I think the standard reference is probably Spivak's 'Calculus on Manifolds' but this never really did it for me.
If you have a background in physics then some combination of Nakahara's 'Geometry, Topology and Physics' and Baez and Muniain's 'Gauge Fields, Knots and Gravity' might be good (I haven't included relativity textbooks as I assume it you have a background in GR then you have enough differential geometry).
An unusual recommendation that I think is really nice is 'Stochastic Models, Information Theory and Lie Groups' by Chirikjian. It covers a few other topics mentioned in this thread and is really nice. It's _extremely_ concrete and spells out a lot of calculations in great detail. Plus, the connection to engineering applications is much more obvious.
Chirikjian's book looks really cool! Its website says that in volume 1 "The author reviews stochastic processes and basic differential geometry in an accessible way for applied mathematicians, scientists, and engineers." And I can't tell if that means 'brief review because this is a prereq to the book' or if this is a good first take on it. Do you know which it is?
This is a little difficult for me since I learned it mostly in the sense of general relativity (which is why I said some differential geometry). For that course, I mostly used the instructor's lecture notes. However, the books for the course were:
Hartle, James B., Gravity: An Introduction to Einstein's General
Schutz, Bernard, A First Course in General Relativity
The first few chapters would be all you need, but they don't include the nice things I learned from the lecture notes like how to derive the gradient, divergence, and curl in any curvilinear coordinate system by using the Christoffel symbols.
If you have a background in physics then some combination of Nakahara's 'Geometry, Topology and Physics' and Baez and Muniain's 'Gauge Fields, Knots and Gravity' might be good (I haven't included relativity textbooks as I assume it you have a background in GR then you have enough differential geometry).
An unusual recommendation that I think is really nice is 'Stochastic Models, Information Theory and Lie Groups' by Chirikjian. It covers a few other topics mentioned in this thread and is really nice. It's _extremely_ concrete and spells out a lot of calculations in great detail. Plus, the connection to engineering applications is much more obvious.