Differential Geometry of Manifolds: (Textbooks in Mathematics)

From the coauthor of Differential Geometry of Curves and Surfaces, this companion book presents the extension of differential geometry from curves and surfaces to manifolds in general. It provides a broad introduction to the field of differentiable and Riemannian manifolds, tying together the classical and modern formulations. The three appendices provide background information on point set topology, calculus of variations, and multilinear algebra--topics that may not have been covered in the prerequisite courses of multivariable calculus and linear algebra. Differential Geometry of Manifolds takes a practical approach, containing extensive exercises and focusing on applications of differential geometry in physics, including the Hamiltonian formulation of dynamics (with a view toward symplectic manifolds), the tensorial formulation of electromagnetism, some string theory, and some fundamental concepts in general relativity.

Table of Contents:
Analysis of Multivariable Functions. Coordinates, Frames, and Tensor Notation. Differentiable Manifolds. Analysis on Manifolds. Introduction to Riemannian Geometry. Applications of Manifolds to Physics. Point Set Topology. Calculus of Variations. Multilinear Algebra.

About the Author :
Stephen Lovett is an associate professor of mathematics at Wheaton College in Illinois. Lovett has also taught at Eastern Nazarene College and has taught introductory courses on differential geometry for many years. Lovett has traveled extensively and has given many talks over the past several years on differential and algebraic geometry, as well as cryptography.

Review :
Stephen Lovett's book, Differential Geometry of Manifolds, a sequel to Differential Geometry of Curves and Surfaces, which Lovett co-authored with Thomas Banchoff, looks to be the right book at the right time. ... We live in an age when borders between mathematical disciplines (and even between parts of mathematics and parts of physics) are being re-drawn --or erased altogether-- and differential geometry is a major player in all this Perestroika. Thus, the teaching of the subject to rookies should perhaps be restructured, too, at least in the sense of getting to the more avant garde stuff more quickly, and it looks like a major aim of Lovett's book is exactly that. ... I think this is going to be a very successful textbook especially for rookie graduate students (and the zealous undergraduate would-be differential geometer, of course), as well as a very popular self-study source. It is a very nice book indeed. -- Michael Berg, Professor of Methematics, Layola Marymount University, Los Angeles, CA.