Hermitian manifolds with flat Gauduchon connections

Abstract

We complete the classification of compact Hermitian manifolds admitting a flat Gauduchon connection. In particular, we establish a conjecture of Yang and Zheng, showing that apart from the cases of a flat Chern or Bismut connection, such manifolds are Kähler. More generally, we prove the same result holds when the flatness assumption is replaced by the so-called Kähler-like condition, proving a conjecture of Angella, Otal, Ugarte and Villacampa. We also treat the non-compact case.

11 pages. Significant changes (including title) as the main theorem (Theorem 3.1) has been improved: The compactness assumption can be dropped with two possible exceptions and the flatness assumption has been relaxed to the Kaehler-like condition in all cases. Proof simplified and notation updated