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Monatshefte für Mathematik

, Volume 174, Issue 3, pp 329–355 | Cite as

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

  • Henri AnciauxEmail author
  • Pascal Romon
Article

Abstract

It is a classical fact that the cotangent bundle \(T^* {\mathcal {M}}\) of a differentiable manifold \({\mathcal {M}}\) enjoys a canonical symplectic form \(\Omega ^*\). If \(({\mathcal {M}},\mathrm{J} ,g,\omega )\) is a pseudo-Kähler or para-Kähler \(2n\)-dimensional manifold, we prove that the tangent bundle \(T{\mathcal {M}}\) also enjoys a natural pseudo-Kähler or para-Kähler structure \(({\tilde{\hbox {J}}},\tilde{g},\Omega )\), where \(\Omega \) is the pull-back by \(g\) of \(\Omega ^*\) and \(\tilde{g}\) is a pseudo-Riemannian metric with neutral signature \((2n,2n)\). We investigate the curvature properties of the pair \(({\tilde{\hbox {J}}},\tilde{g})\) and prove that: \(\tilde{g}\) 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 \(({\tilde{\hbox {J}}},\tilde{g})\) is not constant unless \(g\) is flat, and (ii) in \(n=1\) case, that \(\tilde{g}\) is never anti-self-dual, unless conformally flat.

Keywords

Tangent bundle Pseudo-Kähler geometry Para-Kähler geometry Self-duality Anti-self-duality 

Mathematics Subject Classification (2010)

32Q15 53D05 

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Copyright information

© Springer-Verlag Wien 2014

Authors and Affiliations

  1. 1.Universidade de São PauloSão PauloBrazil
  2. 2.Université Paris-Est Marne-la-ValléeMarne-la-ValléeFrance

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