Suppose $F\to M\stackrel{\pi}{\to} B$ is a Riemannian submersion with totally geodesic fibers, all manifolds compact. In general, unless $M=B\times F$ is a Riemannian product, the isometry groups of these manifolds is not related in any obvious way. In this direction:

**Question 1.** Give sufficient conditions so that the isometry group $Iso(B)$ of the base also acts by isometries on the total space $M$.

**Question 2.** Same for the isometry group $Iso(F)$ of the fiber.

A few remarks:

(i) In general, an isometry of the base (or of the fiber) does not extend to an isometry of the total space. Consider e.g. a Klein bottle $K$ as a $S^1$-bundle and note that any rotation of a fiber $S^1$ cannot extend to a continuous map of $K$.

(iii) By a result of Hermann (see e.g. Besse's book on Einstein mflds), since the fibers are assumeed totally geodesic, the holonomy group is a subgroup of the isometry group of the fiber.

[edit: Please disregard the following remark; see V Kapovitch's answer below.]

(ii) A possible answer for (Q1), ignoring the compactness requirement, is when $\sec_B\leq 0$ and $\pi_1(B)=1$. Note this implies $B$ diffeomorphic to the Euclidean space, hence noncompact. With these assumptions it is not hard to see that an isometry $f:B\to B$ can be lifted to an isometry $\tilde f:M\to M$ by letting $\tilde f(x)$ be the endpoint of the lift of the unique minimal geodesic from $p=\pi(x)\in B$ to $f(p)\in B$. Non-positive curvature is used to guarantee uniqueness of the minimal geodesic from $p$ to $f(p)$ and simply- connectedness is used to guarantee continuity of the lifted map. [incorrect -- see Kapovitch's answer below -- I left the text only so readers can see this previous mistake.]

Question 1: Even in the setting of principal bundles, given a continuous action on the base space, it is not always possible to lift it to the total space. Not even when the structure group of the principal bundle is a torus. In this direction there are obstructions given by Hattori and Yoshida inLifting compact group actions in fiber bundles[ Japan. J. Math. (N.S.) 2 (1976), no. 1, 13–25, [MR0461538]](mathscinet. $\endgroup$