Geometry III: Theory of Surfaces (Encyclopaedia of Mathematical Sciences)

The theory of two-dimensional surfaces in Euclidean spaces is remarkably rich in deep results and applications, for example in the theory of non-linear partial differential equations, physics and mechanics. This theory has great clarity and intrinsic beauty, and differs in many respects from the theory of multidimensional submanifolds. A separate volume of the Encyclopaedia is therefore devoted to surfaces. It is concerned mainly with the connection between the theory of embedded surfaces and two-dimensional Riemannian geometry (and its generalizations), and, above all, with the question of the influence of properties of intrinsic metrics on the geometry of surfaces. In the first article Yu.D.Burago and S.Z.Shefel' give an extended survey of surfaces from a non-traditional viewpoint stressing the connection between classes of metrics and classes of surfaces in En. A number of conjectures are included. The article of E.R.Rozendorn considers the state of the art of the still incomplete theory of the geometry of surfaces of negative curvature in three-dimensional Euclidean space, and the article of I.Kh.Sabitov considers subtle questions of local bendability and rigidity of surfaces. These articles reflect the development of the results of N.V.Efimov and also include statements of unsolved problems.