= Curvature comparison for a geodesically ruled surface
{title2=$K_N(T,E)=K_M(T,E)-\|II(T,E)\|^2$}
A surface locally swept out by ambient <geodesics> has $II(T,T)=0$ in its unit ruling direction $T$. For a perpendicular unit tangent $E$, the <Gauss equation in a curved ambient manifold> therefore gives the displayed identity. Its <Gaussian curvature> is at most the ambient <sectional curvature> in its tangent plane, with equality precisely when $II(T,E)=0$. An independent variation argument applies the <Jacobi field> equation to the length of a perpendicular connecting field; its ambient derivative has the extra normal component $II(T,E)$. The Euclidean surface $(t,s,ts)$ is ruled by straight lines and has <Gaussian curvature> $-(1+t^2+s^2)^{-2}<0$.
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