Graded metrics adapted to splittings

Homogeneous graded metrics over split Z(2)-graded manifolds whose Levi-Civita connection is adapted to a given splitting, in the sense recently introduced by Koszul, are completely described. A subclass of such is singled out by the vanishing of certain components of the graded curvature tensor, a condition that plays a role similar to the closedness of a graded symplectic form in graded symplectic geometry: It amounts to determining a graded metric by the data (g, w, del'), where g is a metric tensor on M, w is a fibered nondegenerate skewsymmetric bilinear form on the Batchelor bundle E --> M, and del' is a connection on E satisfying del'w = 0. Odd metrics are also studied under the same criterion and they are specified by the data {kappa, del'}, with kappa is an element of Hom(TM, E) invertible, and del'kappa = 0. It is shown in general that even graded metrics of constant graded curvature can be supported only over a Riemannian manifold of constant curvature, and the curvature of del' on E satisfies R-del'(X, Y)(2) = 0. It is shown that graded Ricci flat even metrics are supported over Ricci flat manifolds and the curvature of the connection del' satisfies a specific set of equations. Finally, graded Einstein even metrics can be supported only over Ricci flat Riemannian manifolds. Related results for graded metrics on Omega(M) are also discussed.