Global Differential Geometry of Surfaces

ISBN-13: 9781402003189 / Angielski / Miękka / 2001 / 146 str.

A. Svec

Opis

Writing this book, I had in my mind areader trying to get some knowledge of a part of the modern differential geometry. I concentrate myself on the study of sur- faces in the Euclidean 3-space, this being the most natural object for investigation. The global differential geometry of surfaces in E3 is based on two classical results: (i) the ovaloids (i.e., closed surfaces with positive Gauss curvature) with constant Gauss or mean curvature are the spheres, (u) two isometrie ovaloids are congruent. The results presented here show vast generalizations of these facts. Up to now, there is only one book covering this area of research: the Lecture Notes 3] written in the tensor slang. In my book, I am using the machinary of E. Cartan's calculus. It should be equivalent to the tensor calculus; nevertheless, using it I get better results (but, honestly, sometimes it is too complicated). It may be said that almost all results are new and belong to myself (the exceptions being the introductory three chapters, the few classical results and results of my post- graduate student Mr. M. AFWAT who proved Theorems V.3.1, V.3.3 and VIII.2.1-6).

Spis treści

I. Multilinear algebra.- II. Differentiable manifolds.- III. Methods of global differential geometry.- IV. Local differential geometry of surfaces in E3.- V. Global differential geometry of Weingarten surfaces.- VI. Global differential geometry of isometries.- VII. Surfaces in E4 and E5.- VIII. Global differential geometry of hypersurfaces in En+1.