Local rigidity of covering constructions and Weil–Petersson subvarieties of the moduli space of curves

We show that totally geodesic subvarieties of the moduli space Mg,n of genus g curves with n marked points, endowed with the Weil–Petersson metric, are locally rigid. This implies that covering constructions—examples of totally geodesic subvarieties of Mg,n endowed with the Teichmüller metric—are locally rigid. We deduce the local rigidity statement from a more general rigidity result for a class of orbifold maps to Mg,n.