= Differential geometry of surfaces
{wiki}
The differential geometry of <smooth surfaces> studies their induced <Riemannian metrics> and intrinsic or extrinsic <curvature>. The <first fundamental form> measures <lengths>, while the <second fundamental form> and <shape operator> describe bending in an ambient <Euclidean space>. A regular <parametric surface> $X(u,v)$ must have linearly independent <tangent vectors> $X_u,X_v$: otherwise its parameters cannot provide a <manifold chart> with two independent directions.
For a <sphere> of radius $R>0$, use colatitude $0<\theta<\pi$ and longitude $\phi$:
$$
X(\theta,\phi)=R(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta),\qquad I=R^2d\theta^2+R^2\sin^2\theta\,d\phi^2.
$$
Thus a change $d\phi$ has <arc length> $R\sin\theta\,|d\phi|$: equally spaced longitudes draw closer together toward a pole. The coordinate degeneracy at the poles belongs to this parametrization, not to the <smooth surface>.
Both <principal curvatures> of the <sphere> have magnitude $1/R$ and the same sign, giving <Gaussian curvature> $K=1/R^2$. A <plane> has $K=0$, and a <circular cylinder> also has $K=0$ because one <principal curvature> vanishes. These three familiar shapes show why visible bending alone cannot measure intrinsic <Gaussian curvature>.
\Image[/surface-intrinsic-curvature.png]
{title=Visible bending and Gaussian curvature are different measurements}
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