If you need some Lie groups and algebras the book by Kirilov "An Introduction to Lie Groops and Lie Algebras" is nice for applications to geometry the best is Helgason's "Differential Geometry - Lie Groups and Symmetric Spaces".įOR TONS OF SOLVED PROBLEMS ON DIFFERENTIAL GEOMETRY the best book by far is the recent volume by Gadea/Muñoz - "Analysis and Algebra on Differentiable Manifolds: a workbook for students and teachers". A nice introduction for Symplectic Geometry is Cannas da Silva "Lectures on Symplectic Geometry" or Berndt's "An Introduction to Symplectic Geometry". For Riemannian Geometry I would recommend Jost's "Riemannian Geometry and Geometric Analysis" and Petersen's "Riemannian Geometry". In particular, Nicolaescu's is my favorite. Both are deep, readable, thorough and cover a lot of topics with a very modern style and notation. Lee "Manifolds and Differential Geometry" and Liviu Nicolaescu's "Geometry of Manifolds". Other nice classic texts are Kreyszig "Differential Geometry" and Struik's "Lectures on Classical Differential Geometry".įor modern differential geometry I cannot stress enough to study carefully the books of Jeffrey M. To really understand the classic and intuitive motivations for modern differential geometry you should master curves and surfaces from books like Toponogov's "Differential Geometry of Curves and Surfaces" and make the transition with Kühnel's "Differential Geometry - Curves, Surfaces, Manifolds". Graduate level standard references are Hatcher's "Algebraic Topology" and Bredon's "Topology and Geometry", tom Dieck's "Algebraic Topology" along with Bott/Tu "Differential Forms in Algebraic Topology." To start Algebraic Topology these two are of great help: Croom's "Basic Concepts of Algebraic Topology" and Sato/Hudson "Algebraic Topology an intuitive approach". Although it is always nice to have a working knowledge of general point set topology which you can quickly learn from Jänich's "Topology" and more rigorously with Runde's "A Taste of Topology". If you want to learn Differential Topology study these in this order: Milnor's "Topology from a Differentiable Viewpoint", Jänich/Bröcker's "Introduction to Differential Topology" and Madsen's "From Calculus to Cohomology". If you want to have an overall knowledge Physics-flavored the best books are Nakahara's "Geometry, Topology and Physics" and above all: Frankel's "The Geometry of Physics" (great book, but sometimes his notation can bug you a lot compared to standards). In particular the books I recommend below for differential topology and differential geometry I hope to fill in commentaries for each title as I have the time in the future. Suggestions about important theorems and concepts to learn, and book references, will be most helpful.ĪDDITION: I have compiled what I think is a definitive collection of listmanias at Amazon for a best selection of books an references, mostly in increasing order of difficulty, in almost any branch of geometry and topology. What are the most important and basic theorems here? Are there concise books which can teach me the stuff faster than the voluminous Spivak books?Īlso finally I want to read into some algebraic geometry and Hodge/Kähler stuff. What is a connection? Which notion should I use? I want to know about parallel transport and holonomy. For a start, for differential topology, I think I must read Stokes' theorem and de Rham theorem with complete proofs.ĭifferential geometry is a bit more difficult. Towards this purpose I want to know what are the most important basic theorems in differential geometry and differential topology. I must teach myself all the stuff by reading books. Unfortunately I cannot attend a course right now. I have decided to fix this lacuna once for all. I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology.
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