Metric compatible connection on a riemannian vector bundle

Let $h$ be a supplied smooth positive-definite bundle metric, as in Smooth bundle metrics. A connection is metric compatible if, for all local smooth sections $s,t$ and local vector fields $X$, $X(h(s,t))=h({\mathrm{\nabla }}_{X}s,t)+h(s,{\mathrm{\nabla }}_{X}t)\mathrm{.}$ Local connection operations are provided by Connection laws in directional form. Equivalently, the induced connection on $E^{\ast }\otimes E^{\ast }$ annihilates $h$: by Product connection on tensor and hom bundles, its evaluation is exactly the left side minus the right side. Thus $\mathrm{\nabla }h=0$ is a specified identity of tensors, not an independent assumption about an endomorphism.

In a local frame write $h(e_i,e_j)=H_{ij}$. Testing on frame sections gives $X(H)=\omega (X{)}^{T}H+H\omega (X)$; conversely expanding $s=eu,t=ev$ and using the scalar product rule gives the full identity from this matrix equation. This makes the condition pointwise testable in the direction and the two section values. Rank zero satisfies it vacuously. A metric is supplied, so metric existence and its choice requirements are not used.