"Comprehensive advanced introduction to manifolds, with a thorough treatment of every fundamental topic, from tensors and differential forms, to connections and curvature of fiber bundles, through Riemannian geometry."
"Specially suitable for anyone tending to global analysis. Formal and advanced but very didactic and insightful, with many examples an exercises. He manages to cover many topics in great detail. The last chapters are unique making it an ideal reading before attempting titles on Dirac operators, spin geometry and index theorems."
"GREAT highly advanced, formal and algebraic treatment of differential geometry, with an emphasis on functoriality and naturality in the differential categories. Very detailed and comprehensive results on many topics at a very general level, and with a growing web supplement pdf."
"These two volumes are the classical monographic reference on differential geometry. They cover almost any important topic, and are one of the few book references where the Newlander-Nirenberg theorem is explicitly proved (at least for the analytic case)."
"A true gem for geometric and global analysis. It treats the real and complex aspects from the beginning (review connections and curvature), fundamental development of Hodge theory and with even an introduction to Seiberg-Witten theory."
"Great brief introduction to differential operators of Dirac type on manifolds. It serves as a good continuation to Nicolaescu's final chapters, as an introduction to spin geometry, as needed to more advanced results like the Atiyah-Index theorem and Seiberg-Witten theory (which even introduces)."
"This is short classic introduces the Atiyah-Singer index theorem and its main corollaries (Hirzebruch, Chern-GauĆ-Bonnet...) using characteristic classes and developing a K-theoretic proof. Also dealing with the subsequent fixed point theorems."
"This book is a must have for any student, above all for self-studying. It is a collection of solved problems for any of the standard chapters of a typical book on differential geometry. It contains a detailed appendix full of important formulas and results to use."