Riemannian geometry a beginners guideAuthor: Morgan F.
Publishers: Jones and Bartlett
Year of publication: 1993
Number of pages: 121
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A BEGINNER GUIDE
Department of Mathematics Williams College Williamstown, Massachusetts
Illustrated by James F. Bred
Department of Mechanical Engineering Massachusetts Institute of Technology
Jones and Bartlett Publishers
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Manuscript typed by Dan Robb
Copyright © 1993 by Frank Morgan
All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any informational storage and retrieval system,
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Library of Congress Cataloging-in-Publication Data
Riemannian geometry : a beginner’s guide / Frank Morgan, p. cm.
Includes bibliographical references and index.
1. Geometry, Riemannian. I. Title.
QA611.M674 1992 516.3'73—dc20
Printed in the United States of America
96 95 94 93 92 10 9 8 7 6 5 4 3 2 1
Photograph courtesy of the Morgan family; taken by the author’s grandfather, Dr. Charles Selemeyer.
This book is dedicated to my teachers—notably Fred Almgren, Clem Collins, Arthur Mattuck, Mabel , my mom, and my dad. Here as a child I got an early
geometry lesson from my dad.
CURVES IN R" 5
SURFACES IN R3 11
SURFACES IN R" 25
m-DIMENSIONAL SURFACES IN R" 31
INTRINSIC RIEMANNIAN GEOMETRY
GENERAL RELATIVITY 55
THE GAUSS-BONNET THEOREM 65
GEODESICS AND GLOBAL GEOMETRY
GENERAL NORMS 89
SELECTED FORMULAS 101
SOLUTIONS TO SELECTED EXERCISES
SYMBOL INDEX 113
NAME INDEX 115
SUBJECT INDEX 117
The complicated formulations of Riemannian geometry present a daunting aspect to the student. This little book focuses on the central concept—curvature. It gives a naive treatment of Riemannian geometry, based on surfaces in R" rather than on abstract Riemannian manifolds.
The more sophisticated intrinsic formulas follow naturally. Later chapters treat hyperbolic geometry, general relativity, global geometry, and some current research on energy-minimizing curves and the isoperimetric problem. Proofs, when given at all, emphasize the main ideas and suppress the details that otherwise might overwhelm the student.
This book grew out of graduate courses I taught on tensor analysis at MIT in 1977 and on differential geometry at Stanford in 1987 and Princeton in 1990, and out of my own need to understand curvature better for my work. The last chapter includes research by Williams undergraduates. I want to thank my students, notably Alice Underwood; Paul Siegel, my teaching assistant for tensor analysis; and participants in a seminar at Washington and Lee led by Tim Murdoch.
Other books I have found helpful include Laugwitz’s Differential and Riemannian Geometry [L], Hicks’s Notes on Dijferential Geometry [Hi] (unfortunately out of print), Spivak’s Comprehensive Introduction to Differential Geometry [S], and Stoker’s Differential Geometry [St].
I am currently using this book and Geometric Measure Theory: A Beginner's Guide [M], both so happily edited by Klaus Peters and illustrated by Jim Bredt, as texts for an advanced, one-semester undergraduate course at Williams.
Frank. Morgan@williams. edu
The central concept of Riemannian geometry is curvature. It describes the most important geometric features of racetracks and of universes. We will begin by defining the curvature of a racetrack. Chapter 7 uses general relativity’s interpretation of mass as curvature to predict the mysterious precession of Mercury’s orbit.
The curvature k of a racetrack is defined as the rate at which the direction vector T of motion is turning, measured in radians or degrees per meter. The curvature is big on sharp curves, zero on straightaways. See Figure 1.1.
A two-dimensional surface, such as the surface of Figure 1.2, can curve different amounts in different directions, perhaps upward in some directions, downward in others, and along straight lines in between. The principal curvatures kx and k2 are the most upward (positive) and the most downward (negative), respectively. For the saddle of Figure 1.2, it appears that at the origin kx = 4 and k2=- 1. The mean curvature H = kx + k2 = — 4. The Gauss curvature G =
*1*2 = ~4-
At the south pole of the unit sphere of Figure 1.3, kx = k2 = 1, H = 2, and G — 1.
Since KX and k2 measure the amount that the surface is curving in space, they could not be measured by a bug confined to the surface. They are “extrinsic” properties. Gauss made the astonishing discovery, however, that the Gauss curvature G = kxk2 can, in prin-
2 CHAPTER 1
Figure 1.1. Curvature k is defined as the rate of change of the direction vector T.