Let $\pi: M\to B$ be a fiber bundle of smooth manifolds with $B$ connected and each fiber of $\pi$ is a compact manifold. Let $G$ be a compact Lie group acting smoothly on $M$ such that
$\pi(g\cdot m)=\pi(m)$. It is clear that $G$ acts smoothly on each fiber $M\_b$ for $b\in B$.


Noe fix a $g\in G$. For each $b\in B$ we consider the fixed point submanifold $M\_b^g\subset M\_b$.


My question is: when $b$ varies, does the diffeomorphic type of $M\_b^g$ unchanged?



Does a compact Lie group action on a family of compact manifolds have diffeomorphic fixed point submanifolds?
I think the answer is yes.


Since $G$ is compact, there is a $G$-invariant Riemannian metric on $M$ (by averaging any metric). The orthogonal distribution to the fiber for this metric is a $G$-invariant Ehresmann connection, and the parallel transport for this connection thus commutes with the $G$-action.


This shows that the action of $G$ on any two fibers are conjugate. In particular, the fixed loci are diffeomorphic.


