On almost complex structures on tangent bundles
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ON ALMOST COMPLEX STRUCTURES WITH NORDEN METRICS ON TANGENT BUNDLES Zbigniew Olszak (Wroclaw) [Communicated by: J´ anos Szenthe ]
Abstract Let (M, J, g) be a K¨ ahler–Norden manifold. Using the notions of the horizontal and vertical lifts, a class of almost complex structures J is defined on the tangent bundle TM , and necessary and sufficient conditions for such a structure to be integrable (complex) are described. Next, a class of pseudo-Riemannian metrics g of Norden type is defined on TM , for which J is an antiisometry. Thus, the g ) becomes an almost complex structure with Norden metric on TM . It is pair (J, g ) is K¨ checked whether the structure (J, ahler–Norden itself.
1. Almost complex Norden manifolds An almost complex Norden manifold (M, J, g) (an almost complex manifold with a Norden metric) is defined to be a real 2m-dimensional differentiable manifold M endowed with an almost complex structure J and a pseudo-Riemannian metric g for which J is an antiisometry ([6]). Thus, J is a (1, 1)-tensor field and J 2 = −Id|M ,
g(JX, JY ) = −g(X, Y ),
X, Y ∈ X(M ),
where Id |M is the identity tensor field on M and X(M ) the Lie algebra of vector fields on M . A metric g realizing the antiisometry condition must have signature (m, m) and it is called of Norden type. A pair (J, g) as above will be called an almost complex Norden structure on M . In the case when J is integrable (comes from a complex structure on M ), (M, J, g) (resp., (J, g)) is called a complex Norden manifold (resp., a complex Norden structure). For an almost complex Norden manifold (M, J, g), define a (0, 2)-tensor field h by h(X, Y ) = g(X, JY ), X, Y ∈ X(M ). Then h is non-degenerate, symmetric and Mathematics subject classification number: Primary 53C56, Secondary 53C15, 53C50. Key words and phrases: Almost complex manifold with Norden metric, K¨ ahler manifold with Norden metric, tangent bundle, horizontal lift, vertical lift. Research supported by State Committee for Scientific Research (KBN) grant 2P03A00617. 0031-5303/2005/$20.00 c Akad´ emiai Kiad´ o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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z. olszak
also realizes the antiisometry condition. Thus, (J, h) becomes an almost complex Norden structure on M too. An almost complex Norden manifold (M, J, g) is said to be K¨ ahler–Norden (a K¨ ahlerian manifold with a Norden metric) if J is parallel with respect to the Levi– Civita connection ∇ of g, ∇J = 0 ([6]). A K¨ ahler–Norden manifold is necessarily complex Norden. If (M, J, g) is a K¨ ahler–Norden manifold, then so is (M, J, h). In the complex convention, a complex Norden manifold can be interpreted as a complex Riemann manifold and a K¨ ahler–Norden manifold as a holomorphic Riemann manifold; for details, see [2, 3, 7, 8]. Almost complex Norden structures on tangent bundles of differentiable manifolds were studied in the papers [1, 12, 13].
2. Lifts to tangent bundles ([9, 16]) Let M be an n-dimensional differentiable manifold, TM its tangent bundle and π :TM → M the natural projection. Local charts of M w
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