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?



Isometries between spherical space forms
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.


