Geodesic curvature

In Riemannian geometry, the geodesic curvature ${\displaystyle k_{g}}$ of a curve ${\displaystyle \gamma }$ 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 ${\displaystyle {\bar {M}}}$, the geodesic curvature is just the usual curvature of ${\displaystyle \gamma }$ (see below). However, when the curve ${\displaystyle \gamma }$ is restricted to lie on a submanifold ${\displaystyle M}$ of ${\displaystyle {\bar {M}}}$ (e.g. for curves on surfaces), geodesic curvature refers to the curvature of ${\displaystyle \gamma }$ in ${\displaystyle M}$ and it is different in general from the curvature of ${\displaystyle \gamma }$ in the ambient manifold ${\displaystyle {\bar {M}}}$. The (ambient) curvature ${\displaystyle k}$ of ${\displaystyle \gamma }$ depends on two factors: the curvature of the submanifold ${\displaystyle M}$ in the direction of ${\displaystyle \gamma }$ (the normal curvature ${\displaystyle k_{n}}$), which depends only on the direction of the curve, and the curvature of ${\displaystyle \gamma }$ seen in ${\displaystyle M}$ (the geodesic curvature ${\displaystyle k_{g}}$), which is a second order quantity. The relation between these is ${\displaystyle k={\sqrt {k_{g}^{2}+k_{n}^{2}}}}$. In particular geodesics on ${\displaystyle M}$ have zero geodesic curvature (they are "straight"), so that ${\displaystyle k=k_{n}}$, which explains why they appear to be curved in ambient space whenever the submanifold is.

Definition

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

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

Example

Let ${\displaystyle M}$  be the unit sphere ${\displaystyle S^{2}}$  in three-dimensional Euclidean space. The normal curvature of ${\displaystyle S^{2}}$  is identically 1, independently of the direction considered. Great circles have curvature ${\displaystyle k=1}$ , so they have zero geodesic curvature, and are therefore geodesics. Smaller circles of radius ${\displaystyle r}$  will have curvature ${\displaystyle 1/r}$  and geodesic curvature ${\displaystyle k_{g}={\frac {\sqrt {1-r^{2}}}{r}}}$ .

Some results involving geodesic curvature

• The geodesic curvature is none other than the usual curvature of the curve when computed intrinsically in the submanifold ${\displaystyle M}$ . It does not depend on the way the submanifold ${\displaystyle M}$  sits in ${\displaystyle {\bar {M}}}$ .
• Geodesics of ${\displaystyle M}$  have zero geodesic curvature, which is equivalent to saying that ${\displaystyle DT/ds}$  is orthogonal to the tangent space to ${\displaystyle M}$ .
• On the other hand the normal curvature depends strongly on how the submanifold lies in the ambient space, but marginally on the curve: ${\displaystyle k_{n}}$  only depends on the point on the submanifold and the direction ${\displaystyle T}$ , but not on ${\displaystyle DT/ds}$ .
• In general Riemannian geometry, the derivative is computed using the Levi-Civita connection ${\displaystyle {\bar {\nabla }}}$  of the ambient manifold: ${\displaystyle DT/ds={\bar {\nabla }}_{T}T}$ . It splits into a tangent part and a normal part to the submanifold: ${\displaystyle {\bar {\nabla }}_{T}T=\nabla _{T}T+({\bar {\nabla }}_{T}T)^{\perp }}$ . The tangent part is the usual derivative ${\displaystyle \nabla _{T}T}$  in ${\displaystyle M}$  (it is a particular case of Gauss equation in the Gauss-Codazzi equations), while the normal part is ${\displaystyle \mathrm {I\!I} (T,T)}$ , where ${\displaystyle \mathrm {I\!I} }$  denotes the second fundamental form.
• The Gauss–Bonnet theorem.