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Article Dans Une Revue Monatshefte für Mathematik Année : 2014

A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold

Résumé

It is a classical fact that the cotangent bundle TM 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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Dates et versions

hal-00778411 , version 1 (20-01-2013)
hal-00778411 , version 2 (27-03-2013)
hal-00778411 , version 3 (02-09-2013)

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Henri Anciaux, Pascal Romon. A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold. Monatshefte für Mathematik, 2014, 174 (3), pp.329-355. ⟨10.1007/s00605-014-0630-6⟩. ⟨hal-00778411v3⟩
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