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A Hermitian manifold is a complex manifold whose tangent bundle carries a smoothly varying positive-definite Hermitian form. Its real part is a Riemannian metric invariant under the complex structure $J$.

A Hermitian metric on a holomorphic vector bundle $E\to X$ is a smoothly varying positive-definite Hermitian form $h_x$ on every complex fiber $E_x$.

A Hermitian vector bundle is a complex vector bundle with a smoothly varying positive definite Hermitian form on each fibre. When the bundle is holomorphic, this is a Hermitian metric on a holomorphic vector bundle. Orthogonal projection to a smooth subbundle produces its induced metric and the quotient Hermitian metric.

An indefinite Hermitian form takes both positive and negative values on $⟨v,v⟩$ for nonzero vectors. For example, $-|v_0{|}^2+|v_1{|}^2+|v_2{|}^2+|v_3{|}^2$ is indefinite. It is not a positive inner product defining a Hilbert space, and a nonzero vector can have zero norm without being orthogonal to all vectors. A positive semidefinite restricted form becomes a positive inner product only after quotienting its radical of a Hermitian form.

On $\mathbb{C}/(\mathbb{Z}+\mathbb{Z}i)$, a Hermitian form linear in its first argument has integral imaginary part on the period lattice exactly when its real scalar coefficient is integral. Positivity is equivalent to $t>0$. The associated holomorphic line bundle has degree $t$ because its First Chern class is represented by $(it/2)dz\wedge d\overline{z}$. In particular the oriented Chern pairing is minus the imaginary part under this convention.

For a basis $(v_i)$, the matrix $A$ of a Hermitian form has entries $A_{ij}=H(v_i,v_j)$. With the convention that $H$ is linear in its first argument, coordinate columns $x,y$ satisfy $H(x,y)=x^{T}A\overline{y}$.

With a Hermitian form linear in its first slot, $H(s\otimes \bar{t})=h(s,t)$ is a complex-linear dual tensor. The original connection, its conjugate connection, the tensor product connection and the dual connection induce $D_0$. Its derivative is $(D_{0,V}H)(s\otimes \bar{t})=Vh(s,t)-h(D_{V}s,t)-h(s,D_{V}t)$. Consequently $D_{0}H=0$ is exactly metric compatibility.