Bismut K\"ahler-like manifolds of dimension 4 and 5

TL;DR

This paper advances the understanding of Bismut K"ahler-like manifolds by proving a conjecture relating Bismut Ricci flatness to Bismut flatness and classifying such manifolds in dimensions 4 and 5.

Contribution

It proves that Bismut Ricci flat BKL manifolds are Bismut flat and provides a complete classification of BKL manifolds in dimensions 4 and 5.

Findings

Bismut Ricci flat BKL manifolds are Bismut flat.

Complete classifications of BKL manifolds in dimensions 4 and 5.

Structural theorems for BKL manifolds.

Abstract

This paper is a sequel to our studies \cite{ZZ} and \cite{YZZ} on Bismut K\"ahler-like manifolds, or {\em BKL} manifolds for short. We will study the structural theorems for {\em BKL} manifolds, prove a conjecture raised in \cite{YZZ} which states that any {\em BKL} manifold that is Bismut Ricci flat must be Bismut flat, and give complete classifications of {\em BKL} manifolds in dimension $4$ and $5$.