Curvature

Curvature is the amount by which a curve, surface, or other manifold deviates from a straight line or (hyper)plane.

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.

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 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

<math>

K = \lim_{r \rarr 0} (2 \pi r - C(r)) \cdot \frac{3}{\pi r^3}</math>

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.

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 Biancci 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