Israel Journal of Mathematics

, Volume 99, Issue 1, pp 231–270 | Cite as

Graded metrics adapted to splittings

  • J. Monterde
  • O. A. Sánchez-Valenzuela


Homogeneous graded metrics over split ℤ2-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, ω, Δ′}, whereg is a metric tensor onM, ω 0 is a fibered nondegenerate skewsymmetric bilinear form on the Batchelor bundleE → M, and Δ′ is a connection onE satisfying Δ′ω = 0. Odd metrics are also studied under the same criterion and they are specified by the data {κ, Δ′}, with κ ∈ Hom (TM, E) invertible, and Δ′κ = 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 Δ′ onE satisfiesR Δ′ (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 Δ′ satisfies a specific set of equations. 0 Finally, graded Einstein even metrics can be supported only over Ricci flat Riemannian manifolds. Related results for graded metrics on Ω(M) are also discussed.


Riemannian Manifold Vector Bundle Sectional Curvature Conformal Transformation Civita Connection 
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Copyright information

© Hebrew University 1984

Authors and Affiliations

  1. 1.Departament de Geometria i Topologia Facultat de MatemàtiquesUniversitat de ValènciaBurjassot (València)Spain
  2. 2.Centro de Investigación en MatemáticasGuanajuatoMéxico

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