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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.

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|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|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====
<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) {{MR|0216434}} {{ZBL|0146.44103}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Rinow, "Die innere Geometrie der metrischen Räume" , Springer (1961) {{MR|0123969}} {{ZBL|0096.16302}} </TD></TR></table>