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

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2 Curvature of surfaces 3 Higher dimensions |

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 diopterss (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*.

For a plane curve given parametrically as the curvature is

where the dots denote differentiation respect toFor a plane curve given implicitly as the curvature is

For two-dimensional surfaces embedded in **R**^{3}, 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/*R _{1}* and 1/

The **Gaussian curvature** is equal to the product 1/*R _{1}R_{2}*. It has the dimension of 1/length

The above definition of Gaussian curvature is *extrinsic* in that it uses the surface's embedding in **R**^{3}, 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*) = 2π*r*. 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 **Mean curvature** is equal to the sum of the main curvatures 1/*R _{1}*+1/

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 vectorss (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-formss.

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: