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 $\overline{M}$, the geodesic curvature is just the usual curvature of $\gamma$ (see below). However, when the curve $\gamma$ is restricted to lie on a submanifold $M$ of $\overline{M}$ (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 $\overline{M}$. 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 $\gamma$ 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 $\overline{M}$, parametrized by arclength, with unit tangent vector $T=d\gamma /ds$. Its curvature is the norm of the covariant derivative of $T$: $k=\Vert DT/ds\Vert$. 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 $dT/ds$.

If $\gamma$ is unit-speed, i.e. $\Vert {\gamma }^{\prime }(s)\Vert =1$, and $N$ designates the unit normal field of $M$ along $\gamma$, the geodesic curvature is given by

$k_g={\gamma }^{″}(s)\cdot (N(\gamma (s))\times {\gamma }^{\prime }(s))=[\frac{{\mathrm{d}}^2\gamma (s)}{\mathrm{d}{s}^2},N(\gamma (s)),\frac{\mathrm{d}\gamma (s)}{\mathrm{d}s}],$

where the square brackets denote the scalar triple product.

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 $M$ sits in $\overline{M}$.
• 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 $\overline{\mathrm{\nabla }}$ of the ambient manifold: $DT/ds={\overline{\mathrm{\nabla }}}_{T}T$. It splits into a tangent part and a normal part to the submanifold: ${\overline{\mathrm{\nabla }}}_{T}T={\mathrm{\nabla }}_{T}T+({\overline{\mathrm{\nabla }}}_{T}T{)}^{\perp }$. The tangent part is the usual derivative ${\mathrm{\nabla }}_{T}T$ in $M$ (it is a particular case of Gauss equation in the Gauss-Codazzi equations), while the normal part is $\mathrm{I}\mathrm{I}(T,T)$, where $\mathrm{I}\mathrm{I}$ denotes the second fundamental form.
• The Gauss–Bonnet theorem.

See also

• Curvature
• Darboux frame
• Gauss–Codazzi equations

References

• do Carmo, Manfredo P. (1976), Differential Geometry of Curves and Surfaces, Prentice-Hall, ISBN 0-13-212589-7 
• Guggenheimer, Heinrich (1977), "Surfaces", Differential Geometry, Dover, ISBN 0-486-63433-7 .
• Hazewinkel, Michiel, ed. (2001), "Geodesic curvature", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=G/g044070 .

External links

• Weisstein, Eric W.. "Geodesic curvature". http://mathworld.wolfram.com/GeodesicCurvature.html.