Suppose we have a hypersurface $f(x^\mu)=0$ (with $\mu=0,1,...,d-1$) in a manifold with metric tensor $g_{\mu\nu}$. The transverse metric is defined as
$$h_{\mu\nu}=g_{\mu\nu}-n_\mu n_\nu,$$ with
$$n_\mu=\frac{\partial_\mu f}{\sqrt{|\partial_\nu f \partial^\nu f|}}.$$
It is used to define the extrinsic curvature of the hypersurface: $$K_{\mu\nu}=h^\alpha_{~~\mu} h^\beta_{~~\nu}\nabla_\alpha n_\beta$$ Does it have any sense computing the Riemann (intrinsic) curvature tensor for $h_{\mu\nu}$? What would it mean? Would it be related to the extrinsic curvature?