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?
  • 1
    $\begingroup$ 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. $\endgroup$ Jan 18, 2021 at 20:19
  • $\begingroup$ usually at least 3 days! $\endgroup$
    – C.F.G
    Jan 18, 2021 at 20:31
  • $\begingroup$ Oh ok thanks! I didn't know about this etiquette, I'll remember this! $\endgroup$
    – Dinisaur
    Jan 19, 2021 at 8:17


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.