Manifolds | Geodesic (mathematics)

Geodesic curvature

In Riemannian geometry, the geodesic curvature of a curve 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 , the geodesic curvature is just the usual curvature of (see below). However, when the curve is restricted to lie on a submanifold of (e.g. for curves on surfaces), geodesic curvature refers to the curvature of in and it is different in general from the curvature of in the ambient manifold . The (ambient) curvature of depends on two factors: the curvature of the submanifold in the direction of (the normal curvature ), which depends only on the direction of the curve, and the curvature of seen in (the geodesic curvature ), which is a second order quantity. The relation between these is . In particular geodesics on have zero geodesic curvature (they are "straight"), so that , which explains why they appear to be curved in ambient space whenever the submanifold is. (Wikipedia).

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Related pages

Covariant derivative | Manifold | Gauss–Codazzi equations | Darboux frame | Gauss–Bonnet theorem | Geodesic | Curvature | Levi-Civita connection | Riemannian geometry | Second fundamental form