I am a senior undergraduate math and physics student applying to graduate school in math for this upcoming fall. I have had classes in: Abstract Algebra (Rings, Fields, and Groups), Point-Set Topology, Analysis, and advanced courses in differential equations . I currently do research in symmetry reduction methods in Non-linear Partial Differential Equations. However, this seems like a very limited field, so my research adviser suggested that I look into Differential Geometry and related fields for graduate study.
I am planning on doing an independent study in differential geometry and closely related topics in physics with a professor at my university, because my university does not offer a differential geometry course. He recommended The Geometry of Physics by Ted Frankel. I think this is mainly because he studied under Dr. Frankel some years ago. My question is: Will this prepare me for Introductory courses in Differential Geometry in graduate school? If not any suggestions for an introductory text?
Frankel looks like a good choice. I'm a third year grad student in differential geometry and did my undergrad in math and physics, too. Sitting here thumbing through Frankel, it covers a lot of ground and does so in the modern language of differential geometry and algebraic topology. He emphasizes definitions and computations rather than proofs. The omission of most proofs allows him to cover a lot of definitions and concepts in one book. That said, it's a good fit for your situation because of the vast exposure to the subject. In grad school you'll have to pick at all the details and prove everything. That's slow going. I think that's nice pedagogy: big picture first, clean up the details later. Frankel also has several exercises. I don't know how good they are, but it's good to have something you can sink your teeth into.