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Elementary Topics in Differential Geometry

Elementary Topics in Differential Geometry - 79 edition

ISBN13: 978-0387903576

Cover of Elementary Topics in Differential Geometry 79 (ISBN 978-0387903576)
ISBN13: 978-0387903576
ISBN10: 0387903577
Cover type: Hardback
Edition/Copyright: 79
Publisher: Springer-Verlag New York
Published: 1979
International: No

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Elementary Topics in Differential Geometry - 79 edition

ISBN13: 978-0387903576

John A. Thorpe

ISBN13: 978-0387903576
ISBN10: 0387903577
Cover type: Hardback
Edition/Copyright: 79
Publisher: Springer-Verlag New York

Published: 1979
International: No
Summary

This introductory text develops the geometry of n-dimensional oriented surfaces in Rn+1. By viewing such surfaces as level sets of smooth functions, the author is able to introduce global ideas early without the need for preliminary chapters developing sophisticated machinery. The calculus of vector fields is used as the primary tool in developing the theory. Coordinate patches are introduced only after preliminary discussions of geodesics, parallel transport, curvature, and convexity. Differential forms are introduced only as needed for use in integration. The text, which draws significantly on students' prior knowledge of linear algebra, multivariate calculus, and differential equations, is designed for a one-semester course at the junior/senior level.

Author Bio

Thorpe, J. A. : City University of New York Queens College

Table of Contents

Graphs and Level Sets.
Vector Fields.
The Tangent Space.
Surfaces.
Vector Fields on Surfaces; Orientation.
The Gauss Map.
Geodesics.
Parallel Transport.
The Weingarten Map.
Curvature of Plane Curves.
Arc Length and Line Integrals.
Curvature of Surfaces.
Convex Surfaces.
Parametrized Surfaces.
Local Equivalence of Surfaces and Parametrized Surfaces.
Focal Points.
Surface Area and Volume.
Minimal Surfaces.
The Exponential Map.
Surfaces with Boundary.
The Gauss-Bonnet Theorem.
Rigid Motions and Congruence.
Isometries.
Riemannian Metrics.

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