Submanifolds of the manifold of metrics $\M$ appear in several contexts in differential geometry such as in the theory of Einstein metrics, the Yamabe problem and Teichmuller theory. Using the natural family of pseudometrics $G^c$ on the manifold of metrics from Ref. GMN92, I have tried to describe the pseudo-riemannian geometry of the relevant submanifolds of $\M$ . They will be described as maximal integral submanifolds. Submanifold charts and formulas for the second fundamental forms and the induced connections will be given. In the conformal class, which is geodesically closed, also the geodesic distance is studied. Most of these theories cited above use some variation principle on submanifolds of $\M$. I have used the pseudoriemannian structure to derive gradients, Hessians and conditions for the ellipticity of the Hessians of the relevant functionals.