Glueing method

in the theory of surfaces

A method for constructing surfaces isometric to a given one. The glueing method has applications in proving the realizability of abstractly-defined convex metrics, in questions of bending convex surfaces, and in quantitative estimates of bendability. Its strength lies in the fact that it works in situations where differential equations are powerless.

Aleksandrov's theorem: Let $G_1,\ldots,G_n$ be closed domains in a manifold with an intrinsic metric of positive curvature, bounded by a finite number of curves with bounded rotational variation. Let $G$ be the manifold consisting of the domains $G_i$ with their boundaries identified in such a way that:

1) the identified segments of the boundaries of $G_i$ and $G_j$ have equal length;

2) the sum of the rotations of the identified segments of the boundaries of $G_i$ and $G_j$ from the sides of these domains is non-negative;

3) the sum of the angles made by the sectors at the identified points of the domains $G_k$ with the sides of these domains does not exceed $2\pi$.

Then $G$ has an intrinsic metric of positive curvature that coincides with the metrics of the domains in neighbourhoods of corresponding points.

References

[1]	A.D. Aleksandrov, V.A. Zalgaller, "Two-dimensional manifolds of bounded curvature" Proc. Steklov Inst. Math. , 76 (1962) Trudy Mat. Inst. Steklov. , 76 (1962)
[2]	A.V. Pogorelov, "Die Verbiegung konvexer Flächen" , Akademie Verlag (1957) (Translated from Russian)