On structure-preserving connections
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On structure-preserving connections Arif Salimov1
© Akadémiai Kiadó, Budapest, Hungary 2018
Abstract In this paper we find the formula of connections under which an almost complex structure is covariantly constant. These types of connections on anti-Kähler–Codazzi manifolds are described. Also, twin metric-preserving connections are analyzed for quasi-Kähler manifolds. Finally, anti-Hermitian Chern connections are investigated. Keywords Almost complex structure · Semi-Riemannian metric · Anti-Hermitian structure · Anti-Kähler–Codazzi manifold · Anti-Kähler manifold · Quasi-Kähler manifold · Chern connection Mathematics Subject Classification 53C15 · 53C05
1 Introduction On an almost complex manifold (M, J ), a connection ∇ is called a J -connection if the almost complex structure J is covariantly constant with respect to ∇, i.e. ∇ J = 0. Also, in this case we say that the connection ∇ is structure-preserving. Naturally, if a triple (M, J, g) is a real Kähler manifold with Kählerian metric g, then the Levi-Civita connection of g is a J -connection. On the other hand, in order to determine the class of J -connections, in [1,2] the notion of a complex conjugate connection associated to an initial connection ∇ is introduced by ∇ (J ) = ∇ − J ◦ ∇ J. Similar problems are investigated in [3,4,6] for apbc, product and tangent structures, which are defined by −J12 = J22 = J32 = J1 ◦ J2 ◦ J3 = −id M (J1 = ±id M ), J 2 = id M and J 2 = 0, respectively. Many results on J -connections and their holomorphicity on manifolds with nilpotent structures were obtained by Vishnevskii [15].
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Arif Salimov [email protected] Department of Algebra and Geometry, Baku State University, 1148 Baku, Azerbaijan
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A. Salimov
We now introduce a connection ∇ˆ on an almost complex manifold (M, J ) by [16, p. 255] 1 ∇ˆ X Y = ∇ X Y − J ((∇ X J )Y ) 2
(1.1)
for all vector fields X, Y. We say that ∇ˆ is a J -translation of ∇, i.e. ∇ˆ = J (∇). An important field in complex geometry is the search for anti-Hermitian metrics having special properties. A semi-Riemannian metric g with signature (n, n) on an almost complex manifold (M, J ) is called anti-Hermitian (also known as Norden [7,10] or B-manifold [15]) if g(J X, J Y ) = −g(X, Y ) for every vector field X and Y on M. The pair (g, J ) is usually called an almost anti-Hermitian structure (simply anti-Hermitian when J is integrable) and G(X, Y ) = g(J X, Y ) = (g ◦ J )(X, Y ) is the twin anti-Hermitian metric. A Levi-Civita connection of g which preserves J is usually called an anti-Kähler connection. In such case the anti-Kähler connection of g coincides with the Levi-Civita connection of the twin metric G. It is a remarkable fact that the anti-Kähler connection of g is complex-holomorphic, because there exists a one-to-one correspondence between anti-Kähler (Kähler-Norden) manifolds and holomorphic Riemannian manifolds [9]. An anti-Hermitian metric connection of type I on an almost anti-Hermitian manifold (M, g, J ) is the unique connection ∇˜ with torsion defined by [12] ∇˜ = ∇ g −
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