# Lie group actions on $S^n$ with some invariant hypersphere but no totally geodesic ones

Does there exist a compact connected Lie group $$G$$ acting smoothly as isometries on the standard sphere $$S^n$$ for some $$n\ge 3$$, so that no totally geodesic hypersphere $$S^{n-1}$$ is $$G$$-invariant, but there exists embedded $$G$$-invariant diffeomorphic $$S^{n-1}$$?

By $$G$$-invariant, I mean the submanifold is a union of orbits. I've searched the literature, and a direct search using some combination of the keywords didn't give anything immediately relevant. My guess is that if the principal orbits are of low enough dimension, then we would have enough degree of freedom to produce an invariant diffeomorphic hypersphere. However, I don't know whether this is the case.

I'm aware that Hsiang and Lawson have classified some low cohomogeneity actions back in the 1970s in their paper Minimal Submanifolds of Low Cohomogeneity. However, a quick look at those classifications doesn't seem to produce an example.

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• Did you mean "there exists embedded $G$-invariant diffeomorphic $S^{n - 1}$"at the end of the first paragraph? – user44191 Nov 9 at 3:22
• @user44191 Yes, of course. Thank you. – Zhenhua Liu Nov 9 at 3:27