Let $F\to S\overset{\pi}{\to} B$ a fiber bundle or Riemannian manifolds with totally geodesic fibers (i.e. $F_x\cong\pi^{-1}(x)\subset S$ is a totally geodesic submanifold for each $x\in B$).
Question: How much information about the isometries of $S$ we have if we know the isometries of $F$ and $B$? For example:
- Does the short exact sequence $$1\to\operatorname{Isom}(F)\to\operatorname{Isom}(S)\to\operatorname{Isom}(B)\to 1$$ ever hold? In which cases?
- Even if we don't have such sequence, do we have part of it? For example I guess we always have a map $\operatorname{Isom}(F)\to\operatorname{Isom}(S)$ just sending an isometry of $F$ to a map of $S$ that applies the same isometry to each fiber, but is this an isometry?
- Can we define a map $\operatorname{Isom}(S)\to\operatorname{Isom}(B)$? I guess that if the isometries of $S$ are fiber-preserving we might do so just composing with $\pi$, but is this an isometry? Do we have conditions under which we know that the isometries of $S$ are fiber-preserving?