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In a local orthonormal frame $(e_i)$, write $\nabla e_j=A^i{}_j{e}_i$. Metric compatibility and $d⟨e_i,e_j⟩=0$ giveThus the connection one-form of a real metric connection takes values in the skew-symmetric matrices.

For $g=-d{t}^2+A(u{)}^{2}d{x}^2+B(u{)}^{2}d{y}^2+d{z}^2$, $u=t-z$, with nowhere-zero smooth $A,B$, take $e^0=dt$, $e^1=Adx$, $e^2=Bdy$, $e^3=dz$, $\ell =du$. In signature $(-,+,+,+)$, Cartan's first structure equation and metric compatibility give the lower-index connection 1-forms ${\omega }_{01}={\omega }_{13}=-(A^{\prime }/A)e^1$ and ${\omega }_{02}={\omega }_{23}=-(B^{\prime }/B)e^2$. Cartan's second structure equation gives ${\Theta }_{01}={\Theta }_{13}=-(A^{\prime \prime }/A)\ell \wedge e^1$ and ${\Theta }_{02}={\Theta }_{23}=-(B^{\prime \prime }/B)\ell \wedge e^2$. All other independent curvature 2-forms vanish: in particular the $12$ component cancels between the time and longitudinal terms. Contracting $R^a{}_{bad}$ yields $\mathrm{Ric}=-(A^{\prime \prime }/A+B^{\prime \prime }/B)du\otimes du$. Thus the Vacuum Einstein equations are $A^{\prime \prime }/A+B^{\prime \prime }/B=0$, while nonzero $A^{\prime \prime }/A=-B^{\prime \prime }/B$ gives opposite transverse tidal curvatures and plus polarization relative to these axes.

With derivative-last connection coefficients and $N_{abc}={\nabla }_c{g}_{ab}$, the displayed tensor is the symmetric-in-$a,b$ correction associated with the nonmetricity tensor. Raising its final slot gives the correction to connection coefficients. It vanishes for metric compatibility. Reversing the sign used to define nonmetricity reverses this formula.

In a holomorphic local frame of a Hermitian holomorphic vector bundle, let $H$ be the matrix of its Hermitian metric on a holomorphic vector bundle. With the convention $\nabla e=eA$, the Chern connection hasThe condition ${\nabla }^{0,1}={\bar{\partial }}_E$ forces $A$ to have type $(1,0)$, while metric compatibility forces this formula, proving both existence and uniqueness.

Metric compatibility is ${\nabla }_{\alpha }{g}_{\mu \nu }=0$, allowing the metric to pass through covariant derivatives and index operations.

For a unit timelike normal $n$, the induced metric is $h_{\mu \nu }=g_{\mu \nu }+n_{\mu }{n}_{\nu }$. The spatial covariant derivative projects the derivative and all free indices with the spatial projection tensor $P^{\mu }{}_{\nu }={\delta }^{\mu }{}_{\nu }+n^{\mu }{n}_{\nu }$. Metric compatibility gives ${\nabla }_{\lambda }{h}_{\mu \nu }=({\nabla }_{\lambda }{n}_{\mu })n_{\nu }+n_{\mu }{\nabla }_{\lambda }{n}_{\nu }$; projecting kills the normal factor in each term. Thus the spatial covariant derivative is compatible with the induced metric.

Given a metric tensor and an affine connection, $N_{abc}={\nabla }_c{g}_{ab}$ measures failure of metric compatibility. It is symmetric in $a,b$. Some authors instead define nonmetricity as $-\nabla g$, so formulas must specify the sign. Together with the torsion tensor, it determines the difference from the Levi-Civita connection.

For a geodesic variation of unit timelike geodesics, proper time normalization holds across the entire variation: $g(T,T)=-1$. Metric compatibility and ${\nabla }_{T}S={\nabla }_{S}T$ then give $dg(S,T)/d\tau =\frac{1}{2}S(g(T,T))=0$. Consequently an initially orthogonal deviation vector stays orthogonal. The common normalization is essential: an arbitrary family of affinely parametrized curves with different tangent norms does not imply this conclusion.