# Does a compact Lie group action on a family of compact manifolds have diffeomorphic fixed point submanifolds?

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?

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.