Gaussian curvature

Curvature of curves

For a plane curve C, the curvature at a given point P has magnitude equal to the reciprocal of the radius of an osculating circle (a circle that "kisses" or closely touches the curve at the given point), and is a vector pointing in the direction of that circle's center. The magnitude of curvature at points on physical curves can be measured in diopters (alternative spelling: dioptre); a diopter is one per meter.

The smaller the radius r of the osculating circle, the larger the magnitude of the curvature (1/r) will be; so that where a curve is "nearly" straight, the curvature will be close to zero, and where the curve undergoes a tight turn, the curvature will be large in magnitude.

A straight line has everywhere curvature 0; a circle of radius r has everywhere curvature of magnitude 1/r.

Local expressions

For a plane curve given parametrically as the curvature is

where the dots denote differentiation respect to t.

For a plane curve given implicitly as the curvature is

that is, the divergence of the direction of the gradient of f. This last formula also gives the mean curvature of an hypersurface in euclidean space.

Curvature of surfaces

For two-dimensional surfaces embedded in R³, there are two kinds of curvature: Gaussian (or scalar) curvature, and Mean curvature. To compute these at a given point of the surface, consider the intersection of the surface with a plane containing a fixed normal vector at the point. This intersection is a plane curve and has a curvature; if we vary the plane, this curvature will change, and there are two extremal values - the maximal and the minimal curvature, called the main curvatures, 1/R₁ and 1/R₂. Here we adopt the convention that a curvature is taken to be positive if its vector points in the same direction as the surface's chosen normal, otherwise negative.

The Gaussian curvature is equal to the product 1/R₁R₂. It has the dimension of 1/length² and is everywhere positive for spheres, everywhere negative for hyperboloids and everywhere zero for planes. It determines whether a surface has elliptic (when it is positive) or hyperbolic (when it is negative) geometry at a point.

The above definition of Gaussian curvature is extrinsic in that it uses the surface's embedding in R³, normal vectors, external planes etc. Gaussian curvature is however in fact an intrinsic property of the surface, meaning it does not depend on the particular embedding of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature. Formally, Gaussian curvature only depends on the surface's structure as a Riemannian manifold. This is Gauss' celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking.

An intrinsic definition of the Gaussian curvature at a point P is the following: imagine an ant which is tied to P with a short thread of length r. She runs around P while the thread is completely stretched and measures the length C(r) of one complete trip around P. If the surface were flat, she would find C(r) = 2πr. On curved surfaces, the formula for C(r) will be different, and the Gaussian curvature K at the point P can be computed as

The integral of the Gaussian curvature over the whole surface is closely related to the surface's Euler characteristic; see the Gauss-Bonnet theorem.

The Mean curvature is equal to the sum of the main curvatures 1/R₁+1/R₂. It has the dimension of 1/length. A minimal surface like a soap film or soap bubble has mean curvature zero. The mean curvature depends on the embedding and is not an intrinstic property of a surface - for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.

Higher dimensions

In the case of higher-dimensional manifolds curvature is defined as a tensor, which depends on a connection. A connection gives a way to transport vectors (and therefore also tensors) parallelly along a given path on a manifold. Given a metric (or first fundamental form) on a manifold, there is a unique connection which preserves this metric, the Levi Civita connection, and a corresponding curvature tensor.

The curvature tensor tells you what happens if you transport a vector around a small loop. If a loop is approximated by a small parallelogram spanned by two tangent vectors, then transporting a vector around this loop results in a linear transformation of this vector - for each pair of vectors defining a parallelogram, there is a matrix which tells you what change in a tangent space results from the parallel transport along this parallelogram. Thus, curvature is a tensor of type (1,3).

The curvature tensor has the special property that it is antisymmetric in the indices giving a loop (if you reverse your loop you will get the inverse transformation) and is thus a matrix of 2-forms.

The sectional curvature, which depends on the plane of the section, determines curvature tensor completely, and is a good way to think of curvature.

Curvature is intimately related to the holonomy group which is the group of all linear transformations of the tangent space at a point which can result from a parallel transport around a loop. The Bianchi identities restrict the possibilities for these groups, and with the exception of symmetric spaces there are few possibilities given by the Berger list.

Contraction of a full curvature tensor gives the two-valent Ricci-curvature and the scalar curvature. The Ricci-curvature can be used to define Chern classes of a manifold, which are topological invariants independent of the metric. The Einstein equations of general relativity are given in terms of scalar and Ricci curvatures.

See also:

• flatness

Referenced By