Gaussian curvature


In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and, at the given point:
For example, a sphere of radius has Gaussian curvature everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.
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. This is the content of the Theorema egregium.
Gaussian curvature is named after Carl Friedrich Gauss, who published the Theorema egregium in 1827.

Informal definition

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 intersection of a normal plane and the surface will form a curve called a normal section and the curvature of this curve is the normal curvature. For most points on most surfaces, different normal sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures, call these,. The Gaussian curvature is the product of the two principal curvatures.
The sign of the Gaussian curvature can be used to characterise the surface.
Most surfaces will contain regions of positive Gaussian curvature and regions of negative Gaussian curvature separated by a curve of points with zero Gaussian curvature called a parabolic line.

Relation to geometries

When a surface has a constant zero Gaussian curvature, then it is a developable surface and the geometry of the surface is Euclidean geometry.
When a surface has a constant positive Gaussian curvature, then it is a sphere and the geometry of the surface is spherical geometry.
When a surface has a constant negative Gaussian curvature, then it is a pseudospherical surface and the geometry of the surface is hyperbolic geometry.

Relation to principal curvatures

The two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different directions at that point. We represent the surface by the implicit function theorem as the graph of a function,, of two variables, in such a way that the point is a critical point, that is, the gradient of vanishes. Then the Gaussian curvature of the surface at is the determinant of the Hessian matrix of . This definition allows one immediately to grasp the distinction between a cup/cap versus a saddle point.

Alternative definitions

It is also given by
where is the covariant derivative and is the metric tensor.
At a point on a regular surface in, the Gaussian curvature is also given by
where is the shape operator.
A useful formula for the Gaussian curvature is Liouville's equation in terms of the Laplacian in isothermal coordinates.

Total curvature

The surface integral of the Gaussian curvature over some region of a surface is called the total curvature. The total curvature of a geodesic triangle equals the deviation of the sum of its angles from. The sum of the angles of a triangle on a surface of positive curvature will exceed, while the sum of the angles of a triangle on a surface of negative curvature will be less than. On a surface of zero curvature, such as the Euclidean plane, the angles will sum to precisely radians.
A more general result is the Gauss–Bonnet theorem.

Important theorems

''Theorema egregium''

Gauss's Theorema egregium states that Gaussian curvature of a surface can be determined from the measurements of length on the surface itself. 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. Equivalently, the determinant of the second fundamental form of a surface in can be so expressed. The "remarkable", and surprising, feature of this theorem is that although the definition of the Gaussian curvature of a surface in certainly depends on the way in which the surface is located in space, the end result, the Gaussian curvature itself, is determined by the intrinsic metric of the surface without any further reference to the ambient space: it is an intrinsic invariant. In particular, the Gaussian curvature is invariant under isometric deformations of the surface.
In contemporary differential geometry, a "surface", viewed abstractly, is a two-dimensional differentiable manifold. To connect this point of view with the classical theory of surfaces, such an abstract surface is embedded into and endowed with the Riemannian metric given by the first fundamental form. Suppose that the image of the embedding is a surface in. A local isometry is a diffeomorphism between open regions of whose restriction to is an isometry onto its image. Theorema egregium is then stated as follows:
For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the "unrolled" tube. On the other hand, since a sphere of radius has constant positive curvature and a flat plane has constant curvature 0, these two surfaces are not isometric, even locally. Thus any planar representation of even a part of a sphere must distort the distances. Therefore, no cartographic projection is perfect.

Gauss–Bonnet theorem

The Gauss–Bonnet theorem links the total curvature of a surface to its Euler characteristic and provides an important link between local geometric properties and global topological properties.

Surfaces of constant curvature