Geodesic curvature

In Riemannian geometry, the geodesic curvature \( k_{g} \) of a curve \( \gamma \) 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 \( {\bar {M}} \), the geodesic curvature is just the usual curvature of \( \gamma \) (see below). However, when the curve \( \gamma \) is restricted to lie on a submanifold M {\displaystyle M} M of M ¯ {\displaystyle {\bar {M}}} {\bar {M}} (e.g. for curves on surfaces), geodesic curvature refers to the curvature of \( \gamma \) in M and it is different in general from the curvature of \( \gamma \) in the ambient manifold \( {\bar {M}} \). The (ambient) curvature k of \( \gamma \( depends on two factors: the curvature of the submanifold M in the direction of\( \gamma \) (the normal curvature \( k_{n}) \), which depends only on the direction of the curve, and the curvature of γ {\displaystyle \gamma } \gamma seen in M (the geodesic curvature \( k_{g}) \), which is a second order quantity. The relation between these is \( k={\sqrt {k_{g}^{2}+k_{n}^{2}}} \). 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.

Definition

Consider a curve \( \gamma \) in a manifold \( {\bar {M}} \) , parametrized by arclength, with unit tangent vector \( T=d\gamma /ds \). Its curvature is the norm of the covariant derivative of } T: \( k=\|DT/ds\| \). If \( \gamma \) lies on M, the geodesic curvature is the norm of the projection of the covariant derivative \( DT/ds \) on the tangent space to the submanifold. Conversely the normal curvature is the norm of the projection of \( DT/ds \) on the normal bundle to the submanifold at the point considered.

If the ambient manifold is the euclidean space \( \mathbb {R} ^{n} \), then the covariant derivative \( DT/ds \) is just the usual derivative \( dT/ds. \)

Example

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 \( k_{g}={\frac {\sqrt {1-r^{2}}}{r}} \).

Some results involving geodesic curvature

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 \( {\bar {M}} \).
Geodesics of M have zero geodesic curvature, which is equivalent to saying that \( DT/ds \) 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 \( DT/ds \).
In general Riemannian geometry, the derivative is computed using the Levi-Civita connection \( {\bar {\nabla }} \) of the ambient manifold: \( DT/ds={\bar {\nabla }}_{T}T \). It splits into a tangent part and a normal part to the submanifold: \( {\bar {\nabla }}_{T}T=\nabla _{T}T+({\bar {\nabla }}_{T}T)^{\perp } \). The tangent part is the usual derivative \( \nabla _{T}T \) in M (it is a particular case of Gauss equation in the Gauss-Codazzi equations), while the normal part is \( \mathrm {I\!I} (T,T) \), where \( \mathrm {I\!I} \) denotes the second fundamental form.

The Gauss–Bonnet theorem.