ and a triangle $ T ^ {k} $ with sides of the same lengths in a space of constant curvature $ K $. Let $ \alpha _ {k} , \beta _ {k} , \gamma _ {k} $ be the angles of such a triangle; the relative upper excess of the triangle $ T $ is defined as $ \overline \delta \; _ {k} ( T) = ( \widetilde \alpha + \widetilde \beta + \widetilde \gamma ) - ( \alpha _ {k} + \beta _ {k} + \gamma _ {k} ) $. Condition $ A ^ {-} $) in the definition of a space of curvature $ \leq K $ can be replaced by the following condition:

$ A _ {1} ^ {-} $) any point has a neighbourhood $ R _ {k} ^ {-} $ in which $ \overline \delta \; _ {k} ( T) \leq 0 $ for any triangle $ T $.

An even stronger property of concavity of the metric also holds. Namely, let $ L $ and $ M $ be shortests starting at the same point $ O $ and let $ \gamma _ {k} ( x, y) $ be the angle in the triangle $ T ^ {k} $ with sides $ x = \rho ( O, X) $, $ y = \rho ( O, Y) $, $ z = \rho ( X, Y) $, $ X \in L $, $ Y \in M $, in a space of constant curvature $ K $, opposite to the side $ z $. In $ R _ {k} ^ {-} $( locally) the angle $ \gamma _ {k} ( x, y) $ turns out to be a non-decreasing function ( $ \gamma _ {k} ( x _ {1} , y _ {1} ) \leq \gamma _ {k} ( x _ {2} , y _ {2} ) $ for $ x _ {1} \leq x _ {2} $, $ y _ {1} \leq y _ {2} $, a $ k $- 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" $ \alpha _ {C} = \lim\limits _ {x,y \rightarrow 0 } \gamma _ {k} ( x, y) $( so that, in particular, if $ y = \textrm{ const } $, $ \lim\limits _ {x\rightarrow} 0 ( y- z)/x = \cos \alpha _ {C} $);

II) for the angles $ \alpha , \beta , \gamma $ of a triangle in $ R _ {k} ^ {-} $ and the corresponding triangle $ T ^ {k} $,

$$ \alpha \leq \alpha _ {k} ,\ \beta \leq \beta _ {k} ,\ \gamma \leq \gamma _ {k} ; $$

III) in $ R _ {k} ^ {-} $, if $ A _ {n} \rightarrow A $, $ B _ {n} \rightarrow B $, then the shortests $ A _ {n} B _ {n} \rightarrow AB $( thus, a shortest with given end points is unique in $ R _ {k} $).

Dual to spaces of curvature $ \leq K $ are the spaces of curvature $ \geq K $ subject to the condition dual to $ K $- concavity:

$ A _ {1} ^ {+} $) each point has a neighbourhood $ R _ {k} ^ {+} $ in which the angle $ \gamma _ {k} ( x, y) $ for two shortests $ L, M $ is a non-increasing function (a $ K $- concave metric, cf. also Convex metric).

Similarly to spaces of curvature $ \leq K $, for spaces of curvature $ \geq K $ the following (local) properties analogous to I) and II) are valid: Between two shortests there exists an angle in the strong sense; $ \alpha \geq \alpha _ {k} $, $ \beta \geq \beta _ {k} $, $ \gamma \geq \gamma _ {k} $ for any triangle in $ R _ {k} ^ {+} $. Instead of III) the condition of non-overlapping of shortests or, which is the same, uniqueness of their extension holds: If $ AC \supset AB $ and $ AC _ {1} \supset AB $ in $ R _ {k} ^ {+} $, then either $ AC \supset AC _ {1} $ or $ AC _ {1} \supset AC $.

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 $ \underline \delta $). Condition $ A $) can be replaced, similar to $ A _ {1} ^ {-} $), by the condition:

$ A _ {1} $) each point has a neighbourhood $ R _ {kk ^ \prime } $, where $ \delta _ {k} ( T) \leq 0 $, $ \delta _ {k ^ \prime } ( T) \geq 0 $ for any triangle $ T $.

The above turns out to be equivalent to the following:

$ A _ {2} $) for any quadruple of points in $ R _ {kk ^ \prime } $ there exists a quadruple of points with the same pairwise distances in a space of constant curvature $ k $, where $ K ^ \prime \leq k \leq K $ and $ k $ depends, in general, on the chosen quadruple of points in $ R _ {kk ^ \prime } $( see [10]).

An example of a generalized Riemannian space of curvature $ \leq K $ $ (\geq K ^ \prime ) $ 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 $ K $( from below by $ K ^ \prime $).

A set $ V $ in a space with an internal metric is called convex if any two points $ X, Y \in V $ can be connected by a shortest $ XY $ and if every such shortest lies in $ V $.

The following result [7] has been established: If a space $ R $ with an internal metric is obtained by glueing together of two spaces $ R ^ \prime , R ^ {\prime\prime} $ of curvatures $ \leq K $ along convex sets $ V ^ \prime \subset R ^ \prime $ and $ V ^ {\prime\prime} \subset R ^ {\prime\prime} $, then $ R $ itself is a space of curvature $ \leq K $. The glueing condition is that $ R = R ^ \prime \cup R ^ {\prime\prime} $, $ V ^ \prime = V ^ {\prime\prime} = R ^ \prime \cup R ^ {\prime\prime} $ and the metrics of $ R ^ \prime , R ^ {\prime\prime} $ are induced by that of the space $ R $.

By definition, two curves $ L $, $ M $ starting at a point $ O $ have the same direction at $ O $ if the upper angle between them is equal to zero (if $ L = M $, $ L $ is said to have a definite oriented direction at $ O $). A direction at the point $ O $ is defined as a class of curves having the same direction at $ O $. The directions at the point $ O $ 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 $ O $.

The following has been proved [5]: If the point $ O $ lies in a neighbourhood of a space of curvature $ \leq K $ homeomorphic to $ E ^ {n} $, then the space of directions at the point $ O $ has curvature $ \leq 1 $. In the general case it is not homeomorphic to the $ ( n- 1) $- dimensional sphere.

In the two-dimensional case, the theory of manifolds of curvature $ \leq K $ 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 $ \leq K $ is a ruled surface in $ R _ {k} $ provided with an internal metric, i.e. the surface formed by the interior parts of shortests whose ends cut out two rectifiable curves $ L _ {1} , L _ {2} $. If the curve $ L _ {2} $ degenerates to a point $ O $, the surface is called the cone of shortests spanned from the point $ O $ over the curve $ L _ {1} $. If $ L _ {1} $ is a triangle $ OAB $, then such a cone is called a surface triangle (see [3]).

A mapping $ \phi : M _ {1} \rightarrow M _ {2} $ of metric spaces is called non-stretching if $ \rho _ {M _ {1} } ( X, Y) \geq \rho _ {M _ {2} } ( \phi ( X), \phi ( Y)) $ for any $ X, Y \in M _ {1} $. A mapping $ \phi : \Gamma _ {1} \rightarrow \Gamma _ {2} $ of a closed curve $ \Gamma _ {1} $ in $ M _ {1} $ onto a closed curve $ \Gamma _ {2} $ in $ M _ {2} $ is called length-preserving if the lengths of corresponding arcs of $ \Gamma _ {1} $ and $ \Gamma _ {2} $ coincide under $ \phi $. Let $ V $ be a convex domain in a space of constant curvature $ K $ and $ L $ be the boundary contour of $ V $. The domain $ V $ is said to majorize a closed curve $ \Gamma $ in a metric space $ M $ if there exists a non-stretching mapping from $ V $ into $ M $ that is length-preserving from $ L $ to $ \Gamma $. The mapping itself is called majorizing.

Let $ R _ {k} $ be a convex space with an internal metric; let $ C $ be the cone of shortests spanned over a closed rectifiable curve $ \Gamma $ in $ R _ {k} $ from a point $ O \in \Gamma $, and, moreover, let, if $ K > 0 $, the length $ l $ of $ \Gamma $ be less than $ 2 \pi / \sqrt K $. Then in a space of constant curvature $ K $ there exists a convex domain $ V $ majorizing $ \Gamma $ and such that $ \phi ( V) = C $ for the corresponding majorizing mapping $ \phi $. This property is characteristic for spaces of curvature $ \leq K $. The existence of a length-preserving non-stretching mapping of the contour $ L $ of $ V $ onto $ \Gamma $ is already sufficient (see [8]).

A continuous mapping $ f $ from a disc $ B $ into a metric space $ M $ is called a surface in $ M $. Let $ P $ be a triangulated polygon, i.e. a complex of triangles $ T _ {i} $ inscribed in $ B $. To the triangle $ T _ {i} $ with vertices $ X, Y, Z $ there corresponds the Euclidean triangle $ T _ {i} ^ {0} $ with sides equal to the distances between points $ f( X), f( Y), f( Z) $. Let $ S _ {0} ( P) $ be the sum of the areas $ S( T _ {i} ^ {0} ) $ of all triangles $ T _ {i} ^ {0} $; then the area $ S( f ) $ of the surface $ f $ is defined (see [3]) as the limes inferior of $ S _ {0} ( f ) $ under the condition that the vertices of $ P $ unboundedly contract in $ B $: $ S( f ) = \lim\limits S _ {0} ( P) $. This definition is modified as follows (see [6]). Instead of $ f( X), f( Y), f( Z) $, the vertices $ X, Y, Z $ of the triangle $ T _ {i} $ of the complex $ P $ are put into correspondence with points $ X ^ {P} , Y ^ {P} , Z ^ {P} $ in $ M $, where, moreover, to vertices of the complex $ P $ correspond the same points if and only if the images of the vertices under $ f $ coincide. For the area $ L( f ) $ of the surface $ f $ one takes the limes inferior of the sums of the areas of the Euclidean triangles $ T _ {i} ^ {0} $ with sides equal to the distances between $ X ^ {P} , Y ^ {P} , Z ^ {P} $, under the additional assumption that $ \rho ( f( X _ {k} ), X _ {k} ^ {P} ) $ tends to zero for all vertices $ X _ {k} $ of the complex $ P $. One always has $ L( f ) \leq S( f ) $.

$ \alpha $) If a sequence of surfaces $ f _ {n} $ in $ R _ {k} $ converges uniformly to a surface $ f $, then

$$ L( f ) \leq \lim\limits L( f _ {n} ) \ ( \textrm{ semi"\AAh"continuity } ). $$

$ \beta $) If $ p $ is a non-stretching mapping from $ R _ {k} $ into $ R _ {k} $ and $ f $ is a surface in $ R _ {k} $, then

$$ L( p \circ f ) \leq L( f ) \ ( \textrm{ Kolmogorov\prime s principle } ). $$

$ \delta $) The area $ S( f ) $ of a surface triangle $ T $ in $ R _ {k} $ is not larger than the area of the corresponding triangle $ T ^ {k} $ and is equal to it if and only if $ T $ is isometric to $ T ^ {k} $( the local property).

$ \gamma $) Under the conditions of the existence theorem for a majorizing mapping (see above), the area $ S( G) $ is not larger than the area of the disc of perimeter 1 in a space of constant curvature $ K $( 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 $ \Gamma $ in $ R _ {k} $ is solved. The following has been proved. Let $ R _ {k} $ be a metrically-complete space of curvature $ \leq K $( for $ K > 0 $, the diameter $ d( R _ {k} ) < \pi /2 \sqrt K $) and let $ \Gamma $ be a closed Jordan curve in $ R _ {k} $. Then there exists a surface $ f $ of minimal area $ L( f ) $ spanned over the curve $ \Gamma $. Let $ \Gamma , \Gamma _ {n} $, $ n = 1, 2 \dots $ be closed Jordan curves in such a space and let $ a( \Gamma ) $, $ a( \Gamma _ {n} ) $ be the minimal areas of the surfaces spanned over $ \Gamma $ and $ \Gamma _ {n} $, respectively. If the $ \Gamma _ {n} $ converge under some parametrizations uniformly to $ \Gamma $, then $ a( \Gamma ) \geq \lim\limits a( \Gamma _ {n} ) $.

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 " Sib. Math. J. , 19 : 6 (1978) pp. 944–948 Sibirsk. Math. Zh. , 19 : 6 (1978) pp. 1341–1348
[6] I.G. Nikolaev, "Solution of Plateau's problem in spaces of curvature not greater than " Sib. Math. J. , 20 : 2 (1979) pp. 246–251 Sibirsk. Mat. Zh. , 20 : 2 (1979) pp. 345–353
[7] Yu.G. Reshetnyak, "To the theory of spaces with curvature not greater than " Mat. Sb. , 52 : 3 (1960) pp. 789–798 (In Russian)
[8] Yu.G. Reshetnyak, "Inextensible mappings in a space of curvature no greater than " Sib. Math. J. , 9 : 4 (1968) pp. 683–689 Sibirsk. Mat. Zh. , 9 : 4 (1968) pp. 918–927
[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)
How to Cite This Entry:
Riemannian space, generalized. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemannian_space,_generalized&oldid=19196
This article was adapted from an original article by A.D. AleksandrovV.N. Berestovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article