I am currently reading Spivak Calculus on Manifolds and Munkres Analysis on Manifolds. I am looking for a more advanced text, especially on vector fields as they relate to the great conserved fields of physics. Thanks for your help.
This may not move in the directions of your greatest interests, but it is one option. (Not my field so I cannot assess comparable books.)
Eschrig, Helmut. Topology and Geometry in Physics. Vol. 822. Springer Science & Business Media, 2011. Springer link. Review by Jan Jerzy Sławianowski. Publisher summary: "Written as a set of largely self-contained extensive lectures, the book gives an introduction to topological concepts in gauge theories, BRST quantization, chiral anomalies, supersymmetric solitons and noncommutative geometry."
Note: Not to be confused with Topology and Geometry for Physicists by Charles Nash and Siddhartha Sen.