Complex-linear and metric-compatible bundle connections

Let $E\to M$ be a smooth real vector bundle of rank $2r$, where $r\ge 0$, over a smooth manifold that may have boundary. A smooth complex structure on $E$ is a smooth bundle endomorphism $J:E\to E$ with $J^2=-I$. It makes each fiber a complex vector space by $(a+ib)v=av+bJv$. Equivalently, $E$ has local smooth complex frames with transition maps in ${\mathrm{GL}}_r(\mathbb{C})$. In one direction, multiplication by $i$ in complex bundle charts gives such a smooth $J$ because the transition maps are complex-linear. Conversely, near any point choose a complex basis in its fiber and extend its vectors to local smooth real sections; those sections together with their $J$-images remain a real frame after shrinking the neighborhood, and give a local complex frame. Forgetting smoothness in these charts gives the associated rank-$r$ complex topological vector bundle of Real and complex topological vector bundles.

For a smooth complex bundle $(E,J)$, write ${\mathrm{\Gamma }}_{\mathbb{C}}(E)$ for its smooth complex sections. A complex connection is a covariant derivative $\mathrm{\nabla }:{\mathrm{\Gamma }}_{\mathbb{C}}(E)\to {\mathrm{\Omega }}^1(M;E)$ that is $\mathbb{C}$-linear and satisfies $\mathrm{\nabla }(fs)=df\otimes s+f\mathrm{\nabla }s$ for every smooth complex-valued function $f$ and section $s$. This extends the real bundle-connection convention of Connection on a smooth vector bundle. A complex connection is Hermitian for a supplied Hermitian metric $h$, taken linear in its first variable and conjugate-linear in its second, when $Xh(s,t)=h({\mathrm{\nabla }}_{X}s,t)+h(s,{\mathrm{\nabla }}_{X}t)$ for all smooth sections $s,t$ and vector fields $X$. A real connection on a Euclidean vector bundle is Euclidean-compatible when it obeys the same identity for the real bundle metric, as in Metric compatible connection on a riemannian vector bundle.

In a local complex frame $e=(e_1,\dots ,e_r)$ write $\mathrm{\nabla }{e}_j={\sum }_i{e}_i{\omega }^i{}_j$. Applying Hermitian compatibility to the frame sections gives $dH={\omega }^{\mathsf{T}}H+H\bar{\omega },H_{ij}=h(e_i,e_j)\mathrm{.}$ After smooth Gram–Schmidt, a local unitary frame has $H=I$, so ${\omega }^{\ast }=-\omega$. In a real orthonormal frame the corresponding equation is ${\omega }^{\mathsf{T}}=-\omega$. With the curvature convention $\mathrm{\Omega }=d\omega +\omega \wedge \omega$ of Curvature two-form structure equation, these identities imply ${\mathrm{\Omega }}^{\ast }=-\mathrm{\Omega }$ and ${\mathrm{\Omega }}^{\mathsf{T}}=-\mathrm{\Omega }$, respectively: exterior differentiation preserves the adjoint relation, and for a matrix of one-forms $(\omega \wedge \omega {)}^{\ast }=-\omega \wedge \omega$ because transposition reverses the matrix order while one-forms anticommute. The published local structure-equation calculation is coefficientwise, so it also applies to complex frame coefficients. Thus curvature matrices of Hermitian or Euclidean-compatible connections are skew-Hermitian or skew-symmetric in the corresponding frames. For rank zero these frame identities are vacuous. In boundary charts the same identities hold up to the boundary by restriction of the smooth half-space coefficient formulas.