Suppose M is a compact manifold and Rim(M) the space of all Riemannian metrics. Consider $L^2$ metric $ G_g(h,k) = \int_M tr(g^{-1}hg^{-1}k)vol(g)$ where is a Riemannian metric and $ h,k \in T_gM$ . What is known about the injectivity of this metric? for example is it uniformly positive?