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A metric space which is a [[Two-dimensional manifold|two-dimensional manifold]] with a metric, for which analogues of the concepts of two-dimensional Riemannian geometry such as the length and the total curvature of a curve, the area and the total Gaussian curvature of a set have been defined.
 
A metric space which is a [[Two-dimensional manifold|two-dimensional manifold]] with a metric, for which analogues of the concepts of two-dimensional Riemannian geometry such as the length and the total curvature of a curve, the area and the total Gaussian curvature of a set have been defined.
  
 
Special cases of two-dimensional manifolds of bounded curvature are two-dimensional Riemannian spaces and polyhedral surfaces in three-dimensional Euclidean space. In the general case the class of two-dimensional manifolds of bounded curvature may be regarded as the closure of the class of two-dimensional Riemannian manifolds with respect to an appropriate limit process.
 
Special cases of two-dimensional manifolds of bounded curvature are two-dimensional Riemannian spaces and polyhedral surfaces in three-dimensional Euclidean space. In the general case the class of two-dimensional manifolds of bounded curvature may be regarded as the closure of the class of two-dimensional Riemannian manifolds with respect to an appropriate limit process.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t0945401.png" /> be a two-dimensional Riemannian manifold, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t0945402.png" /> be the [[Gaussian curvature|Gaussian curvature]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t0945403.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t0945404.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t0945405.png" /> be the area of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t0945406.png" />; then the total curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t0945407.png" /> will be
+
Let $  M $
 +
be a two-dimensional Riemannian manifold, let $  K( x) $
 +
be the [[Gaussian curvature|Gaussian curvature]] of $  M $
 +
at a point $  x $
 +
and let $  \sigma ( E) $
 +
be the area of a set $  E \subset  M $;  
 +
then the total curvature of $  E \subset  M $
 +
will be
  
<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/t/t094/t094540/t0945408.png" /></td> </tr></table>
+
$$
 +
\omega ( E)  = {\int\limits \int\limits } _ { E } K ( x)  d \sigma ( x) ,
 +
$$
  
 
its total absolute curvature will be
 
its total absolute curvature will be
  
<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/t/t094/t094540/t0945409.png" /></td> </tr></table>
+
$$
 +
| \omega | ( E)  = {\int\limits \int\limits } _ { E } | K ( x) |  d \sigma ( x) ;
 +
$$
  
and the positive part of the total curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454010.png" /> will be
+
and the positive part of the total curvature of $  E $
 +
will be
  
<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/t/t094/t094540/t09454011.png" /></td> </tr></table>
+
$$
 +
\omega  ^ {+} ( E)  = {\int\limits \int\limits } _ { E } K  ^ {+} ( x)  d \sigma ( x) .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454014.png" /> are two points in a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454015.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454016.png" /> be the lower bound of the lengths of the curves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454017.png" /> which connect the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454019.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454020.png" /> is an [[Internal metric|internal metric]]; it is known as the natural metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454021.png" />.
+
where $  K  ^ {+} ( x) = \max \{ 0 , K( x) \} $.  
 +
If $  x $
 +
and $  y $
 +
are two points in a Riemannian space $  M $,  
 +
let $  \rho ( x , y ) $
 +
be the lower bound of the lengths of the curves on $  M $
 +
which connect the points $  x $
 +
and $  y $.  
 +
The function $  \rho $
 +
is an [[Internal metric|internal metric]]; it is known as the natural metric of $  M $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454022.png" /> be an arbitrary two-dimensional manifold with metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454023.png" />. One says that the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454024.png" /> is Riemannian if the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454025.png" /> with the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454026.png" /> is isometric to some two-dimensional Riemannian space with its natural metric.
+
Let $  M $
 +
be an arbitrary two-dimensional manifold with metric $  \rho $.  
 +
One says that the metric $  \rho $
 +
is Riemannian if the manifold $  M $
 +
with the metric $  \rho $
 +
is isometric to some two-dimensional Riemannian space with its natural metric.
  
A two-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454027.png" /> with a metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454028.png" /> is a two-dimensional manifold of bounded curvature if the following condition is met: There exists a sequence of Riemannian metrics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454030.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454031.png" />, such that for any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454032.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454033.png" /> uniformly (i.e. the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454034.png" /> uniformly converge to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454035.png" /> on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454036.png" />) and the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454038.png" /> is bounded; here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454039.png" /> is the total absolute curvature of the Riemannian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454040.png" />. Two-dimensional manifolds of bounded curvature can be defined axiomatically.
+
A two-dimensional manifold $  M $
 +
with a metric $  \rho $
 +
is a two-dimensional manifold of bounded curvature if the following condition is met: There exists a sequence of Riemannian metrics $  \rho _ {n} $,
 +
$  n = 1 , 2 \dots $
 +
defined on $  M $,  
 +
such that for any compact set $  A \subset  M $
 +
one has $  \rho _ {n} \rightarrow \rho $
 +
uniformly (i.e. the functions $  \rho _ {n} ( x , y) $
 +
uniformly converge to the function $  \rho ( x , y) $
 +
on the set $  A \times A $)  
 +
and the sequence $  | \omega _ {n} | ( A) $,
 +
$  n= 1 , 2 \dots $
 +
is bounded; here, $  | \omega _ {n} | $
 +
is the total absolute curvature of the Riemannian metric $  \rho _ {n} $.  
 +
Two-dimensional manifolds of bounded curvature can be defined axiomatically.
  
The sufficient conditions in the definition of a two-dimensional manifold of bounded curvature given above may be partially weakened. Namely, a two-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454041.png" /> with a metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454042.png" /> will be a two-dimensional manifold of bounded curvature if for any of its points it is possible to determine neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454044.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454045.png" />, and a sequence of Riemannian metrics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454047.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454048.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454049.png" /> uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454050.png" /> and the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454051.png" /> is bounded.
+
The sufficient conditions in the definition of a two-dimensional manifold of bounded curvature given above may be partially weakened. Namely, a two-dimensional manifold $  M $
 +
with a metric $  \rho $
 +
will be a two-dimensional manifold of bounded curvature if for any of its points it is possible to determine neighbourhoods $  U $
 +
and $  V $,  
 +
where $  V \subset  U $,  
 +
and a sequence of Riemannian metrics $  \rho _ {n} $,
 +
$  n= 1 , 2 \dots $
 +
defined on $  U $,  
 +
such that $  \rho _ {n} \rightarrow \rho $
 +
uniformly on $  V $
 +
and the sequence $  \{ \omega _ {n}  ^ {+} ( V) \} $
 +
is bounded.
  
For any two-dimensional manifold of bounded curvature there are defined totally-additive set functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454053.png" />, viz., the area and the curvature of a set, respectively. In contrast to the Riemannian case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454054.png" /> need not be absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454055.png" />. For two-dimensional manifolds of bounded curvature the concept of the rotation of a curve is also defined; it is the analogue of the concept of the total geodesic curvature of a curve.
+
For any two-dimensional manifold of bounded curvature there are defined totally-additive set functions $  \sigma ( E) $
 +
and $  \omega ( E) $,  
 +
viz., the area and the curvature of a set, respectively. In contrast to the Riemannian case, $  \omega ( E) $
 +
need not be absolutely continuous with respect to $  \sigma ( E) $.  
 +
For two-dimensional manifolds of bounded curvature the concept of the rotation of a curve is also defined; it is the analogue of the concept of the total geodesic curvature of a curve.
  
 
Any convex surface in three-dimensional Euclidean space is a two-dimensional manifold of bounded curvature. In such a case the total curvature of a set is always non-negative.
 
Any convex surface in three-dimensional Euclidean space is a two-dimensional manifold of bounded curvature. In such a case the total curvature of a set is always non-negative.
  
Two-dimensional manifolds of bounded curvature can have singularities like conical points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454056.png" /> (for such points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094540/t09454057.png" /> is non-zero), edges, borders with a cylindrical base, etc.
+
Two-dimensional manifolds of bounded curvature can have singularities like conical points $  p $(
 +
for such points $  \omega ( \{ p \} ) $
 +
is non-zero), edges, borders with a cylindrical base, etc.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.D. Aleksandrov, V.A. Zalgaller, "Two-dimensional surfaces of bounded curvature" , Moscow-Leningrad (1962) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.D. Aleksandrov, V.A. Zalgaller, "Two-dimensional surfaces of bounded curvature Part 2" ''Proc. Steklov Inst. Math.'' , '''76''' (1967) ''Trudy Mat. Inst. Steklov.'' , '''76''' (1965)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.D. Aleksandrov, V.A. Zalgaller, "Two-dimensional surfaces of bounded curvature" , Moscow-Leningrad (1962) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.D. Aleksandrov, V.A. Zalgaller, "Two-dimensional surfaces of bounded curvature Part 2" ''Proc. Steklov Inst. Math.'' , '''76''' (1967) ''Trudy Mat. Inst. Steklov.'' , '''76''' (1965)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:26, 6 June 2020


A metric space which is a two-dimensional manifold with a metric, for which analogues of the concepts of two-dimensional Riemannian geometry such as the length and the total curvature of a curve, the area and the total Gaussian curvature of a set have been defined.

Special cases of two-dimensional manifolds of bounded curvature are two-dimensional Riemannian spaces and polyhedral surfaces in three-dimensional Euclidean space. In the general case the class of two-dimensional manifolds of bounded curvature may be regarded as the closure of the class of two-dimensional Riemannian manifolds with respect to an appropriate limit process.

Let $ M $ be a two-dimensional Riemannian manifold, let $ K( x) $ be the Gaussian curvature of $ M $ at a point $ x $ and let $ \sigma ( E) $ be the area of a set $ E \subset M $; then the total curvature of $ E \subset M $ will be

$$ \omega ( E) = {\int\limits \int\limits } _ { E } K ( x) d \sigma ( x) , $$

its total absolute curvature will be

$$ | \omega | ( E) = {\int\limits \int\limits } _ { E } | K ( x) | d \sigma ( x) ; $$

and the positive part of the total curvature of $ E $ will be

$$ \omega ^ {+} ( E) = {\int\limits \int\limits } _ { E } K ^ {+} ( x) d \sigma ( x) . $$

where $ K ^ {+} ( x) = \max \{ 0 , K( x) \} $. If $ x $ and $ y $ are two points in a Riemannian space $ M $, let $ \rho ( x , y ) $ be the lower bound of the lengths of the curves on $ M $ which connect the points $ x $ and $ y $. The function $ \rho $ is an internal metric; it is known as the natural metric of $ M $.

Let $ M $ be an arbitrary two-dimensional manifold with metric $ \rho $. One says that the metric $ \rho $ is Riemannian if the manifold $ M $ with the metric $ \rho $ is isometric to some two-dimensional Riemannian space with its natural metric.

A two-dimensional manifold $ M $ with a metric $ \rho $ is a two-dimensional manifold of bounded curvature if the following condition is met: There exists a sequence of Riemannian metrics $ \rho _ {n} $, $ n = 1 , 2 \dots $ defined on $ M $, such that for any compact set $ A \subset M $ one has $ \rho _ {n} \rightarrow \rho $ uniformly (i.e. the functions $ \rho _ {n} ( x , y) $ uniformly converge to the function $ \rho ( x , y) $ on the set $ A \times A $) and the sequence $ | \omega _ {n} | ( A) $, $ n= 1 , 2 \dots $ is bounded; here, $ | \omega _ {n} | $ is the total absolute curvature of the Riemannian metric $ \rho _ {n} $. Two-dimensional manifolds of bounded curvature can be defined axiomatically.

The sufficient conditions in the definition of a two-dimensional manifold of bounded curvature given above may be partially weakened. Namely, a two-dimensional manifold $ M $ with a metric $ \rho $ will be a two-dimensional manifold of bounded curvature if for any of its points it is possible to determine neighbourhoods $ U $ and $ V $, where $ V \subset U $, and a sequence of Riemannian metrics $ \rho _ {n} $, $ n= 1 , 2 \dots $ defined on $ U $, such that $ \rho _ {n} \rightarrow \rho $ uniformly on $ V $ and the sequence $ \{ \omega _ {n} ^ {+} ( V) \} $ is bounded.

For any two-dimensional manifold of bounded curvature there are defined totally-additive set functions $ \sigma ( E) $ and $ \omega ( E) $, viz., the area and the curvature of a set, respectively. In contrast to the Riemannian case, $ \omega ( E) $ need not be absolutely continuous with respect to $ \sigma ( E) $. For two-dimensional manifolds of bounded curvature the concept of the rotation of a curve is also defined; it is the analogue of the concept of the total geodesic curvature of a curve.

Any convex surface in three-dimensional Euclidean space is a two-dimensional manifold of bounded curvature. In such a case the total curvature of a set is always non-negative.

Two-dimensional manifolds of bounded curvature can have singularities like conical points $ p $( for such points $ \omega ( \{ p \} ) $ is non-zero), edges, borders with a cylindrical base, etc.

References

[1] A.D. Aleksandrov, V.A. Zalgaller, "Two-dimensional surfaces of bounded curvature" , Moscow-Leningrad (1962) (In Russian)
[2] A.D. Aleksandrov, V.A. Zalgaller, "Two-dimensional surfaces of bounded curvature Part 2" Proc. Steklov Inst. Math. , 76 (1967) Trudy Mat. Inst. Steklov. , 76 (1965)

Comments

Instead of natural metric the terminology induced metric and intrinsic metric is also used.

References

[a1] A.D. Aleksandrov, V.A. Zalgaller, "Intrinsic geometry of surfaces" , Transl. Math. Monogr. , Amer. Math. Soc. (1967) (Translated from Russian) MR0216434 Zbl 0146.44103
[a2] W. Rinow, "Die innere Geometrie der metrischen Räume" , Springer (1961) MR0123969 Zbl 0096.16302
How to Cite This Entry:
Two-dimensional manifold of bounded curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Two-dimensional_manifold_of_bounded_curvature&oldid=24584
This article was adapted from an original article by Yu.G. Reshetnyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article