Английская Википедия:Geodesic curvature
In Riemannian geometry, the geodesic curvature <math>k_g</math> of a curve <math>\gamma</math> 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 <math>\bar{M}</math>, the geodesic curvature is just the usual curvature of <math>\gamma</math> (see below). However, when the curve <math>\gamma</math> is restricted to lie on a submanifold <math>M</math> of <math>\bar{M}</math> (e.g. for curves on surfaces), geodesic curvature refers to the curvature of <math>\gamma</math> in <math>M</math> and it is different in general from the curvature of <math>\gamma</math> in the ambient manifold <math>\bar{M}</math>. The (ambient) curvature <math>k</math> of <math>\gamma</math> depends on two factors: the curvature of the submanifold <math>M</math> in the direction of <math>\gamma</math> (the normal curvature <math>k_n</math>), which depends only on the direction of the curve, and the curvature of <math>\gamma</math> seen in <math>M</math> (the geodesic curvature <math>k_g</math>), which is a second order quantity. The relation between these is <math>k = \sqrt{k_g^2+k_n^2}</math>. In particular geodesics on <math>M</math> have zero geodesic curvature (they are "straight"), so that <math>k=k_n</math>, which explains why they appear to be curved in ambient space whenever the submanifold is.
Definition
Consider a curve <math>\gamma</math> in a manifold <math>\bar{M}</math>, parametrized by arclength, with unit tangent vector <math>T=d\gamma/ds</math>. Its curvature is the norm of the covariant derivative of <math>T</math>: <math>k = \|DT/ds \|</math>. If <math>\gamma</math> lies on <math>M</math>, the geodesic curvature is the norm of the projection of the covariant derivative <math>DT/ds</math> on the tangent space to the submanifold. Conversely the normal curvature is the norm of the projection of <math>DT/ds</math> on the normal bundle to the submanifold at the point considered.
If the ambient manifold is the euclidean space <math>\mathbb{R}^n</math>, then the covariant derivative <math>DT/ds</math> is just the usual derivative <math>dT/ds</math>.
If <math>\gamma</math> is unit-speed, i.e. <math>\|\gamma'(s)\|=1</math>, and <math>N</math> designates the unit normal field of <math>M</math> along <math>\gamma</math>, the geodesic curvature is given by
- <math>
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]\,,
</math> where the square brackets denote the scalar triple product.
Example
Let <math>M</math> be the unit sphere <math>S^2</math> in three-dimensional Euclidean space. The normal curvature of <math>S^2</math> is identically 1, independently of the direction considered. Great circles have curvature <math>k=1</math>, so they have zero geodesic curvature, and are therefore geodesics. Smaller circles of radius <math>r</math> will have curvature <math>1/r</math> and geodesic curvature <math>k_g = \frac{\sqrt{1-r^2}}{r}</math>.
Some results involving geodesic curvature
- The geodesic curvature is none other than the usual curvature of the curve when computed intrinsically in the submanifold <math>M</math>. It does not depend on the way the submanifold <math>M</math> sits in <math>\bar{M}</math>.
- Geodesics of <math>M</math> have zero geodesic curvature, which is equivalent to saying that <math>DT/ds</math> is orthogonal to the tangent space to <math>M</math>.
- On the other hand the normal curvature depends strongly on how the submanifold lies in the ambient space, but marginally on the curve: <math>k_n</math> only depends on the point on the submanifold and the direction <math>T</math>, but not on <math>DT/ds</math>.
- In general Riemannian geometry, the derivative is computed using the Levi-Civita connection <math>\bar{\nabla}</math> of the ambient manifold: <math>DT/ds = \bar{\nabla}_T T</math>. It splits into a tangent part and a normal part to the submanifold: <math>\bar{\nabla}_T T = \nabla_T T + (\bar{\nabla}_T T)^\perp</math>. The tangent part is the usual derivative <math>\nabla_T T</math> in <math>M</math> (it is a particular case of Gauss equation in the Gauss-Codazzi equations), while the normal part is <math>\mathrm{I\!I}(T,T)</math>, where <math>\mathrm{I\!I}</math> denotes the second fundamental form.
- The Gauss–Bonnet theorem.
See also
References
External links
