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
  • 1
    @user44191 Yes, of course. Thank you. – Zhenhua Liu Nov 9 at 3:27

See

Minimal cones and the spherical Bernstein problem, II. Wu-Yi Hsiang Inventiones mathematicae (1983) Volume: 74, page 351-370

https://link.springer.com/article/10.1007%2FBF01394241

  • If I get it right, then by their construction, there exists one totally geodesic hypersphere. Just note the representation is something direct sum the trivial representation on the last axis, so the zero slice of last coordinate axis will produce an invariant totally geodesic hypersphere, which is what I want to exclude entirely. – Zhenhua Liu Nov 9 at 18:52

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