In Riemannian geometry, the geodesic curvature k_g of a curve \gamma measures how far the curve is from being a geodesic. For example, for 1D curves on a 2D surface embedded in 3D space, it is the curvature of the curve projected onto the surface's tangent plane. More generally, in a given manifold \bar{M}, the geodesic curvature is just the usual curvature of \gamma (see below). However, when the curve \gamma is restricted to lie on a submanifold M of \bar{M} (e.g. for curves on surfaces), geodesic curvature refers to the curvature of \gamma in M and it is different in general from the curvature of \gamma in the ambient manifold \bar{M}. The (ambient) curvature k of \gamma depends on two factors: the curvature of the submanifold M in the direction of \gamma (the normal curvature k_n), which depends only on the direction of the curve, and the curvature of \gamma seen in M (the geodesic curvature k_g), which is a second order quantity. The relation between these is. In particular geodesics on M have zero geodesic curvature (they are "straight"), so that k=k_n, which explains why they appear to be curved in ambient space whenever the submanifold is.

Definition

Consider a curve \gamma in a manifold \bar{M}, parametrized by arclength, with unit tangent vector. Its curvature is the norm of the covariant derivative of T:. If \gamma lies on M, the geodesic curvature is the norm of the projection of the covariant derivative DT/ds on the tangent space to the submanifold. Conversely the normal curvature is the norm of the projection of DT/ds on the normal bundle to the submanifold at the point considered. If the ambient manifold is the euclidean space, then the covariant derivative DT/ds is just the usual derivative dT/ds. If \gamma is unit-speed, i.e., and N designates the unit normal field of M along \gamma, the geodesic curvature is given by where the square brackets denote the scalar triple product.

Example

Let M be the unit sphere S^2 in three-dimensional Euclidean space. The normal curvature of S^2 is identically 1, independently of the direction considered. Great circles have curvature k=1, so they have zero geodesic curvature, and are therefore geodesics. Smaller circles of radius r will have curvature 1/r and geodesic curvature.

Some results involving geodesic curvature