Ehresmann's theorem says that a proper smooth submersion is a fiber bundle. The proofs I know rely on the existence of connections locally on the base, and this is furnished by partitions of unity.


[This question](https://mathoverflow.net/q/56737/69037) gives a counterexample in the holomorphic category which is probably classic (elliptic curves), but I don't know any of the story.


I am wondering about the real analytic category. Are the fibers of a proper real analytic submersion isomorphic? If not, will it be locally trivial (in the real analytic category) when they are isomorphic, as in the linked question?



In the real analytic category, are the fibers of a proper submersion isomorphic?
There exists an analytic Riemann metric on a real-analytic manifold, which follows from embeddability of real analytic manifolds (see The Analytic Embedding of Abstract Real-Analytic Manifolds Charles B. Morrey, Jr.) - maybe can also be proved easier.


So, you can probably just take orthogonal connection.


