In Riemannian geometry, the geodesic curvature
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
is restricted to lie on a submanifold
(e.g. for curves on surfaces), geodesic curvature refers to the curvature of
and it is different in general from the curvature of
in the ambient manifold
depends on two factors: the curvature of the submanifold
), which depends only on the direction of the curve, and the curvature of
), which is a second order quantity.
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
, parametrized by arclength, with unit tangent vector
Its curvature is the norm of the covariant derivative of
, 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
designates the unit normal field of
, the geodesic curvature is given by where the square brackets denote the scalar triple product.
be the unit sphere
in three-dimensional Euclidean space.
is identically 1, independently of the direction considered.
Great circles have curvature
Smaller circles of radius