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The connection is orthogonal, or metric-compatible, when $\nabla g=0$; equivalently, its parallel transport preserves the Riemannian metric. Compatibility of the induced connections with tensor contraction givesfor all vector fields $U,V$. Taking $U={\partial }_j$ and $V={\partial }_k$ yieldsConversely, this coordinate identity makes every component of $\nabla g$ vanish, so it is equivalent to orthogonality.

For a connected Riemannian manifold, the full holonomy group consists of parallel transport maps around all piecewise smooth loops based at $p$. Metric compatibility makes them orthogonal. Basepoint changes conjugate this group by parallel transport. This is full holonomy, not just the subgroup using contractible loops.

The square is a rank-two Killing tensor. It is symmetric because it pairs the covectors $Y_{a\cdot }$ and $Y_{b\cdot }$ using the metric. Along every affinely parametrized geodesic, its contraction $K(v,v)$ is the squared norm of the carried by parallel transport covector $Y(v,\cdot )$. Metric compatibility makes that norm constant, proving the Killing tensor equation. On a Riemannian manifold this tensor is positive semidefinite.

With gauge covariant derivative $D=d+A$, a Wilson loop is the matrix trace in a specified group representation of parallel transport around a closed curve. The holonomy of a connection changes by conjugation under gauge equivalence of principal connections, so its trace is gauge-invariant. It detects global connection data even when the local gauge curvature is zero.