# What is curvature?

• (noun): The rate of change (at a point) of the angle between a curve and a tangent to the curve.

## 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.

### Some articles on curvature:

Schur's Lemma (from Riemannian Geometry)
... 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 - 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 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 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