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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.
Consider a curve in a manifold , parametrized by arclength, with unit tangent vector . Its curvature is the norm of the covariant derivative of : . If lies on , the geodesic curvature is the norm of the projection of the covariant derivative on the tangent space to the submanifold. Conversely the normal curvature is the norm of the projection of on the normal bundle to the submanifold at the point considered.
If the ambient manifold is the euclidean space , then the covariant derivative is just the usual derivative .
Let be the unit sphere in three-dimensional Euclidean space. The normal curvature of is identically 1, independently of the direction considered. Great circles have curvature , so they have zero geodesic curvature, and are therefore geodesics. Smaller circles of radius will have curvature and geodesic curvature .
Some results involving geodesic curvature
The geodesic curvature is none other than the usual curvature of the curve when computed intrinsically in the submanifold . It does not depend on the way the submanifold sits in .
Geodesics of have zero geodesic curvature, which is equivalent to saying that is orthogonal to the tangent space to .
On the other hand the normal curvature depends strongly on how the submanifold lies in the ambient space, but marginally on the curve: only depends on the point on the submanifold and the direction , but not on .
In general Riemannian geometry, the derivative is computed using the Levi-Civita connection of the ambient manifold: . It splits into a tangent part and a normal part to the submanifold: . The tangent part is the usual derivative in (it is a particular case of Gauss equation in the Gauss-Codazzi equations), while the normal part is , where denotes the second fundamental form.