Two-dimensional manifold of bounded curvature

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 be a two-dimensional Riemannian manifold, let be the Gaussian curvature of at a point and let be the area of a set ; then the total curvature of will be

its total absolute curvature will be

and the positive part of the total curvature of will be

where . If and are two points in a Riemannian space , let be the lower bound of the lengths of the curves on which connect the points and . The function is an internal metric; it is known as the natural metric of .

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

A two-dimensional manifold with a metric is a two-dimensional manifold of bounded curvature if the following condition is met: There exists a sequence of Riemannian metrics , defined on , such that for any compact set one has uniformly (i.e. the functions uniformly converge to the function on the set ) and the sequence , is bounded; here, is the total absolute curvature of the Riemannian metric . 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 with a metric will be a two-dimensional manifold of bounded curvature if for any of its points it is possible to determine neighbourhoods and , where , and a sequence of Riemannian metrics , defined on , such that uniformly on and the sequence is bounded.

For any two-dimensional manifold of bounded curvature there are defined totally-additive set functions and , viz., the area and the curvature of a set, respectively. In contrast to the Riemannian case, need not be absolutely continuous with respect to . 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 (for such points 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)
[a2]	W. Rinow, "Die innere Geometrie der metrischen Räume" , Springer (1961)