Hale "Asymptotic behavior of dissipative systems".Guckenheimer and Holmes "Nonlinear oscillations, dynamical systems, and bifurcations of vector fields".Temam "Infinite-dimensional dynamical systems in mechanics and physics".Katok and Hasselblatt "Introduction to the modern theory of dynamical systems".Mawhin and Willem "Critical point theory and Hamiltonian systems".Rabinowitz "Minimax methods in critical point theory with applications to differential equations".Olver "Applications of Lie groups to differential equations".Helgason "Differential geometry, Lie groups, and symmetric spaces".Hawking and Ellis "The large scale structure of spacetime".O'Neill "Semi-Riemannian geometry with applications to relativity".Arnold "Mathematical methods of classical mechanics".Abraham and Marsden "Foundations of mechanics".Kobayashi and Nomizu "Foundations of differential geometry".Gromov "Metric structures for Riemannian and non-Riemannian structures".Burago, Burago, and Ivanov "A course in metric geometry".Bridson and Haefliger "Metric spaces of non-positive curvature".I've roughly grouped them by subject area: I offer that differential geometry may be a much broader field than algebraic topology, and so it is impossible to have textbooks analogous to Switzer or Whitehead.** So, although it isn't precisely an answer to your question, these are the most widely cited differential geometry textbooks according to MathSciNet. I'm wondering whether that advanced volume exists.Īny recommendations for great textbooks/monographs would be much appreciated! Where he lists some of these other topics and almost implies that they would take another volume. This was inspired by page viii of Lee's excellent book: link The only book I have found that is sort of along these lines is Nicolaescu's Lectures on the Geometry of Manifolds, but this book misses many topics. But none of these topics completely, just as Switzer does with a unifying perspective and proofs of legitimate results done at an advanced level, but really as an introduction to each of the topics (Switzer does this with K-theory, spectral sequences, cohomology operations, Spectra.). I am looking for a book that covers topics like Characteristic Classes, Index Theory, the analytic side of manifold theory, Lie groups, Hodge theory, Kahler manifolds and complex geometry, symplectic and Poisson geometry, Riemmanian Geometry and geometric analysis, and perhaps some relations to algebraic geometry and mathematical physics. They just aren't the most efficient way to learn modern differential geometry (or so I've heard). But the age of those books is showing in terms of what people are really doing today compared to what you learn from using those books. Now you might be thinking that Kobayashi/Nomizu seems natural. These are both excellent books that (theoretically) give you overviews and introduction to most of the main topics that you need for becoming a modern researcher in algebraic topology.ĭifferential Geometry seems replete with excellent introductory textbooks. They are Switzer Algebraic Topology: Homology and Homotopy and Whitehead Elements of Homotopy Theory. The general theory is illustrated and expanded using the examples of curves and surfaces.In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual graduate courses. The authors begin with the necessary tools from analysis and topology, including Sard's theorem, de Rham cohomology, calculus on manifolds, and a degree theory. This book is an introduction to modern differential geometry. Differential Geometry: Manifolds, Curves, and Surfacesīackground - Differential Equations - Differentiable Manifolds - Partitions of Unity, Densities and Curves - Critical Points - Differential Forms - Integration of Differential Forms - Degree Theory - Curves: The Local Theory - Plane Curves: The Global Theory - A Guide to the Local Theory of Surfaces in R3 - A Guide to the Global Theory of Surfaces - Bibliography - Index of Symbols and Notations - Index
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |