it was believed that the flat Euclidean geometry is the geometry of physical space (regarded by Immanuel Kant as being necessarily true as an « a priori synthetic » proposition) until Einstein’s great discovery that space-time, though locally flat, is in fact curve
From "Non Local Aspects of Quantum Phases" by J. ANANDAN, also noticing:
the electromagnetic field strength of a magnetic monopole belongs to a Chern class that is an element of the second de Rham cohomology group.
More generally I'd say that a differential structure with a tangent bundle is almost always assumed in physics (both classical and quantum, and e.g. including here even Penrose spinor bundles) and I can hardly imagine a generalization to other bundles than those ones.