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?