A step in the proof of Proposition 3.3 (p.6) in this paper argues the following:

Let $M^n$ be a topological sphere with a Riemannian metric $g$. Now assume further $(M^n,g)$ is locally symmetric, then it must be the standard round sphere by some results from a paper by Borel. I just can't follow Borel's paper, it might be something like 'For the sphere in the form $G/H$ with $G$ the isometric group, even-dim sphere has Euler number 2, which after some arguments can determine G and H'. But I am not sure how it really works or what happens for odd-dim sphere. Maybe the whole question follow from the classification theory for symmetric space of compact type, which I know almost zero.

Now I was wondering if a geometric way possible, or if anyone knows how exactly Borel's aurgment works.