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But when you define differential forms on a differentiable manifold, you by definition use inner product induced by trivialization. After that what kind of Riemannian (or Lorenzian or whatever) metric you endow on the manifold you have is up to you. And this additional structure is sometimes called PHYSICS, isn't it? (At least, I think. :P )

In Newtonian mechanics, your differentiable manifold has the simplest metric.

In Special Relativity your differential manifold has Lorenzian metric and called the Minkowski space.

In Quantum mechanics, it is still unknown, right?

As for the case involving temperature or other physical quantities, like momentum (in case of Hamiltonian mechanics), I don't know. Aren't they studying this kind of things in Dynamical System?

When you use diff mfds in physics to yield new results, since it is basically an application of a mathematical theory, I think the manifold corresponding to your physical phenamena should have a nice mathematical theory, or it will be just a new form of writing down equations. (I am not specializing on differential geometry, so I really cannot say such a thing, but I think if you don't fix an inner product of a manifold, the technique you can use to investigate it is very limited.)