On canonical-type connections on almost contact complex Riemannian manifolds

TL;DR

This paper surveys the differential geometry of canonical-type connections on almost contact complex Riemannian manifolds, highlighting their properties and recent results in the context of Norden and B-metric structures.

Contribution

It provides a comprehensive overview of canonical-type connections on these manifolds, including new results and insights into their geometric properties.

Findings

Characterization of canonical-type connections on almost contact complex Riemannian manifolds

Identification of algebraic identities satisfied by these connections

Extension of known results to new classes of manifolds

Abstract

We consider a pair of smooth manifolds, which are the counterparts in the even-dimensional and odd-dimensional cases. They are separately an almost complex manifold with Norden metric and an almost contact manifolds with B-metric, respectively. They can be combined as the so-called almost contact complex Riemannian manifold. This paper is a survey with additions of results on differential geometry of canonical-type connections (i.e. metric connections with torsion satisfying a certain algebraic identity) on the considered manifolds.