In Riemannian geometry, the geodesic curvature k g of a curve γ measures how far the curve is from being a geodesic. In a given manifold M ¯ , the geodesic curvature is just the usual curvature of γ (see below), but when γ is restricted to lie on a submanifold M of M ¯ (e.g. for curves on surfaces), geodesic curvature refers to the curvature of γ in M and it is different in general from the curvature of γ in the ambient manifold M ¯ . The (ambient) curvature k of γ depends on two factors: the curvature of the submanifold M in the direction of γ (the normal curvature k n ), which depends only on the direction of the curve, and the curvature of γ seen in M (the geodesic curvature k g ), which is a second order quantity. The relation between these is k = k g 2 + k n 2 . 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.
Consider a curve γ in a manifold M ¯ , parametrized by arclength, with unit tangent vector T = d γ / d s . Its curvature is the norm of the covariant derivative of T : k = ∥ D T / d s ∥ . If γ lies on M , the geodesic curvature is the norm of the projection of the covariant derivative D T / d s on the tangent space to the submanifold. Conversely the normal curvature is the norm of the projection of D T / d s on the normal bundle to the submanifold at the point considered.
If the ambient manifold is the euclidean space R n , then the covariant derivative D T / d s is just the usual derivative d T / d s .
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 k g = 1 − r 2 r .
The geodesic curvature is none other than the usual curvature of the curve when computed intrinsically in the submanifold M . It does not depend on the way the submanifold M sits in M ¯ .Geodesics of M have zero geodesic curvature, which is equivalent to saying that D T / d s is orthogonal to the tangent space to M .On the other hand the normal curvature depends strongly on how the submanifold lies in the ambient space, but marginally on the curve: k n only depends on the point on the submanifold and the direction T , but not on D T / d s .In general Riemannian geometry, the derivative is computed using the Levi-Civita connection ∇ ¯ of the ambient manifold: D T / d s = ∇ ¯ T T . It splits into a tangent part and a normal part to the submanifold: ∇ ¯ T T = ∇ T T + ( ∇ ¯ T T ) ⊥ . The tangent part is the usual derivative ∇ T T in M (it is a particular case of Gauss equation in the Gauss-Codazzi equations), while the normal part is I I ( T , T ) , where I I denotes the second fundamental form.The Gauss–Bonnet theorem.