First steps. Here are books for different areas that I sometimes recommend as the initial reading:
Intro to Diff Geometry:
- William Burke, Applied Differential Geometry, Cambridge, 1985 (436 p) [Amz, pdf]
- Loring Tu: Introduction to manifolds, Springer, 2010 [Amz, pdf]
- R.W.R. Darling: Differential forms and connections, Cambridge, 1994
- H. Flanders: Differential forms (with application to the physical science), Dover., $6
- Shiing-Shen Chern, W.H. Chen, K.S. Lan: Lectures on Differential Geometry, World Sci., (1999)
- Shlomo Sternberg: Lectures on Differential Geometry: 2nd ed., AMS, 1983, 442 pp
- R. Bishop & S. Goldberg: Tensor analysis on manifolds, Dover, $11.95 (price)
- John Lee, Introduction to Smooth Manifolds (Grad Texts in Math, Vol. 218), Springer 2012
- Shigeyuki Morita: Geometry of Differential Forms, AMS, 2001, 321 pp.
- Andrew McInerney: First Steps in Differential Geometry: Riemannian, Contact, Symplectic (Undergrad Texts in Math) Springer, 2013 [did not see, but looks promising]
- Tevian Dray: Differential Forms and the Geometry of General Relativity, CTC, 2014 [also worth checking]
- Nicolaescu, Lectures on the Geometry of Manifolds [Amz, pdf]
Geometry, topology and physics:
- Abraham, Marsden, Ratiu: Manifolds, Tensor Analysis, and Applications, (Springer)
- Theodore Frankel: The Geometry of Physics, (CRC, 2nd ed., 2003), 600 pp [Amz, djvu]
- Mikio Nakahara: Geometry, Topology, and Physics, (Inst of Phys Pub), 520pp
- Gregory L. Naber: Topology, Geometry, and Gauge Fields: Foundations, (Springer, 2nd ed., 2011), 400 pp
- Gregory L. Naber: Topology, Geometry and Gauge Fields: Interactions (Applied Math Sci,, Vol 141), Springer (2nd ed, 2012), 400 pages
- A P Balachandran , G Marmo B S Skagerstam A Stern: Classical Topology And Quantum States, 376pp,
- Charles Nash & Siddhartha Sen: Topology and Geometry for Physicists, (Dover, 2011) only $10!
- Peter Szekeres: A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry, (Cambridge, 2004) (I have not seen this one)
- Brian Hall: Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, (Springer, 2004) [Amz ]
- Louis H. Kauffman: Knots and Physics, (Knots and Everything, 3rd ed, 2001)
Classical mechanics in the modern language:
- V.I. Arnold: Mathematical Methods of Classical Mechanics, 2nd edn, Grad Text in Math 60, Springer, 1989.
- R. Abraham, J. Marsden: Foundation of Mechanics (2nd Ed.), Addisin-Wesley, 1978.
- J.E. Marsden: Lectures on Mechanics, Cambridge, 1992
- J.E. Marsden & T.S. Ratiu: Introduction to mechanics and symmetry, 2nd ed, Springer, 1999
- V.Guillemin & S. Sternberg: Symplectic techniques in Physics, Cambridge, 1984.
- Gaetano Vilasi: Hamiltonian Mechanics, World Scientific, 2001.
- G. Marmo, et al.: Dynamical Systems (a geometric approach to symmetry and reduction), Willey, 1985
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