A note on Riemannian metrics on the moduli space of Riemann surfaces (Q2796720)

Let \(S_{g, n}\) be a compact Riemann surface of genus \(g\) with \(n\) points removed such that \(3g+n\geq 5\). Denote by \(T(g, n)\), \(M(g, n)\), and \(\mathrm{Mod}(g, n)\) the Teichmüller space, Riemann moduli space, and Teichmüller modular group (or mapping class group) of \(S_{g, n}\), respectively. It is well known that \(\mathrm{Mod}(g, n)\) acts discontinuously on \(T(g, n)\) so that \(M(g, n)=T(g, n)/\mathrm{Mod}(g, n)\). In this note the author shows that on the Riemann moduli space \(M(g, n)\) there does not exist any complete Riemannian metric of nonpositive sectional curvature such that the Teichmüller space \(T(g, n)\) is a \(\mathrm{Mod}(g, n)\)-invariant visibility manifold.