  We find all orthogonal metrics where the geodesic Hamilton-Jacobi equation
separates and the Riemann curvature tensor satisfies a certain equation (called
the diagonal curvature condition). All orthogonal metrics of constant curvature
satisfy the diagonal curvature condition. The metrics we find either correspond
to a Benenti system or are warped product metrics where the induced metric on
the base manifold corresponds to a Benenti system. Furthermore we show that
most metrics we find are characterized by concircular tensors; these metrics,
called Kalnins-Eisenhart-Miller (KEM) metrics, have an intrinsic
characterization which can be used to obtain them. In conjunction with other
results, we show that the metrics we found constitute all separable metrics for
Riemannian spaces of constant curvature and de Sitter space.
