Geodesic curvature

In differential geometry, the geodesic curvature vector is a property of curves in a metric space which reflects the deviance of the curve from following the shortest arc length distance along each infinitesimal segment of its length.

The vector is defined as follows: at a point P on a curve C, the geodesic curvature vector k_g is the curvature vector k of the projection of the curve C onto the tangent plane at P.

The scalar magnitude of the geodesic curvature vector is simply called the geodesic curvature k_g. A curve for which the geodesic curvature is everywhere vanishing is called a geodesic.

Some theorems involving geodesic curvature

• At a point P on a curve C, the geodesic curvature vector k_g is the projection of the curvature vector k of C at P onto the tangent plane at P.

• The Gauss-Bonnet theorem.

See also: Geodesic curvature, Arc length, Curvature vector, Curve, Differential geometry, Gauss-Bonnet theorem, Geodesic, Infinitesimal, Metric space, Tangent plane