I am not an expert in Riemannian geometry. I have been told that if $f: (M,g) \rightarrow (N,h)$ is a Riemannian submersion, then if the fibers of $f$ are totally geodesic submanifolds of $(M,g)$, then $f$ preserves the Levi-Civita connection, but that the converse does not hold. So does anybody have an example of a Riemannian submersion that does preserve the Levi-Civita connection but doesn't have totally geodesic fibers?
2026-03-27 05:41:48.1774590108
Riemannian submersion that preserves the Levi-Civita connection where the fibers are not totally geodesic submanifolds?
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