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

作者：Anciaux Henri^*; Romon Pascal

来源：Monatshefte fur Mathematik, 2014, 174(3): 329-355.

DOI：10.1007/s00605-014-0630-6

• Tangent bundle
• Pseudo-Kahler geometry
• Para-Kahler geometry
• Self-duality
• Anti-self-duality

It is a classical fact that the cotangent bundle of a differentiable manifold enjoys a canonical symplectic form . If is a pseudo-Kahler or para-Kahler -dimensional manifold, we prove that the tangent bundle also enjoys a natural pseudo-Kahler or para-Kahler structure , where is the pull-back by of and is a pseudo-Riemannian metric with neutral signature . We investigate the curvature properties of the pair and prove that: is scalar-flat, is not Einstein unless is flat, has nonpositive (resp. nonnegative) Ricci curvature if and only if has nonpositive (resp. nonnegative) Ricci curvature as well, and is locally conformally flat if and only if and has constant curvature, or and is flat. We also check that (i) the holomorphic sectional curvature of is not constant unless is flat, and (ii) in case, that is never anti-self-dual, unless conformally flat.

• 出版日期2014-7