Jeffrey M. Lee: "Manifolds and Differential Geometry", Graduate Studies in Mathematics, Volume 107, AMS, 2009.
Syllabus (keywords):
Part 1:
- smooth manifolds
- smooth maps
- paracompactness and partitions of unity
- tangent spaces and tangent maps
- immersions and submersions
- submanifolds
- regular and critical points and values
- Sard's theorem (proved only in the easiest case)
- transversality
- Whitney embedding theorem (weak version for compact manifolds)
Part 2:
- tangent bundles
- vector fields and Lie brackets
- flows of vector fields
- commuting flows
- Lie groups
- Lie algebra of a Lie group
- adjoint representations
Part 3:
- alternating tensors on a finite-dimensional real vector space
- wedge products and the exterior algebra
- differential forms
- the exterior derivative
- manifolds with boundary
- orientations
- Stokes' theorem