How important is to learn differential geometry to do Physics?
Applications here have been found in relativity and spacetime theories.
Gravity is curvature of spacetime, but for the electromagnetic, electroweak, and strong forces, gauge fields and field strengths are connections and curvatures of abstract internal spaces.
When not to use forms
It is time to correct the impression I may have given that differential forms are the solution to all mathematical problems………..The formalism of differential forms and the exterior calculus is a highly structured language. This structure is both a strength and a limitation. In this language there are things we cannot say……………I must admit that in several places in this book I first had to work things out in “old tensor”.