How important is to learn differential geometry to do Physics?
This is a difficult question to answer. The theories of all the fundamental forces (gravity, electromagnetism, electroweak, and strong) are formulated using differential geometry, but:
1) many calculation involving these forces do not require much understanding of differential geometry;
2) most of physics is applied physics.
Much of physics does not require formal study of differential geometry. I, howver, enjoy learning about differential geometry because of the subject's intrinsic beauty, and because of its relevance to theoretical physics.
It should be pointed out that there are two flavours of differential geometry.
Classic DG originally developed for surveying and engineering applications such as Naval Architecture of hulls.
This flavour is solely concerned with curves and surfaces in Euclidian (3 dimensional) geometry.
It forms a natural extension to vector calculus in the format of grad, div curl etc.
Modern DG developed by pure mathematicians. They have extended DG to other geometries, principally Riemannian and reset in the modern parlance of linear algebra.
Applications here have been found in relativity and spacetime theories.
Of course both flavours refer to the same basic theory, but it is important to know the direction of any proposed course in DG.
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.
Yes modern DG is all about forms and spaces. But I wonder the worth of pointing this out in detail to someone just contemplating a DG course, who may not be that much further along in Physics either.
I refer to article 41 of Burke: Applied Differential Geometry : Cambridge University Press.
Although the source of many useful results, DG has always been considered esoteric. And as the comment working things out in old tensor shows has become even more so.
Separate names with a comma.