The Geometry of Total Curvature on Complete Open Surfaces

Self-contained, richly illustrated account of modern differential geometry applied to integral geometry. For graduates and researchers.

Katsuhiro Shiohama (Author), Takashi Shioya (Author), Minoru Tanaka (Author)

9780521450546, Cambridge University Press

Hardback, published 13 November 2003

294 pages, 45 b/w illus.
23.6 x 16.1 x 2.1 cm, 0.535 kg

'...carefully written ... a very valuable addition to libraries.' Zentralblatt MATH

This is a self-contained account of how some modern ideas in differential geometry can be used to tackle and extend classical results in integral geometry. The authors investigate the influence of total curvature on the metric structure of complete, non-compact Riemannian 2-manifolds, though their work, much of which has never appeared in book form before, can be extended to more general spaces. Many classical results are introduced and then extended by the authors. The compactification of complete open surfaces is discussed, as are Busemann functions for rays. Open problems are provided in each chapter, and the text is richly illustrated with figures designed to help the reader understand the subject matter and get intuitive ideas about the subject. The treatment is self-contained, assuming only a basic knowledge of manifold theory, so is suitable for graduate students and non-specialists who seek an introduction to this modern area of differential geometry.

1. Riemannian geometry
2. Classical results by Cohn-Vossen and Huber
3. The ideal boundary
4. The cut loci of complete open surfaces
5. Isoperimetric inequalities
6. Mass of rays
7. Poles and cut loci of a surface of revolution
8. Behaviour of geodesics.

Subject Areas: Differential & Riemannian geometry [PBMP]