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 ${\displaystyle dT/ds}$.

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

${\displaystyle k_{g}=\gamma ''(s)\cdot {\Big (}N(\gamma (s))\times \gamma '(s){\Big )}=\left[{\frac {\mathrm {d} ^{2}\gamma (s)}{\mathrm {d} s^{2}}},N(\gamma (s)),{\frac {\mathrm {d} \gamma (s)}{\mathrm {d} s}}\right]\,,}$

where the square brackets denote the scalar triple product.

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.

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.
• Slobodyan, Yu.S. (2001) [1994], "Geodesic curvature", Encyclopedia of Mathematics, EMS Press.

External links

• Weisstein, Eric W. "Geodesic curvature". MathWorld.