Locally conformally Hermitian-flat manifolds (Q1891977)

The Hermitian connection on a Hermitian manifold is a connection preserving the holomorphic sections. It coincides with the Levi-Civita connection on Kähler manifolds. The author considers locally conformally Hermitian flat manifolds, i.e. Hermitian manifolds which are locally conformally related to flat (in the Hermitian sense) manifolds. The main purpose of this paper is to derive necessary and sufficient conditions for a Hermitian manifold to be locally conformally Hermitian flat. These conditions are given in terms of a conformally invariant tensor \({\mathcal B}_0\) and scalar curvatures. The author also constructs a family of examples.