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

For any maximal surface group representation into SO0(2, n+1), we introduce a non-degenerate scalar product on the first cohomology group of the surface with values inthe associated flat bundle. In particular, it gives rise to a non-degene Riemannian metric on the smooth locus of the subset consisting of maximal representations inside the character variety. In the case 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, we show the existence of totally geodesic sub-varieties. Finally, in the general case, we explain when a representation with Zariski closure contained in SO0(2, 3)represents a smooth or orbifold point in the maximal SO0(2, n+1)-character variety and we discuss about the inclusion of Hitchin and Gothen components.