Isometric Spherical Space Forms

Question

I have seen the following "theorem" stated in any places, but haven't been able to find a proof of it:

Theorem: Two spherical space forms $S^{2n-1}/G_1$ and $S^{2n-1}/G_2$ are isometric iff $G_1$ and $G_2$ are orthogonal in O(2n).

Can anyone please share a proof of this?

Thanks.

Answer 1

I think the word you want is conjugate, not "orthogonal." (I don't know what it means for subgroups of $O(2n)$ to be "orthogonal.")

A more general version of the theorem you're looking for is proved, for example, in Joe Wolf's Spaces of Constant Curvature:

Lemma 2.5.6: Let $P\colon L\to M$ and $Q\colon L\to N$ be universal pseudo-Riemannian coverings. Let $\Gamma$ and $\Delta$ be the respective groups of deck transformations. Then $M$ and $N$ are isometric if and only if $\Gamma$ and $\Delta$ are conjugate subgroups of the group of all isometries of $L$.