Two weak Riemannian structures on the space of Riemannian metrics (Q1896727)

Let \(\mathcal M\) be the set of all smooth Riemannian structures on a compact, orientable \(n\)-manifold of class \(C^\infty\). Then \(\mathcal M\) can be given the structure of a Fréchet manifold. Two new weak Riemannian structures on \(\mathcal M\) are announced and for each the covariant derivative, curvature tensor and sectional curvature are described. The relevant geodesics are also explained.