Books: Diff Geo & CM

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