Geodesic curvature

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In Riemannian geometry, the geodesic curvature k_g of a curve \gamma measures how far the curve is from being a geodesic. In a given manifold \bar{M}, the geodesic curvature is just the usual curvature of \gamma (see below), but when \gamma is restricted to lie on a submanifold M of \bar{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 \bar{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 from 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[edit]

Consider a curve \gamma in a manifold \bar{M}, parametrized by arclength, with unit tangent vector T=d\gamma/ds. Its curvature is the norm of the covariant derivative of T: k = \|DT/ds \|. 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.

Example[edit]

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 = \sqrt{1-r^2}/r.

Some results involving geodesic curvature[edit]

  • 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 the Levi-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 the Gauss-Codazzi equations), while the normal part is \mathrm{I\!I}(T,T), where \mathrm{I\!I} denotes the second fundamental form.

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