Introduction to Differential Geometry with Tensor Applications

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Dipankar De

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Library of Congress Cataloging-in-Publication Data

ISBN 9781119795629

Cover image: Geometry images provided by De

Background image: Abstract Spiral, Fabian Schmidt | Dreamstime.com

Cover design by Kris Hackerott

Set in size of 11pt and Minion Pro by Manila Typesetting Company, Makati, Philippines

Printed in the USA

10 9 8 7 6 5 4 3 2 1

This book is dedicated to the late Dr. G. Suseendran whose inspiration made this book possible .

Differential Geometry is the study of geometric properties of curves and surfaces and their higher dimensional analogues using the methods of tensor calculus. It has a long and rich history and, in addition to its intrinsic mathematical value and important connections with various other branches of mathematics, it has many applications in various physical sciences. Differential Geometry is a vast subject. A comprehensive introduction would require prerequisites in several related subjects. In this elementary introductory course, we discuss tensor calculus and its applications in details and many of the basic concepts of Differential Geometry in the simpler context of curves and surfaces in ordinary 3-dimensional Euclidean spaces. Our aim is to build both solid mathematical understanding of the fundamental notions of Differential Geometry and sufficient visual and geometric intuition of the subject.