I am looking at definitions of diffeomorphisms. In (Marathe, 2010) the author keeps the chapter (chap 3) on manifolds fairly general - except when defining diffeomorphims and differential maps, between manifolds. In this case they restrict their attention to real differentiable manifolds. My question is does this restriction need to be made? i.e. can you define diffeomorphims between non-real differentiable manifolds?
2026-03-26 12:33:45.1774528425
Can diffeomorphisms only be defined on real manifolds?
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There is indeed a notion of differentiable manifold over any complete field (see here). As long as you can make sense of what differentiability over a field means you can make sense of a differentiable manifold over that field. You just literally carry over the definitions for the real case.
However in some cases you are more interested in analytic manifolds and the notion of analyticity is usually stronger than differentiability. See here for a treatment of the p-adic case.