= Constant-angle geodesic coordinate grids are locally flat
{title2=$F=k\sqrt{EG},\quad E_v=G_u=0,\quad |k|<1$}
If the oriented coordinate tangent vectors meet at a constant <angle> with cosine $k$, the <first fundamental form> has $F=k\sqrt{EG}$. The <geodesic coordinate grid> identities reduce to $k(\sqrt E)_v=(\sqrt G)_u$ and $k(\sqrt G)_u=(\sqrt E)_v$. Since positive definiteness gives $|k|<1$, both derivatives vanish. Thus $E=E(u)$ and $G=G(v)$ locally. Changing coordinates to $U=\int\sqrt E\,du$ and $V=\int\sqrt G\,dv$ gives the constant metric $dU^2+2k\,dU\,dV+dV^2$. It is locally Euclidean after a linear change of coordinates, hence has zero <Gaussian curvature>.
// Target: geometry-and-topology.bigb
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