Geometry III

The theory of two-dimensional surfaces in Euclidean spacesis remarkably rich in deep results and applications, forexample in the theory of non-linear partial differentialequations, physics and mechanics. This theory has greatclarity and intrinsic beauty, and differs in many respectsfrom the theory of multidimensional submanifolds. A separatevolume of the Encyclopaedia is therefore devoted tosurfaces. It is concerned mainly with the connection betweenthe theory of embedded surfaces and two-dimensionalRiemannian geometry (and its generalizations), and, aboveall, with the question of the influence of properties ofintrinsic metrics on the geometry of surfaces.In the first article Yu.D.Burago and S.Z.Shefel' give anextended survey of surfaces from a non-traditional viewpointstressing the connection between classes of metrics andclasses of surfaces in En. A number of conjectures areincluded.The article of E.R.Rozendorn considers the state of the artof the still incomplete theory of the geometry of surfacesof negative curvature in three-dimensional Euclidean space,and the article of I.Kh.Sabitov considers subtle questionsof local bendability and rigidity ofsurfaces.These articles reflect the development of the results ofN. V.Efimov and also include statements of unsolved problems.