# 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)