Elements of Differential Topology: 1st Edition (Hardback) book cover

Elements of Differential Topology

1st Edition

By Anant R. Shastri

CRC Press

319 pages | 43 B/W Illus.

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Derived from the author’s course on the subject, Elements of Differential Topology explores the vast and elegant theories in topology developed by Morse, Thom, Smale, Whitney, Milnor, and others. It begins with differential and integral calculus, leads you through the intricacies of manifold theory, and concludes with discussions on algebraic topology, algebraic/differential geometry, and Lie groups.

The first two chapters review differential and integral calculus of several variables and present fundamental results that are used throughout the text. The next few chapters focus on smooth manifolds as submanifolds in a Euclidean space, the algebraic machinery of differential forms necessary for studying integration on manifolds, abstract smooth manifolds, and the foundation for homotopical aspects of manifolds. The author then discusses a central theme of the book: intersection theory. He also covers Morse functions and the basics of Lie groups, which provide a rich source of examples of manifolds. Exercises are included in each chapter, with solutions and hints at the back of the book.

A sound introduction to the theory of smooth manifolds, this text ensures a smooth transition from calculus-level mathematical maturity to the level required to understand abstract manifolds and topology. It contains all standard results, such as Whitney embedding theorems and the Borsuk–Ulam theorem, as well as several equivalent definitions of the Euler characteristic.


… in Shastri’s treatment, the subject [differential forms] is developed in the larger context of the author’s stated goals, which makes for very good motivation and increased accessibility. Shastri does an excellent job with this foundational material. … It’s altogether a solid introduction to serious themes likely to persuade the reader to go deeper into the subject. Shastri’s exposition is rigorous at the same time that it evinces a light touch, and this of course makes for a very readable book. Examples abound, proofs are done in detail and include discussion along the lines of what one might hear in a good lecture presentation, and there are exercises replete with hints or solutions. Pedagogically, Elements of Differential Topology clearly gets very high marks. It is a good and useful textbook.

MAA Reviews, July 2011

Professor Shastri’s book gives an excellent point of entry to this fascinating area of mathematics by providing the basic motivation and background needed for the study of differential geometry, algebraic topology, and Lie groups. … A major strength of Professor Shastri’s book is that detailed arguments are given in places where other books leave too much for the reader to supply on his/her own. This, together with the large quantity of accessible exercises, makes this book particularly reader friendly as a stable text for an introductory course in differential topology.

—From the Foreword by F. Thomas Farrell, Binghamton, New York, USA

Table of Contents

Review of Differential Calculus

Vector Valued Functions

Directional Derivatives and Total Derivative

Linearity of the Derivative

Inverse and Implicit Function Theorems

Lagrange Multiplier Method

Differentiability on Subsets of Euclidean Spaces

Richness of Smooth Maps

Integral Calculus

Multivariable Integration

Sard’s Theorem

Exterior Algebra

Differential Forms

Exterior Differentiation

Integration on Singular Chains

Submanifolds of Euclidean Spaces

Basic Notions

Manifolds with Boundary

Tangent Space

Special Types of Smooth Maps


Homotopy and Stability

Integration on Manifolds

Orientation on Manifolds

Differential Forms on Manifolds

Integration on Manifolds

De Rham Cohomology

Abstract Manifolds

Topological Manifolds

Abstract Differentiable Manifolds

Gluing Lemma

Classification of One-Dimensional Manifolds

Tangent Space and Tangent Bundle

Tangents as Operators

Whitney Embedding Theorems


Normal Bundle and Tubular Neighborhoods

Orientation on Normal Bundle

Vector Fields and Isotopies

Patching-up Diffeomorphisms

Intersection Theory

Transverse Homotopy Theorem

Oriented Intersection Number

Degree of a Map

Nonoriented Case

Winding Number and Separation Theorem

Borsuk–Ulam Theorem

Hopf Degree Theorem

Lefschetz Theory

Some Applications

Geometry of Manifolds

Morse Functions

Morse Lemma

Operations on Manifolds

Further Geometry of Morse Functions

Classification of Compact Surfaces

Lie Groups and Lie Algebras: The Basics

Review of Some Matrix Theory

Topological Groups

Lie Groups

Lie Algebras

Canonical Coordinates

Topological Invariance

Closed Subgroups

The Adjoint Action

Existence of Lie Subgroups


Hints/Solutions to Select Exercises



Exercises appear at the end of each chapter.

About the Author

Anant R. Shastri is a professor in the Department of Mathematics at the Indian Institute of Technology, Bombay. His research interests encompass topology and algebraic geometry.

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Geometry / General