I think a functional (or CS) perspective is an interesting approach, but I think you can lose some nice results in introductory differential geometry by followi

I think a functional (or CS) perspective is an interesting approach, but I think you can lose some nice results in introductory differential geometry by following just this course. For example, the Gauss-Bonet theorem doesn't appear to be covered, which is an incredibly beautiful result linking the geometry and topology of manifolds. For a more classical introduction to differential geometry requiring only multivariate calculus and some real analysis/point set topology, Do Carmo's "Differential Geometry of Curves and Surfaces" is a great textbook. I put together a summary (key definitions/theorems) from an undergraduate course following Do Carmo at [2]. [1]: http://www.amazon.com/Differential-Geometry-Curves-Surfaces-Manfredo/dp/0132125897 http://www.amazon.com/Differential-Geometry-Curves-Surfaces-... [2]: http://ajtulloch.github.io/PDFs/MATH3968LectureNotes.pdf http://ajtulloch.github.io/PDFs/MATH3968LectureNotes.pdf