  On Hermitian manifolds, the second Ricci curvature tensors of various metric
connections are closely related to the geometry of
  Hermitian manifolds. By refining the Bochner formulas for any Hermitian
complex vector bundle (Riemannain real vector bundle) with an arbitrary metric
connection over a compact Hermitian manifold, we can derive various vanishing
theorems for Hermitian manifolds and complex vector bundles by the second Ricci
curvature tensors. We will also introduce a natural geometric flow on Hermitian
manifolds by using the second Ricci curvature tensor.
