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The displayed $e^a$ are one-forms, so they form an orthonormal coframe, dual to the orthonormal frame $E_0=A^{-1}{\partial }_t$, $E_1=A^{-1}{\partial }_x$, $E_2=A^{-1}{\partial }_y$, $E_3={\partial }_z$. Use ${\eta }_{ab}=\mathrm{diag}(-1,1,1,1)$ and abbreviateFor $i=0,1,2$, $d{e}^i=a{e}^3\wedge e^i$, while $d{e}^3=0$. The Levi-Civita connection is fixed by Cartan's first structure equation together with Lorentzian connection-form antisymmetry, ${\omega }_{ab}=-{\omega }_{ba}$, where ${\omega }_{ab}={\eta }_{ac}{\omega }^c{}_b$. Its nonzero connection 1-forms areIn particular, the time-index pair has the same sign with one index raised; incorrectly imposing ${\omega }^a{}_b=-{\omega }^b{}_a$ would lose the Lorentzian sign. All diagonal and all other entries vanish.

Apply Cartan's second structure equation. For $i=0,1,2$,For distinct $i,j$ in $\{0,1,2\}$, putting $s_0=-1$ and $s_1=s_2=1$ gives ${\Theta }^i{}_j={\omega }^i{}_3\wedge {\omega }^3{}_j=-s_j{a}^2{e}^i\wedge e^j$. Thus a complete independent set of curvature 2-forms isAll diagonal entries vanish. The remaining entries follow from ${\Theta }_{ab}=-{\Theta }_{ba}$: ${\Theta }^j{}_i=-s_i{s}_j{\Theta }^i{}_j$ for $i,j\le 2$, and ${\Theta }^3{}_i=-s_i{\Theta }^i{}_3$. These relations and the displayed six entries specify all sixteen curvature 2-forms. This computation is Cartan curvature of a planar warped spacetime.

Treat the printed $e^{\mu }$ as an orthonormal coframe, with frame metric $\eta =\mathrm{diag}(-1,1,1,1)$. Work on a patch $z\ne 0$ and $F=1-\alpha z^3>0$, so the given real coframe exists. Define $f=\sqrt{F}$ and $A=z{f}^{\prime }-f$. The exterior derivatives areThe connection 1-forms satisfying Cartan's first structure equation and metric compatibility areAll other forms vanish. With a Lorentzian frame, it is the lowered forms ${\omega }_{ab}={\eta }_{ac}{\omega }^c{}_b$ that are antisymmetric; the two mixed time–space forms are equal. These displayed forms solve the torsion-free structure equation, and uniqueness of the Levi-Civita connection identifies them as the required connection.

Apply Cartan's second structure equation. For example,The remaining derivatives give ${\Theta }^1{}_3=Af{e}^1\wedge e^3$ and ${\Theta }^1{}_2=-f^2{e}^1\wedge e^2$, with analogous forms for index 2. Put $q=\alpha z^3$ and $B=1+q/2$. From $f^2=1-q$,Therefore all six independent curvature 2-forms areThe other six are fixed by ${\Theta }^b{}_a=-{\eta }_{aa}{\eta }_{bb}{\Theta }^a{}_b$ for $a\ne b$, with no sum: equal for time–space pairs and opposite for spatial pairs. All diagonal forms are zero. The Lorentzian connection-form antisymmetry is essential to these signs.