Geodesic curvature
In Riemannian geometry, the geodesic curvature formula_1 of a curve formula_2 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 formula_3, the geodesic curvature is just the usual curvature of formula_2 (see below). However, when the curve formula_2 is restricted to lie on a submanifold formula_6 of formula_3 (e.g. for curves on surfaces), geodesic curvature refers to the curvature of formula_2 in formula_6 and  it is different in general from the curvature of formula_2 in the ambient manifold formula_3. The (ambient) curvature formula_12 of formula_2 depends on two factors: the curvature of the submanifold formula_6 in the direction of formula_2 (the normal curvature formula_16), which depends only on the direction of the curve, and the curvature of formula_2 seen in formula_6 (the geodesic curvature formula_1), which is a second order quantity. The relation between these is formula_20. In particular geodesics on formula_6 have zero geodesic curvature (they are "straight"), so that formula_22, which explains why they appear to be curved in ambient space whenever  the submanifold is. Definition. Consider a curve formula_2 in a manifold formula_3, parametrized by arclength, with unit tangent vector formula_25. Its curvature is the norm of the covariant derivative of formula_26: formula_27. If formula_2 lies on formula_6, the geodesic curvature is the norm of the projection of the covariant derivative formula_30 on the tangent space to the submanifold. Conversely the normal curvature is the norm of the projection of formula_30 on the normal bundle to the submanifold at the point considered. If the ambient manifold is the euclidean space formula_32, then the covariant derivative formula_30 is just the usual derivative  formula_34. Example. Let formula_6 be the unit sphere formula_36 in three-dimensional Euclidean space. The normal curvature of formula_36 is identically 1, independently of the direction considered. Great circles have curvature formula_38, so they have zero geodesic curvature, and are therefore geodesics. Smaller circles of radius formula_39 will have curvature formula_40 and geodesic curvature formula_41.