Differential geometry plays an increasingly important role in modern theoretical physics and applied mathematics. This textbook gives an introduction to geometrical topics useful in theoretical physics and applied mathematics, covering: manifolds, tensor fields, differential forms, connections, symplectic geometry, actions of Lie groups, bundles, spinors, and so on. Written in an informal style, the author places a strong emphasis on developing the understanding of the general theory through more than 1000 simple exercises, with complete solutions or detailed hints. The book will prepare readers for studying modern treatments of Lagrangian and Hamiltonian mechanics, electromagnetism, gauge fields, relativity and gravitation. Differential Geometry and Lie Groups for Physicists is well suited for courses in physics, mathematics and engineering for advanced undergraduate or graduate students, and can also be used for active self-study. The required mathematical background knowledge does not go beyond the level of standard introductory undergraduate mathematics courses.

â€¢ Complex ideas or computations are divided into a sequence of simple and clear statements which can then be easily grasped â€¢ Much of the theory is illustrated through simple exercises (over 1000 altogether), with detailed hints â€¢ End-of-chapter summaries give important concepts, results and formulas â€¢ Uses both standard mathematical and physical terminology, building a bridge between the jargons involved

### Contents

Introduction; 1. The concept of a manifold; 2. Vector and tensor fields; 3. Mappings of tensors induced by mappings of manifolds; 4. Lie derivative; 5. Exterior algebra; 6. Differential calculus of forms; 7. Integral calculus of forms; 8. Particular cases and applications of Stoke's Theorem; 9. PoincarÃ© Lemma and cohomologies; 10. Lie Groups - basic facts; 11. Differential geometry of Lie Groups; 12. Representations of Lie Groups and Lie Algebras; 13. Actions of Lie Groups and Lie Algebras on manifolds; 14. Hamiltonian mechanics and symplectic manifolds; 15. Parallel transport and linear connection on M; 16. Field theory and the language of forms; 17. Differential geometry on TM and T*M; 18. Hamiltonian and Lagrangian equations; 19. Linear connection and the frame bundle; 20. Connection on a principal G-bundle; 21. Gauge theories and connections; 22. Spinor fields and Dirac operator; Appendices; Bibliography; Index.

### Reviews

Review of the hardback: 'All basic material that is necessary for a young scientist in the field of geometrical formulation of physical theories is included â€¦ ordered and represented in a very appropriate manner â€¦ with a great respect to the reader. â€¦ I truly believe that reading this book will bring a real pleasure to all physically inclined young mathematicians and mathematically inclined young physicists â€¦ a very good high level textbook â€¦ [I] recommend it to all young scientists being interested in finding correspondence between harmony in the physical world and harmony in geometrical structures. â€¦ well written, very well ordered and the exposition is very clear.' Journal of Geometry and Symmetry in Physics

Review of the hardback: 'the contents of this book covers a lot (if not most) of what a theoretical physicist might wish to know about differential geometry and Lie groups. particularly useful may be that the modern formalism is always related to the classical one with tensor indices still mostly used in the physics literature.' American Mathematical Society

Review of the hardback: 'â€¦ the presentation is almost colloquial and this makes reading rather pleasant. The author has made a concerted effort to give intuitive interpretations of complicated ideas such as: the Lie derivative, tensors, the Hodge star operator, Lie group representations, Hamiltonian and Lagrangian mechanics, parallel transport, connections, curvature, gauge theories, spinors and Dirac operators. This will be much appreciated by students (and even researchers, I think). â€¦ an excellent reference for geometers.' Zentralblatt MATH

Review of the hardback: 'The contents of this book cover a lot (if not most) of what a theoretical physicist might wish to know about differential geometry and Lie groups.' Mathematical Reviews Hans-Peter KÃ¼nzle

Review of the hardback: 'Marian Fecko deftly guides you through the material step-by-step, with all the rigour, but without the pain. When going through the chapters, definition by definition, proof by proof and hint by hint, you get an impression of a caring, experienced (and often quirkily funny, but never boring) tutor who really, really wants you to succeed.' Physics in Canada

Review of the hardback: 'From the point of view of presentation the book has a lot going for it. It is written in a pedagogically discursive, conversational style with numerous workable examples and exercises distributed through the text. â€¦ From the UK perspective a student undertaking a level 4 (level 5 in Scotland) or MSc course in theoretical physics would find this book well-pitched to his or her needs.' Contemporary Physics