Over the next two semesters I will be taking Algebraic Geometry courses in which we are supposed to cover almost all of Hartshorne. I have the adequate algebraic background; however, as some people contend, it is instructive to know a bit of differential geometry beforehand in order to create some intuition by relating new abstract concepts in algebraic geometry to their respective, "easier to deal with", analogues in differential geometry.
As it happens, the furthest I have gone in differential geometry was just a multivariable calculus course and a course in curves and surfaces at the level of Shiffrin's notes. I have also studied Lie groups, and in doing so have avoided as much differential geometry as I could, but I know the inevitable - definition of manifolds, submanifolds, constant rank, immersions, submersions, tangent space, but I admit I wish I was more comfortable with those.
Whenever I look into a Differential Geometry textbook - Lee, Spivak, Lang, Kobayashi (the ones so far), they are all either very long, or shorter but too advanced. And the language in differential geometry is, in my opinion, a mess. Wouldn't it be easier if we had something like a category theoretical approach as we have today in algebraic topology?
Sorry for the rant in the last paragraph, back to my original point. I am looking for a textbook in Differential Geometry that is made for someone who is more algebraically intuitioned but still wants to understand the geometric picture. Perhaps it would be:
- A book in Differential Geometry with a view toward Algebraic Geometry, or
- A book in Algebraic Geometry directed at Differential Geometry, but not so advanced that a person with my background could follow, or
- The dreaded answer, there is none and the only way to learn Differential Geometry is by cramming the classics.