Gaussian curvature

Gauss curvaturecurvatureLiebmann's theorem curvaturecurvature of a surfacecurvaturesformula for the Gauss curvatureGaussianGaussian radius of curvature
In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures,wikipedia
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Differential geometry

differentialdifferential geometerdifferential geometry and topology
In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures,
However, the Theorema Egregium of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes strong compatibility conditions on their metrics: the Gaussian curvatures at the corresponding points must be the same.

Hyperboloid

hyperboloid of one sheethyperbolicHyperboloid of two sheets
The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.
It is a connected surface, which has a negative Gaussian curvature at every point.

Theorema Egregium

This is the content of the Theorema egregium. Gaussian curvature is named after Carl Friedrich Gauss, who published the Theorema egregium in 1827.
In other words, the Gaussian curvature of a surface does not change if one bends the surface without stretching it.

Developable surface

developablefully-developable hull formdevelop
When a surface has a constant zero Gaussian curvature, then it is a developable surface and the geometry of the surface is Euclidean geometry.
In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature.

Hyperbolic geometry

hyperbolic planehyperbolichyperbolic surface
When a surface has a constant negative Gaussian curvature, then it is a pseudospherical surface and the geometry of the surface is hyperbolic geometry.
Hyperbolic plane geometry is also the geometry of saddle surfaces and pseudospherical surfaces, surfaces with a constant negative Gaussian curvature.

Curvature

curvednegative curvatureextrinsic curvature
Gaussian curvature is an intrinsic measure of curvature, depending only on distances that are measured on the surface, not on the way it is isometrically embedded in Euclidean space.
The Gaussian curvature, named after Carl Friedrich Gauss, is equal to the product of the principal curvatures,

Principal curvature

principal curvaturesprincipal directionsprincipal radii of curvature
In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures,
The product k 1 k 2 of the two principal curvatures is the Gaussian curvature, K, and the average (k 1 + k 2 )/2 is the mean curvature, H.

Carl Friedrich Gauss

GaussCarl GaussCarl Friedrich Gauß
Gaussian curvature is named after Carl Friedrich Gauss, who published the Theorema egregium in 1827.
Among other things, he came up with the notion of Gaussian curvature.

Asymptotic curve

asymptotic lineasymptoticasymptotic direction
Because one principal curvature is negative, one is positive, and the normal curvature varies continuously if you rotate a plane orthogonal to the surface around the normal to the surface in two directions, the normal curvatures will be zero giving the asymptotic curves for that point.
Asymptotic directions can only occur when the Gaussian curvature is negative (or zero).

Liouville's equation

Liouville equationLiouville equations
A useful formula for the Gaussian curvature is Liouville's equation in terms of the Laplacian in isothermal coordinates.
on a surface of constant Gaussian curvature K:

Torus

toroidaltoriflat torus
The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.
Its surface has zero Gaussian curvature everywhere.

Saddle point

saddle surfacesaddlesaddle points
, then the Gaussian curvature is negative and the surface is said to have a hyperbolic or saddle point.
Saddle surfaces have negative Gaussian curvature which distinguish them from convex/elliptical surfaces which have positive Gaussian curvature.

Parabolic line

Most surfaces will contain regions of positive Gaussian curvature (elliptical points) and regions of negative Gaussian curvature separated by a curve of points with zero Gaussian curvature called a parabolic line.
In differential geometry, a smooth surface in three dimensions has a parabolic point when the Gaussian curvature is zero.

Normal plane (geometry)

normal sectionnormal plane
At any point on a surface, we can find a normal vector that is at right angles to the surface; planes containing the normal vector are called normal planes.
The product of the principal curvatures is the Gaussian curvature of the surface.

Gauss–Bonnet theorem

Gauss-Bonnet theoremGauss–Bonnet formulaChern–Gauss–Bonnet formula
A more general result is the Gauss–Bonnet theorem.
Suppose M is a compact two-dimensional Riemannian manifold with boundary \partial M. Let K be the Gaussian curvature of M, and let k_g be the geodesic curvature of \partial M.

Differential geometry of surfaces

surfaceshape operatorsmooth surface
The two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point.
One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

Sphere

sphericalhemisphereglobose
When a surface has a constant positive Gaussian curvature, then it is a sphere and the geometry of the surface is spherical geometry.

Pseudosphere

pseudospherical surfaceTractricoidpseudo-sphere
When a surface has a constant negative Gaussian curvature, then it is a pseudospherical surface and the geometry of the surface is hyperbolic geometry.
In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.

Hilbert's theorem (differential geometry)

Hilbert's theorem
*Hilbert's theorem (1901) states that there exists no complete analytic (class
In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface S of constant negative gaussian curvature K immersed in.

Umbilical point

umbilicumbilic pointumbilic points
A standard proof uses Hilbert's lemma that non-umbilical points of extreme principal curvature have non-positive Gaussian curvature.
Umbilic points generally occur as isolated points in the elliptical region of the surface; that is, where the Gaussian curvature is positive.

Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
and endowed with the Riemannian metric given by the first fundamental form.
Informally, the theorem says that the curvature of a surface can be determined entirely by measuring distances along paths on the surface.

Sectional curvature

curvaturecurvature tensorsmanifolds with constant sectional curvature
It is the Gaussian curvature of the surface which has the plane σ p as a tangent plane at p, obtained from geodesics which start at p in the directions of σ p (in other words, the image of σ p under the exponential map at p).

Isothermal coordinates

Beltrami equationisothermal
A useful formula for the Gaussian curvature is Liouville's equation in terms of the Laplacian in isothermal coordinates.
In the isothermal coordinates (u, v), the Gaussian curvature takes the simpler form

Hilbert's lemma

A standard proof uses Hilbert's lemma that non-umbilical points of extreme principal curvature have non-positive Gaussian curvature.
It may be used to prove Liebmann's theorem that a compact surface with constant Gaussian curvature must be a sphere.

First fundamental form

firstfirst quadratic forms
In fact, it can be found given the full knowledge of the first fundamental form and expressed via the first fundamental form and its partial derivatives of first and second order.
The Gaussian curvature of a surface is given by