## Curvature

In mathematics, **curvature** refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being *flat,* or *straight* in the case of a line, but this is defined in different ways depending on the context. There is a key distinction between **extrinsic curvature**, which is defined for objects embedded in another space (usually a Euclidean space) in a way that relates to the radius of curvature of circles that touch the object, and *intrinsic curvature*, which is defined at each point in a Riemannian manifold. This article deals primarily with the first concept.

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### Some articles on curvature:

Schur's Lemma (from Riemannian Geometry)

... Then if the sectional

... Then if the sectional

**curvature**is pointwise constant, that is, there exists some function such that for all and all then is constant, and the manifold has constant ...Lipid Bilayer Mechanics -

... which structure a given lipid forms is its shape (i.e.- its intrinsic

**Curvature**... which structure a given lipid forms is its shape (i.e.- its intrinsic

**curvature**) ... Intrinsic**curvature**is defined by the ratio of the diameter of the head group to that of the tail group ... this ratio is nearly one so the intrinsic**curvature**is nearly zero ...Basic Results of Small Cancellation Theory - More General

... More generally, it is possible to define various sorts of local "

**Curvature**... More generally, it is possible to define various sorts of local "

**curvature**" on any van Kampen diagram to be - very roughly - the average excess of vertices + faces ...**Curvature**- Generalizations

... The mathematical notion of

**curvature**is also defined in much more general contexts ... Many of these generalizations emphasize different aspects of the

**curvature**as it is understood in lower dimensions ... The

**curvature**of a curve can naturally be considered as a kinematic quantity, representing the force felt by a certain observer moving along the curve analogously ...

Lipid Bilayer Mechanics - Bending Modulus

... as the energy required to deform a membrane from its intrinsic

... as the energy required to deform a membrane from its intrinsic

**curvature**to some other**curvature**... For an ideal bilayer the intrinsic**curvature**is zero, so this expression is somewhat simplified ... the more each face must deform to accommodate a given**curvature**(see bending moment) ...### More definitions of "curvature":

- (
*noun*): (medicine) a curving or bending; often abnormal.

Example:*"Curvature of the spine"*

- (
*noun*): The property possessed by the curving of a line or surface.

Synonyms: curve