Gaussian curvature

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From left to right: a surface of negative Gaussian curvature (hyperboloid), a surface of zero Gaussian curvature (cylinder), and a surface of positive Gaussian curvature (sphere). Gaussian curvature.svg
From left to right: a surface of negative Gaussian curvature (hyperboloid), a surface of zero Gaussian curvature (cylinder), and a surface of positive Gaussian curvature (sphere).
Some points on the torus have positive, some have negative, and some have zero Gaussian curvature. Torus Positive and negative curvature.png
Some points on the torus have positive, some have negative, and some have zero Gaussian curvature.

In differential geometry, the Gaussian curvature or Gauss curvatureΚ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point:

Contents

The Gaussian radius of curvature is the reciprocal of Κ. For example, a sphere of radius r has Gaussian curvature 1/r2 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

Saddle surface with normal planes in directions of principal curvatures Minimal surface curvature planes-en.svg
Saddle surface with normal planes in directions of principal curvatures

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 κ1, κ2. The Gaussian curvature is the product of the two principal curvatures Κ = κ1κ2.

The sign of the Gaussian curvature can be used to characterise the surface.

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.

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 the geometry of the surface is spherical geometry. Spheres and patches of spheres have this geometry, but there exist other examples as well, such as the football.

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, f, of two variables, in such a way that the point p is a critical point, that is, the gradient of f vanishes (this can always be attained by a suitable rigid motion). Then the Gaussian curvature of the surface at p is the determinant of the Hessian matrix of f (being the product of the eigenvalues of the Hessian). (Recall that the Hessian is the 2×2 matrix of second derivatives.) 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 i = ∇ei is the covariant derivative and g is the metric tensor.

At a point p on a regular surface in R3, the Gaussian curvature is also given by

where S 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 sum of the angles of a triangle on a surface of negative curvature is less than that of a plane triangle. Hyperbolic triangle.svg
The sum of the angles of a triangle on a surface of negative curvature is less than that of a plane triangle.

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 π. The sum of the angles of a triangle on a surface of positive curvature will exceed π, while the sum of the angles of a triangle on a surface of negative curvature will be less than π. On a surface of zero curvature, such as the Euclidean plane, the angles will sum to precisely π radians.

A more general result is the Gauss–Bonnet theorem.

Important theorems

Theorema egregium

Gauss's Theorema egregium (Latin: "remarkable theorem") 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 R3 can be so expressed. The "remarkable", and surprising, feature of this theorem is that although the definition of the Gaussian curvature of a surface S in R3 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 R3 and endowed with the Riemannian metric given by the first fundamental form. Suppose that the image of the embedding is a surface S in R3. A local isometry is a diffeomorphism f : UV between open regions of R3 whose restriction to SU is an isometry onto its image. Theorema egregium is then stated as follows:

The Gaussian curvature of an embedded smooth surface in R3 is invariant under the local isometries.

For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the "unrolled" tube (which is flat). [1] [ page needed ] On the other hand, since a sphere of radius R has constant positive curvature R−2 and a flat plane has constant curvature 0, these two surfaces are not isometric, not even locally. Thus any planar representation of even a small 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

Two surfaces which both have constant positive Gaussian curvature but with either a boundary or singular points. Surfaces of constant positive Gaussian curvature.png
Two surfaces which both have constant positive Gaussian curvature but with either a boundary or singular points.

There are other surfaces which have constant positive Gaussian curvature. Manfredo do Carmo considers surfaces of revolution where , and (an incomplete Elliptic integral of the second kind). These surfaces all have constant Gaussian curvature of 1, but, for either have a boundary or a singular point. do Carmo also gives three different examples of surface with constant negative Gaussian curvature, one of which is pseudosphere. [5]

There are many other possible bounded surfaces with constant Gaussian curvature. Whilst the sphere is rigid and can not be bent using an isometry, if a small region removed, or even a cut along a small segment, then the resulting surface can be bent. Such bending preserves Gaussian curvature so any such bending of a sphere with a region removed will also have constant Gaussian curvature. [6]

Alternative formulas

See also

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