Topology and Geometry for Physics [Paperback]

A concise but self-contained introduction of the central concepts of modern topology and differential geometry on a mathematical level is given specifically with applications in physics in mind. All basic concepts are systematically provided including sketches of the proofs of most statements. Smooth finite-dimensional manifolds, tensor and exterior calculus operating on them, homotopy, (co)homology theory including Morse theory of critical points, as well as the theory of fiber bundles and Riemannian geometry, are treated. Examples from physics comprise topological charges, the topology of periodic boundary conditions for solids, gauge fields, geometric phases in quantum physics and gravitation.

This is a concise but self-contained introduction to the central concepts of modern topology and differential geometry. The book is written on a mathematical level with applications in physics in mind and all basic concepts are systematically provided.

Introduction.- Topology.- Manifolds.- Tensor Fields.- Integration, Homology and Cohomology.- Lie Groups.- Bundles and Connections.- Parallelism, Holonomy, Homotopy and (Co)homology.- Riemannian Geometry.- Compendium.

From the reviews:

It consists of nine main chapters & and the final Compendium which summarizes the volume in thirty two pages the basic geometric concepts and facts used in the book. & the author really has achieved his purpose, namely he wrote a book rigorous in a mathematical sense, nevertheless not over-loaded by the physically non-necessary proofs and mathematical details, and at the same time giving to the reader a rigorous mathematical explanation of physical phenomena, often quite recently discovered and not commonly known. (Jan Jerzy SBawianowski, Journal of Geometry and Symmetry in Physics, Vol. 25, 2012)

The author describes well the purpose of this text. It is to present to the theoretical physicist much of modern mathematics s/he may neelã2