A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold
Résumé
It is a classical fact that the cotangent bundle T∗M of a differentiable manifold M enjoys a canonical symplectic form Ω∗. If (M,j,g,ω) is a pseudo-Kähler or para-Kähler 2n-dimensional manifold, we prove that the tangent bundle TM also enjoys a natural pseudo-Kähler or para-Kähler structure (J,G,Ω), where Ω is the pull-back by g of Ω∗ and G is a pseudo-Riemannian metric with neutral signature (2n,2n). We investigate the curvature properties of the pair (J,G) and prove that: G is scalar-flat, is not Einstein unless g is flat, has nonpositive (resp.\ nonnegative) Ricci curvature if and only if g has nonpositive (resp.\ nonnegative) Ricci curvature as well, and is locally conformally flat if and only if n=1 and g has constant curvature, or n>2 and g is flat. We also check that (i) the holomorphic sectional curvature of (J,G) is not constant unless g is flat, and (ii) in n=1 case, that G is never anti-self-dual, unless conformally flat.
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