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

We investigate how an affine connection del that generally admits torsion interacts with both g and L on an almost (para-)Hermitian manifold (M,g,L), where L denotes either an almost complex structure J with J2=-id or an almost para-complex structure K with K2=id. We show that del becomes (para-)holomorphic and L becomes integrable if and only if the pair (del,L) satisfies a torsion coupling condition. We investigate (para-)Hermitian manifolds M in which this torsion coupling condition is satisfied by the following four connections (all possibly carrying torsion): del,del L,del, and del dagger=del L=del L, where del L and del are, respectively, L-conjugate and g-conjugate transformations of del. This leads to the following special cases (where T stands for torsion): (i) the case of T=T,TL=T dagger, for which all four connections are Codazzi-coupled to g, but d omega not equal 0, whence M is called Codazzi-(para-)Hermitian; (ii) the case of T=-T dagger,TL=-T, for which d omega=0, i.e., the manifold 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.