# Isometries between spherical space forms

Let $$S^n/\Gamma_i\,(i=1,2)$$ be a $$n$$-dimensional spherical space form, where $$\Gamma_i \subset SO(n+1)$$ is a finite subgroup acting freely on $$S^n$$.

Suppose $$S^n/\Gamma_1$$ is diffeomorphic to $$S^n/\Gamma_2$$, can we show they are isometric?

• A big chunk of the details are in Thurston's "Three-Dimensional Geometry and Topology". As Igor mentioned, the primary technicality issue is Lens spaces, and that part isn't as developed in Thurston's book. Bonahon has a lovely proof in that context, and you can also find Bonahon's proof in Hatcher's 3-manifolds notes (on his web page). – Ryan Budney Mar 21 at 16:50
• @RyanBudney Thursron's book and Hatcher's notes are about the three dimensional manifolds while the question is for any $n$. I see something missing here. – Piotr Hajlasz Mar 21 at 16:54
• De Rham's theorem is for every dimension. For 3d lens spaces there are other proofs, I think. – Igor Belegradek Mar 21 at 16:59
• Whoops, I just imagined this was the $n=3$ case. The general case is similar, although the Bonahon proof is particular to $n=3$. – Ryan Budney Mar 21 at 20:24

## 1 Answer

Yes, diffeomorphic spherical space forms are isometric. This famous result of Georges de Rham can be found in [de Rham, G. Complexes à automorphismes et homéomorphie différentiable. Ann. Inst. Fourier Grenoble 2 (1950), 51–67 (1951)]. The main tool is Reidemeister's torsion. For lens spaces the result was proved by W. Franz in 1935.