View source for Two-dimensional manifold of bounded curvature

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.

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

<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>

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>

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

<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>

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" />.

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.

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.

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.

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.

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.

====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>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.D. Aleksandrov,   V.A. Zalgaller,   "Intrinsic geometry of surfaces" , ''Transl. Math. Monogr.'' , Amer. Math. Soc.  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rinow,   "Die innere Geometrie der metrischen Räume" , Springer  (1961)</TD></TR></table>