HOLOMORPHIC AND KILLING VECTOR FIELDS ON COMPACT BALANCED HERMITIAN MANIFOLDS

Abstract

We extend the classical theorems for nonexistence of Killing and holomorphic vector fields on compact Kähler manifolds to compact balanced Hermitian manifolds. We express the obstructions to the existence of Killing and holomorphic vector fields on compact balanced Hermitian manifolds in terms of the Ricci tensors of the Levi–Civita connection as well as in terms of the Ricci tensors of the Chern connection. We show that every affine vector field with respect to the Chern connection on a compact balanced Hermitian manifold is holomorphic.