In mathematics, a Hermitian connection ∇ {\displaystyle \nabla } is a connection on a Hermitian vector bundle E {\displaystyle E} over a smooth manifold M {\displaystyle M} which is compatible with the Hermitian metric ⟨ ⋅, ⋅ ⟩ {\displaystyle \langle \cdot,\cdot \rangle } on E {\displaystyle E}, meaning that v ⟨ s, t ⟩ = ⟨ ∇ v s, t ⟩ + ⟨ s, ∇ v t ⟩ {\displaystyle v\langle s,t\rangle =\langle \nabla _{v}s,t\rangle +\langle s,\nabla _{v}t\rangle } for all smooth vector fields v {\displaystyle v} and all smooth sections s, t {\displaystyle s,t} of E {\displaystyle E}. Wikipedia