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Inspired by comment discussions in this MO post smooth version of splitting principle we ask:
Are there two compact real analytic manifolds $M,N$ of dimension $m>n$ such that there is nonot any analytic surjection $f:M \to N$?
Are there two compact real analytic manifolds $M,N$ of dimension $m>n$ such that there is no any analytic surjection $f:M \to N$?
Are there two compact real analytic manifolds $M,N$ of dimension $m>n$ such that there is not any analytic surjection $f:M \to N$?
