Riemannian geometry of maximal surface group representations acting on pseudo-hyperbolic space

For any maximal surface group representation into \(\textrm{SO}_0(2,n+1)\) ${\text{SO}}_0(2,n+1)$ , we introduce a non-degenerate scalar product on the first cohomology group of the surface with values in the associated flat bundle. In particular, it gives rise to a non-degenerate Riemannian metric on the smooth locus of the subset consisting of maximal representations inside the character variety. In the case \(n=2\) $n=2$ we prove that the Riemannian metric is compatible with the orbifold structure and we compute its restriction to the Fuchsian locus, instead, when \(n=3\) $n=3$ , we show the existence of totally geodesic sub-varieties. Finally, in the general case, we explain when a representation with Zariski closure contained in \(\textrm{SO}_0(2,3)\) ${\text{SO}}_0(2,3)$ represents a smooth or orbifold point in the maximal \(\textrm{SO}_0(2,n+1)\) ${\text{SO}}_0(2,n+1)$ -character variety and we discuss about the inclusion of Hitchin and Gothen components.