Differentiable manifolds : a theoretical physics approach /

This is a concise course resource that provides an overview of standard topics in differentiable manifold theory (e.g., differentiable maps and forms, vector and tensor fields, Lie derivatives, one-parameter subgroups of diffeomorphisms), differential geometry (e.g., Riemannian manifolds and connections, curvature), and introductory Lie group theory. Torres del Castillo (Univ. Autonoma de Puebla, Mexico) aims to demonstrate some of the links between the mathematical formalism presented and topics in theoretical physics. Thus he offers an introduction to Hamiltonian mechanics and geometric optics in the final chapter. However, title notwithstanding, this is very much a mathematics volume. Overall, the treatment is very clean and careful. In addition, the author includes many concrete examples throughout that serve to explicate the mathematical concepts; exercises of varying degrees of sophistication are also distributed throughout the text and similarly illuminate many of the ideas. Although Torres del Castillo states that only multivariable calculus, linear algebra, and differential equations are necessary prerequisites and suggests that the book is suitable for senior undergraduates, it would only be the most advanced of such students who could tackle and appreciate it; it is really better suited for graduate students. Useful for the well-prepared reader. Summing Up: Recommended. Graduate students. S. J. Colley Oberlin College