= Gaussian curvature
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{title2=$K$}
{wiki}
= Gauss curvature
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{synonym}
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>:
$$
K=\det S=k_1k_2=\frac{eg-f^2}{EG-F^2}.
$$
Here $I=E\,du^2+2F\,du\,dv+G\,dv^2$ is the <first fundamental form> and $II=e\,du^2+2f\,du\,dv+g\,dv^2$ 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. https://ghomi.math.gatech.edu/Classes/Math497C/LectureNotes9.pdf[Ghomi, Lecture Notes 9, §2.2: determinant formula and independence of normal choice]{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>. https://ghomi.math.gatech.edu/Classes/Math497C/LectureNotes10.pdf[Ghomi, Lecture Notes 10, §2.3.1: the sign of Gaussian curvature]{ref}
Four worked models make the signs concrete:
* On a <plane>, the <unit normal> is constant, so the <shape operator> is zero and $k_1=k_2=K=H=0$.
* On a <sphere> of <radius> $R$, the outward <unit normal> gives $S=-\operatorname{Id}/R$. Thus $k_1=k_2=-1/R$, $K=1/R^2$, and $H=-1/R$.
* On the lateral surface of a <circular cylinder> of <radius> $R$, the outward <unit normal> gives $k_{\mathrm{circ}}=-1/R$ and $k_{\mathrm{axial}}=0$. Hence $K=0$ but $H=-1/(2R)$: there is <extrinsic curvature> despite intrinsic flatness. The <shape operator> calculation above gives $E=G=1$, $F=f=g=0$, $e=-1/R$, so the determinant formula gives the same result.
* For the upward-oriented saddle $X(x,y)=(x,y,h(x,y))$ with $h=(x^2-y^2)/2$, the <gradient> vanishes at the origin. The <fundamental forms of a graph surface> therefore give
$$
A(0)=\begin{pmatrix}1&0\\0&1\end{pmatrix},\qquad B(0)=\operatorname{Hess}h(0)=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
$$
The <shape operator> has <eigenvalues> $1,-1$, giving $K(0)=-1$ and $H(0)=0$. Away from the origin, <Gaussian curvature of a graph surface> gives $K(x,y)=-1/(1+x^2+y^2)^2$; the label $K=-1$ applies only at the marked point.
\Image[/gaussian-curvature-four-surfaces.png]
{title=Plane, outward cylinder, outward sphere, and upward saddle compared by their principal curvatures and Gaussian curvature}
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. https://ghomi.math.gatech.edu/Classes/Math497C/LectureNotes11.pdf[Ghomi, Lecture Notes 11, §2.5: Theorema Egregium]{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 $k_1k_2$ is determined by the <Riemannian metric>. https://ghomi.math.gatech.edu/Classes/Math497C/LectureNotes12.pdf[Ghomi, Lecture Notes 12, §§2.6–2.7: Gauss formula and metric-based curvature]{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 $I=du^2+dv^2$, 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.
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