Gaussian curvature ($K$)

Rolling a sheet into a circular cylinder bends it without changing its internal lengths and angles. Fitting that same sheet smoothly onto a sphere requires distortion of its first fundamental form. Gaussian curvature distinguishes these situations: it measures the intrinsic curvature of a smooth surface, even though its simplest definition uses how the surface bends in Euclidean space.

The Gaussian curvature at a point is the determinant of the shape operator, or the product of the two principal curvatures:Here $I=Ed{u}^2+2Fdudv+Gd{v}^2$ is the first fundamental form and $II=ed{u}^2+2fdudv+gd{v}^2$ is the second fundamental form, with $S=-dN$. Regularity makes $EG-F^2>0$. Reversing the unit normal negates $e,f,g$ and both principal curvatures, leaving Gaussian curvature unchanged. ^[ref]

The sign describes bending at the point: positive Gaussian curvature means the principal curvatures have the same nonzero sign, as at an elliptic point; negative Gaussian curvature means opposite signs, as on a saddle; zero Gaussian curvature means at least one principal curvature vanishes. Zero Gaussian curvature at a single point does not imply that a neighborhood is a plane. ^[ref]

Four worked models make the signs concrete:

The patches use a common length unit, with $R=1$ for the sphere and circular cylinder. The marked red and blue curves have principal directions at the black point; the arrow is the chosen unit normal. For physical lengths, principal curvature and mean curvature have units of inverse length, while Gaussian curvature has units of inverse squared length. Rescaling a surface by $\lambda >0$ divides its principal curvatures by $\lambda$ and its Gaussian curvature by ${\lambda }^2$. Writing the saddle as $h=(x^2-y^2)/(2L)$ restores an explicit length scale: $K(0)=-1/L^2$.

The Theorema Egregium says that Gaussian curvature depends only on the first fundamental form and its first two coordinate derivatives. A local isometry therefore preserves Gaussian curvature, even if the shape operator and the principal curvatures change. ^[ref]

Here is the outline, rather than a full proof. The Gauss formula splits second derivatives of a parametric surface into tangential Christoffel symbols and the second fundamental form. The Christoffel symbols are determined by the first fundamental form and its first derivatives. Commuting the next partial derivatives gives the Gauss equation, which identifies $\det S$ with the intrinsic sectional curvature computed from the Riemann curvature tensor. Thus the apparently extrinsic product $k_1{k}_2$ is determined by the Riemannian metric. ^[ref]

One can see the distinction directly by rolling a plane strip with coordinates $(u,v)$ into the circular cylinder $X(u,v)=(R\cos (u/R),R\sin (u/R),v)$. Both have $I=d{u}^2+d{v}^2$, so the map is a local isometry, and is one-to-one on an open strip of width less than $2\pi R$. Extending $u$ over all real numbers wraps around periodically; it does not give a global isometry between the entire plane and the entire circular cylinder. A nonempty open patch of a sphere cannot be flattened by a local isometry into a plane, because its Gaussian curvature $1/R^2$ cannot become zero. The obstruction concerns preservation of the first fundamental form, rather than a particular drawing or choice of seam.

For positive twice differentiable $f$, the only nonzero Christoffel symbols are ${\Gamma }_{yy}^x=-f{f}^{\prime }$ and ${\Gamma }_{xy}^y={\Gamma }_{yx}^y=f^{\prime }/f$. In the convention $R^d{}_{abc}={\partial }_b{\Gamma }_{ac}^d-{\partial }_c{\Gamma }_{ab}^d+{\Gamma }_{ac}^e{\Gamma }_{eb}^d-{\Gamma }_{ab}^e{\Gamma }_{ec}^d$, substitution gives $R^x{}_{yxy}=-f{f}^{\prime \prime }$. Dividing $R_{xyxy}$ by $g_{xx}{g}_{yy}-g_{xy}^2=f^2$ gives the Gaussian curvature $K=-f^{\prime \prime }/f$. Thus $f=x$ gives the flat metric in polar coordinates away from its axis, $f=\sin x$ gives unit positive curvature, and $f=\sinh x$ gives unit negative curvature.

A supporting sphere for an embedded surface at $p$ is a sphere tangent at $p$ such that the surface lies locally on one side of the sphere. Comparing second fundamental forms at the tangency controls the surface's normal curvature.

If a compact regular surface lies in a closed Euclidean ball of radius $R$, maximize distance from the ball's centre. At a maximizing point, comparison of the second fundamental form with the tangent supporting sphere makes both principal curvatures have magnitude at least $R^{-1}$ and the same sign. The Gaussian curvature there is therefore at least $R^{-2}$.

An elliptic point of a surface is a point at which the Gaussian curvature is positive, equivalently where the two principal curvatures have the same nonzero sign.

Every regular cone parametrized by $X(r,t)=rc(t)$ has zero Gaussian curvature away from its vertex. Its unit normal is independent of $r$, whileis tangent, so two coefficients of the second fundamental form vanish and $\det II=0$.