Let $F\to S\overset{\pi}{\to} B$ a Riemannian submersion with totally geodesic fibers.

Question: How much information about the isometries of $S$ we have if we know the isometries of $F$ and $B$? For example:

  • Is there some natural condition under which the following short exact sequence hold? $$1\to\operatorname{Isom}(F)\to\operatorname{Isom}(S)\to\operatorname{Isom}(B)\to 1$$
  • 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?
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    $\begingroup$ Hi Dinisaur. When you ask a question on MO that you have previously asked on MSE, you should mention this and include a link. Usually, it is suggested that one should wait a while after asking a question on MSE before posting to MO. $\endgroup$ – Michael Albanese Jan 18 at 20:19
  • $\begingroup$ usually at least 3 days! $\endgroup$ – C.F.G Jan 18 at 20:31
  • $\begingroup$ Oh ok thanks! I didn't know about this etiquette, I'll remember this! $\endgroup$ – Dinisaur Jan 19 at 8:17

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