2013/01/20 by Henri Anciaux, Anciaux, Henri, Pascal Romon +1
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG
paper · pdf · doi:10.48550/arxiv.1301.4638
Clarified the statements on the cotangent bundle. Corrected various typos
openalex publication_date 2013/01/20 · arxiv created 2013/09/02 · arxiv updated 2013/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 T\M 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.