Differential Forms or Tensors for Theoretical Physics Today

In summary, there are different approaches to theoretical physics, such as using tensors with indices or differential forms. Some textbooks, like Kip Thorne's "Modern Classical Physics" use tensors, while others, like Frankel's "Geometry of Physics" and Chris Isham's "Modern differential geometry for physicists" use differential forms. Mathematicians have mostly adopted the use of differential forms, but physicists may have a preference for one or the other depending on the field. It is possible to rewrite a book like Kip Thorne's using differential forms, but it may be shorter. The notation for these approaches is different and it can impact ease of understanding. In general relativity, the differential form notation is often preferred for its computational power, but the
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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.
 

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