Holomorphic and Killing vector fields on compact balanced Hermitian manifolds

We extend the classical theorems for nonexistence of Killing and holomorphi c vector fields on compact Kahler manifolds to compact balanced Hermitian m anifolds. We express the obstructions to the existence of Killing and holom orphic vector fields on compact balanced Hermitian manifolds in terms of th e Ricci tensors of the Levi-Civita connection as well as in terms of the Ri cci tensors of the Chern connection. We show that every affine vector field with respect to the Chern connection on a compact balanced Hermitian manif old is holomorphic.