A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold
نویسندگان
چکیده
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 TM also enjoys a natural pseudo-Kähler or para-Kähler structure (J̃, g̃,Ω), where Ω is the pull-back by g of Ω and g̃ is a pseudoRiemannian metric with neutral signature (2n, 2n). We investigate the curvature properties of the pair (J̃, g̃) and prove that: g̃ 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 (J̃, g̃) is not constant unless g is flat, and (ii) in n = 1 case, that g̃ is never anti-self-dual, unless conformally flat. 2010 MSC : 32Q15, 53D05
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