Differential geometry and topology are crucial for theoretical physicists, particularly in condensed matter physics, gravity, and particle physics. This text introduces differential geometry and topology concepts suitable for postgraduate students and researchers. The second edition features significant updates to cater to readers' needs and reflect advancements in the field. It includes an expanded first chapter that reviews path integral quantization and gauge theories. Chapter 2 covers mathematical concepts such as maps, vector spaces, and topology, while subsequent chapters delve into advanced topics and their applications in liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later sections unify geometry and topology, discussing fiber bundles, characteristic classes, and index theorems, with a new proof of the index theorem via supersymmetric quantum mechanics. The final chapters explore contemporary applications of geometry and topology in physics, focusing on anomalies in gauge field theories and the geometrical analysis of Polakov's bosonic string theory. This edition serves as an excellent introduction to differential geometry and topology for those in theoretical and mathematical physics.
