The geometry of total curvature on complete open surfaces / Katsuhiro Shiohama, Takashi Shioya, Minoru Tanaka.

Format:

Book

Author/Creator:

Shiohama, K. (Katsuhiro), 1940- author.

Shioya, Takashi, 1963- author.

Tanaka, Minoru, 1949- author.

Series:

Cambridge tracts in mathematics ; 159.

Cambridge tracts in mathematics ; 159

Language:

English

Subjects (All):

Riemannian manifolds.

Curves on surfaces.

Global differential geometry.

Physical Description:

1 online resource (ix, 284 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 2003.

Language Note:

English

Summary:

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.

Contents:

1. Riemannian geometry

2. The classical results of Cohn-Vossen and Huber

3. The ideal boundary

4. The cut loci of complete open surfaces

5. Isoperimetric inequalities

6. Mass of rays.

7. The poles and cut loci of a surface of revolution

8. The behavior of geodesics.

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references (p. 275-280) and index.

ISBN:

1-107-12656-8

1-280-41463-4

9786610414635

0-511-20377-2

0-511-17974-X

0-511-06546-9

0-511-30668-7

0-511-54315-8

0-511-06759-3

OCLC:

57172518