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 $\mathcal M_{g,n}$ 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 $\mathcal M_{g,n}$ 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 $\mathcal M_{g,n}$.