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

Henri Anciaux (1) , Pascal Romon (2)

1 USP - Universidade de São Paulo = University of São Paulo
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

It is a classical fact that the cotangent bundle $T^* M$ of a differentiable manifold $M$ enjoys a canonical symplectic form $\Omega^*$. If $(M,j,g,\omega)$ is a pseudo-Kähler or para-Kähler $2n$-dimensional manifold, we prove that the tangent bundle $T M$ also enjoys a natural pseudo-Kähler or para-Kähler structure $(J,G,\Omega)$, where $\Omega$ is the pull-back by $g$ of $\Omega^*$ 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.

Mots clés: pseudo-riemannian metric; complex structure; para-complex structure; tangent bundle

Domaines: Géométrie différentielle [math.DG]

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⟩