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In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.

For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. The curvature of a curve at a point is normally a scalar quantity, that is, it is expressed by a single real number.

For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or manifold. This leads to the concepts of maximal curvature, minimal curvature, and mean curvature.

For Riemannian manifolds (of dimension at least two) that are not necessarily embedded in a Euclidean space, one can define the curvature intrinsically, that is without referring to an external space. See Curvature of Riemannian manifolds for the definition, which is done in terms of lengths of curves traced on the manifold, and expressed, using linear algebra, by the Riemann curvature tensor.

An encapsulation of surface curvature can be found in the shape operator, S, which is a self-adjoint linear operator from the tangent plane to itself (specifically, the differential of the Gauss map).

For a surface with tangent vectors X and normal N, the shape operator can be expressed compactly in index summation notation as

${\displaystyle \partial _{a}\mathbf {N} =-S_{ba}\mathbf {X} _{b}.}$

(Compare the alternative expression of curvature for a plane curve.)

The Weingarten equations give the value of S in terms of the coefficients of the first and second fundamental forms as

S, which is a self-adjoint linear operator from the tangent plane to itself (specifically, the differential of the Gauss map).

For a surface with tangent vectors X and normal N, the shape operator can be expressed compactly in index summation notation as

and normal N, the shape operator can be expressed compactly in index summation notation as

(Compare the alternative expression of curvature for a plane curve.)

The Weingarten equations give the value of S in terms of the coefficients of the first and second fundamental forms as