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Results in Mathematics

, 74:150 | Cite as

(Para-)Holomorphic and Conjugate Connections on (Para-)Hermitian and (Para-)Kähler Manifolds

  • Sergey Grigorian
  • Jun ZhangEmail author
Article
  • 84 Downloads

Abstract

We investigate how an affine connection \(\nabla \) that generally admits torsion interacts with both g and L on an almost (para-)Hermitian manifold \((\mathfrak {M},g,L)\), where L denotes either an almost complex structure J with \(J^{2}=-\hbox {id}\) or an almost para-complex structure K with \(K^{2}=\hbox {id}\). We show that \(\nabla \) becomes (para-)holomorphic and L becomes integrable if and only if the pair \( (\nabla ,L)\) satisfies a torsion coupling condition. We investigate (para-)Hermitian manifolds \(\mathfrak {M}\) in which this torsion coupling condition is satisfied by the following four connections (all possibly carrying torsion): \(\nabla ,\nabla ^{L},\nabla ^{*},\) and \(\nabla ^\dagger = \nabla ^{*L}=\nabla ^{L*}\), where \(\nabla ^{L}\) and \(\nabla ^{*}\) are, respectively, L-conjugate and g-conjugate transformations of \(\nabla \). This leads to the following special cases (where T stands for torsion): (i) the case of \(T = T^*, T^L = T^\dagger \), for which all four connections are Codazzi-coupled to g, but \(d\omega \ne 0\), whence \(\mathfrak {M}\) is called Codazzi-(para-)Hermitian; (ii) the case of \(T = - T^{\dagger }, T^L = - T^{*}\), for which \(d \omega = 0\), i.e., the manifold \(\mathfrak {M}\) becomes (para-)Kähler. In the latter case, quadruples of (para-)holomorphic connections all with non-vanishing torsions may exist in (para-)Kähler manifolds, complementing the result of Fei and Zhang (Results Math 72:2037–2056, 2017) showing the existence of pairs of torsion-free connections, each Codazzi-coupled to both g and L, in Codazzi-(para-)Kähler manifolds.

Keywords

Codazzi coupling conjugate connection torsion Nijenhuis tensor holomorphic connection 

Mathematics Subject Classification

32Q15 32Q60 53B05 53B35 53D05 

Notes

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematical and Statistical SciencesUniversity of Texas Rio Grande ValleyEdinburgUSA
  2. 2.Department of Psychology and Department of MathematicsUniversity of MichiganAnn ArborUSA

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