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
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its total absolute curvature will be
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and the positive part of the total curvature of will be
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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) |
Two-dimensional manifold of bounded curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Two-dimensional_manifold_of_bounded_curvature&oldid=24584