I want to prove that ϕ : Sn/G1 → Sn/G2 is an isometry iff G1 and G2 are conjugate in O(n+1) (where G1 and G2 are groups of isometries acting freely on the sphere Sn ). I know that ϕ lifts to an isometry of Sn and so is in O(n+1). But what to do after that? How does this imply that G1 and G2 are conjugate in O(n+1) and vice versa?
2026-03-26 23:11:06.1774566666
Lifts of Spherical Space Forms and conjugation isometries
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