Geodesic Curvature

In Riemannian geometry, the geodesic curvature of a curve lying on a submanifold of the ambient space measures how far the curve is from being a geodesic. (For instance it applies to curves on surfaces.) The notion of geodesic curvature allows to distinguish the part of the curvature in ambient space that is due to the submanifold (the normal curvature ) and the one that comes from the curve itself. The curvature of the curve is related to these two by . In particular geodesics have zero geodesic curvature (they are "straight"), and that is their definition, so that, which explains why they appear to be curved in ambient space whenever the submanifold is.