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

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-)Kahler. In the latter case, quadruples of (para-)holomorphic connections all with non-vanishing torsions may exist in (para-)Kahler 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-)Kahler manifolds.