A volume devoted to the extremely clear and intrinsically beautiful theory of two-dimensional surfaces in Euclidean spaces. The main focus is on the connection between the theory of embedded surfaces and two-dimensional Riemannian geometry, and the influence of properties of intrinsic metrics on the geometry of surfaces.
|The original version of this article was written more than fiveyears ago with S. Z. Shefel',a profound and original mathematician who died in 1984. Sincethen the geometry of surfaces has continued to be enriched with ideas and results. This has required changes and additions, but has not influenced the character of the article, the design ofwhich originated with Shefel'. Without knowing to what extent Shefel' would have approved the changes, I should nevertheless like to dedicate this article to his memory. (Yu. D. Burago) We are trying to state the qualitative questions of the theory of surfaces in Euclidean spaces in the form in which they appear to the authors at present. This deillegalscription does not entirely correspond to the historical development of the subject. The theory of surfaces was developed in the first place mainly as the 3 theory of surfaces in three-dimensional Euclidean space E ; however, it makes sense to begin by considering surfaces F in Euclidean spaces of any dimension n~ 3. This approach enables us, in particular, to put in a new light some 3 unsolved problems of this developed (and in the case of surfaces in E fairly complete) theory, and in many cases to refer to the connections with the present stage ofdevelopment of the theory of multidimensional submanifolds. The leading question of the article is the problem of the connection between classes of metrics and classes of surfaces in En.|1 e This book deals with the modern theory of surfaces in Euclidean spaces, an active area of mathematics, which is often only treated piecemeal or through examples. The reader should be familiar with the fundamentals of differential geometry, as found, for example, in EMS 28. The material is accessible to students and researchers in mathematics.
I. The Geometry of Surfaces in Euclidean Spaces.- II. Surfaces of Negative Curvature.- III. Local Theory of Bendings of Surfaces.- Author Index.
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.
Inhaltsverzeichnis
I. The Geometry of Surfaces in Euclidean Spaces.- II. Surfaces of Negative Curvature.- III. Local Theory of Bendings of Surfaces.- Author Index.
Klappentext
A volume devoted to the extremely clear and intrinsically beautiful theory of two-dimensional surfaces in Euclidean spaces. The main focus is on the connection between the theory of embedded surfaces and two-dimensional Riemannian geometry, and the influence of properties of intrinsic metrics on the geometry of surfaces.
1 e This book deals with the modern theory of surfaces in Euclidean spaces, an active area of mathematics, which is often only treated piecemeal or through examples. The reader should be familiar with the fundamentals of differential geometry, as found, for example, in EMS 28. The material is accessible to students and researchers in mathematics.
