This textbook delves into the theory behind differentiable manifolds while exploring various physics applications along the way. Included throughout the book are a collection of exercises of varying degrees of difficulty. Differentiable Manifolds is intended for graduate students and researchers interested in a theoretical physics approach to the subject. Prerequisites include multivariable calculus, linear algebra, and differential equations and a basic knowledge of analytical mechanics.
Gerardo Torres del Castillo has published two books previously, both in Birkhauser's Progress in Mathematical Physics series: 3-D Spinors, Spin-Weighted Functions and their Applications, and Spinors in Four-Dimensional Spaces.
Preface.-1 Manifolds.- 2 Lie Derivatives.- 3 Differential Forms.- 4 Integral Manifolds.- 5 Connections .- 6. Riemannian Manifolds.- 7 Lie Groups.- 8 Hamiltonian Classical Mechanics.- References.-Index.
From the reviews:
"The purpose of this book is to present some fundamental notions of differentiable geometry of manifolds and some applications in physics. The topics developed in the book are of interest of advanced undergraduate and graduate students in mathematics and physics. The author succeeded to connect differential geometry with mechanics. The computations are clearly explained and the theory is supported by several examples. Throughout the book there is a large collection of exercises ... which help the reader to fix the obtained knowledge." (Marian Ioan Munteanu, Zentralblatt MATH, Vol. 1237, 2012)
"This book presents an introduction to differential geometry and the calculus on manifolds with a view on some of its applications in physics. ... The book is primarily oriented towards advanced undergraduate and graduate students in mathematics and physics ... . the present author has succeeded in writing a book which has its own flavor and its own emphasis, which makes it certainly a valuable addition to the literature on the subject." (Frans Cantrijn, Mathematical Reviews, Issue 2012 k)
This textbook gives a concise introduction to the theory of differentiable manifolds, focusing on their applications to differential equations, differential geometry, and Hamiltonian mechanics.
The work's first three chapters introduce the basic concepts of the theory, such as differentiable maps, tangent vectors, vector and tensor fields, differential forms, local one-parameter groups of diffeomorphisms, and Lie derivatives. These tools are subsequently employed in the study of differential equations (Chapter 4), connections (Chapter 5), Riemannian manifolds (Chapter 6), Lie groups (Chapter 7), and Hamiltonian mechanics (Chapter 8). Throughout, the book contains examples, worked out in detail, as well as exercises intended to show how the formalism is applied to actual computations and to emphasize the connections among various areas of mathematics.
Differentiable Manifolds is addressed to advanced undergraduate or beginning graduate students in mathematics or physics. Prerequisites include multivariable calculus, linear algebra, differential equations, and (for the last chapter) a basic knowledge of analytical mechanics.