For smooth manifold M, does the set of fixed point for the element in $\sigma \in Diff(M)$ in $M$ become a manifold? How about a sub-manifold of M?
2026-03-25 15:41:08.1774453268
Set of fixed point for the self-diffeomorphism in manifold
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No. Consider the vector field $(x,y)\mapsto (xy,xy)$ on $\mathbb R^2$. The flow of this vector field gives a diffeomorphism of $\mathbb R^2$ whose fixed point set is the union of the coordinate-axes.