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There is a theorem of Whitney showing that a smooth manifold can be endowed with a compatible real-analytic atlas (later, it was proven that this analytic structure is essentially unique).
I am curious, how much stronger structures can be put on a smooth manifold in a compatible way? Most importantly, is it possible to find an atlas such that its transition maps be elements of $GL(n)$ ($n$ being the dimension of the manifold)?
The only related thing that I have found is the concept of piecewise-linear manifold, which seems not to be what I am looking for.