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?

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    $\begingroup$ 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). $\endgroup$ – Ryan Budney Mar 21 at 16:50
  • $\begingroup$ @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. $\endgroup$ – Piotr Hajlasz Mar 21 at 16:54
  • $\begingroup$ De Rham's theorem is for every dimension. For 3d lens spaces there are other proofs, I think. $\endgroup$ – Igor Belegradek Mar 21 at 16:59
  • $\begingroup$ Whoops, I just imagined this was the $n=3$ case. The general case is similar, although the Bonahon proof is particular to $n=3$. $\endgroup$ – Ryan Budney Mar 21 at 20:24

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.


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