Curvature

Curvature measures how much a curve, surface, or space deviates from being straight or flat. A Line has zero curvature; a Circle has constant curvature. Smaller circles bend more sharply, so their curvature is larger.

Curves

For a circle of radius $R$, curvature is

$$\kappa =\frac{1}{R}.$$

More generally, curvature of a curve measures how quickly its tangent direction changes per unit length. A road that changes direction rapidly over a short distance has high curvature.

Surfaces

For a surface, curvature depends on direction. At a point, slice the surface by planes through the normal direction; each slice gives a curve and hence a curvature. Two important summaries are:

• Gaussian curvature $K$: product of the principal curvatures;
• mean curvature $H$: average of the principal curvatures.

Examples:

• A Plane has $K=0$.
• A sphere has positive curvature.
• A saddle has negative curvature.
• A cylinder has zero Gaussian curvature even though it is visibly bent, because one principal curvature is zero.

Intrinsic curvature

Intrinsic curvature can be detected by measurements made inside the space itself. On a positively curved surface, triangle angles can sum to more than $180^{\circ }$; on a negatively curved surface, less.

Curvature and manifolds

On Manifolds, curvature is encoded by the metric and its derivatives. In general relativity, curvature of spacetime replaces the Newtonian idea of a gravitational force; see Gravity as emergent from spacetime curvature and Cosmology index.