Geodesic curvature
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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 kg 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 <math>k_g<math>. 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 kg is the projection of the curvature vector k of C at P onto the tangent plane at P.
- The Gauss-Bonnet theorem.