# Isometries of fiber bundles

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?
• 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. Jan 18, 2021 at 20:19
• usually at least 3 days! Jan 18, 2021 at 20:31
• Oh ok thanks! I didn't know about this etiquette, I'll remember this! Jan 19, 2021 at 8:17