Geometry III

Dieses Buch behandelt die moderne Theorie der Flächen in euklidischen Räumen. Einzigartig ist die umfassende Darstellung, die das Zusammenspiel mit anderen mathematischen Theorien betont. Bislang wurde dieses aktive Gebiet mathematischer Forschung oft nur bruchstückhaft oder mittels isolierter Beispiele dargestellt. Der Leser sollte mit den Grundlagen der Differentialgeometrie vertraut sein, wie sie z.B. in Band 28 der EMS zu finden sind. Der Stoff ist für Studenten verständlich aufbereitet und von großem Nutzen für forschende Mathematiker. The theory of surfaces in Euclidean spaces is remarkably rich in deep results and applications, e.g. 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, above all, with the influence of properties of intrinsic metrics on the geometry of surfaces. 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. §E. R. Rozendorn considers the state of the art of the still incomplete theory of curvature in three-dimensional Euclidean space, and 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 include statements of unsolved problems.