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 $ 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