Geodesic curvature

InRiemannian geometry,thegeodesic curvature $k_{g}$ of a curve $\gamma$ measures how far the curve is from being ageodesic.For example, for1D 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 givenmanifold ${\bar {M}}$ ,thegeodesic curvatureis just the usualcurvatureof $\gamma$ (see below). However, when the curve $\gamma$ is restricted to lie on a submanifold $M$ of ${\bar {M}}$ (e.g. forcurves 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 $\gamma$ (thenormal 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 ${\bar {M}}$ ,parametrized byarclength,with unit tangent vector $T=d\gamma /ds$ .Its curvature is the norm of thecovariant derivativeof $T$ : $k=\|DT/ds\|$ .If $\gamma$ lies on $M$ ,thegeodesic curvatureis the norm of the projection of the covariant derivative $DT/ds$ on the tangent space to the submanifold. Conversely thenormal curvatureis 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. $\|\gamma '(s)\|=1$ ,and $N$ designates the unit normal field of $M$ along $\gamma$ ,the geodesic curvature is given by

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 scalartriple 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 ${\bar {M}}$ .
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 theLevi-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 theGauss-Codazzi equations), while the normal part is $\mathrm {I\!I} (T,T)$ ,where $\mathrm {I\!I}$ denotes thesecond fundamental form.
TheGauss–Bonnet theorem.