Let $X$ be a second-countable Hausdorff smooth manifold, let $\pi:E \to X$ be a smooth complex vector bundle of finite complex rank $r \in \mathbb{N}$, and let $h$ be a smooth Hermitian metric on $E$, with the convention that $h$ is complex-linear in the first argument and conjugate-linear in the second. A complex-linear connection on $E$ means an operator \begin{align*} \nabla:\mathfrak{X}(X)\times \Gamma(E) \to \Gamma(E) \end{align*} that is $C^\infty(X;\mathbb{R})$-linear in the vector-field variable, $\mathbb{C}$-linear in the section variable, and satisfies the Leibniz rule \begin{align*} \nabla_V(fs)=V(f)s+f\nabla_Vs \end{align*} for every $V \in \mathfrak{X}(X)$, $s \in \Gamma(E)$, and $f \in C^\infty(X;\mathbb{C})$. Then there exists a complex-linear connection $\nabla$ on $E$ that is compatible with $h$, meaning that for every smooth real vector field $V \in \mathfrak{X}(X)$ and every pair of smooth sections $s,t \in \Gamma(E)$, \begin{align*} V(h(s,t)) = h(\nabla_V s,t) + h(s,\nabla_V t). \end{align*} Moreover, if $\nabla_0$ is one metric-compatible complex-linear connection on $(E,h)$, then the set of all metric-compatible complex-linear connections on $(E,h)$ is the affine space \begin{align*} \nabla_0 + \Omega^1(X;\mathfrak{u}(E,h)), \end{align*} where $\mathfrak{u}(E,h) \subset \operatorname{End}(E)$ is the smooth real vector subbundle whose fibre at $x \in X$ consists of all complex-linear endomorphisms $A:E_x \to E_x$ satisfying \begin{align*} h_x(Au,v)+h_x(u,Av)=0 \end{align*} for all $u,v \in E_x$, and where $\Omega^1(X;\mathfrak{u}(E,h))$ denotes the space of smooth real one-forms on $X$ with values in this subbundle. Equivalently, every metric-compatible connection $\nabla$ is uniquely of the form \begin{align*} \nabla = \nabla_0 + B, \end{align*} where $B \in \Omega^1(X;\operatorname{End}(E))$ is a smooth real one-form assigning to each $x \in X$ a real-[linear map](/page/Linear%20Map) \begin{align*} B_x:T_xX \to \operatorname{End}_{\mathbb{C}}(E_x) \end{align*} and satisfying \begin{align*} h_x(B_x(V_x)s(x),t(x)) + h_x(s(x),B_x(V_x)t(x))=0 \end{align*} for all $V \in \mathfrak{X}(X)$, $s,t \in \Gamma(E)$, and $x \in X$.