Results in Mathematics

, 74:150 | Cite as

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

  • Sergey Grigorian
  • Jun ZhangEmail author


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.


Codazzi coupling conjugate connection torsion Nijenhuis tensor holomorphic connection 

Mathematics Subject Classification

32Q15 32Q60 53B05 53B35 53D05 



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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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