School of Mathematical & Statistical Sciences Faculty Publications and Presentations

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

Document Type

Article

Publication Date

7-23-2019

Abstract

We investigate how an affine connection ∇ 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 ∇ becomes (para-)holomorphic and L becomes integrable if and only if the pair (∇,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): ∇,∇L,∇∗, and ∇†=∇∗L=∇L∗, where ∇L and ∇∗ are, respectively, L-conjugate and g-conjugate transformations of ∇. This leads to the following special cases (where T stands for torsion): (i) the case of T=T∗,TL=T†, for which all four connections are Codazzi-coupled to g, but dω≠0, whence M is called Codazzi-(para-)Hermitian; (ii) the case of T=−T†,TL=−T∗, for which dω=0, i.e., the manifold 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.

Comments

https://rdcu.be/eLWg4

Publication Title

Results in Mathematics

DOI

10.1007/s00025-019-1071-2

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