In Riemannian geometry, the geodesic curvature of a curve 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 , the geodesic curvature is just the usual curvature of (see below). However, when the curve is restricted to lie on a submanifold of (e.g. for curves on surfaces), geodesic curvature refers to the curvature of in and it is different in general from the curvature of in the ambient manifold . The (ambient) curvature of depends on two factors: the curvature of the submanifold in the direction of (the normal curvature ), which depends only on the direction of the curve, and the curvature of seen in (the geodesic curvature ), which is a second order quantity. The relation between these is . In particular geodesics on have zero geodesic curvature (they are "straight"), so that , which explains why they appear to be curved in ambient space whenever the submanifold is.

Definition

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Consider a curve   in a manifold  , parametrized by arclength, with unit tangent vector  . Its curvature is the norm of the covariant derivative of  :  . If   lies on  , the geodesic curvature is the norm of the projection of the covariant derivative   on the tangent space to the submanifold. Conversely the normal curvature is the norm of the projection of   on the normal bundle to the submanifold at the point considered.

If the ambient manifold is the euclidean space  , then the covariant derivative   is just the usual derivative  .

If   is unit-speed, i.e.  , and   designates the unit normal field of   along  , the geodesic curvature is given by

 

where the square brackets denote the scalar triple product.

Example

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Let   be the unit sphere   in three-dimensional Euclidean space. The normal curvature of   is identically 1, independently of the direction considered. Great circles have curvature  , so they have zero geodesic curvature, and are therefore geodesics. Smaller circles of radius   will have curvature   and geodesic curvature  .

Some results involving geodesic curvature

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  • The geodesic curvature is none other than the usual curvature of the curve when computed intrinsically in the submanifold  . It does not depend on the way the submanifold   sits in  .
  • Geodesics of   have zero geodesic curvature, which is equivalent to saying that   is orthogonal to the tangent space to  .
  • On the other hand the normal curvature depends strongly on how the submanifold lies in the ambient space, but marginally on the curve:   only depends on the point on the submanifold and the direction  , but not on  .
  • In general Riemannian geometry, the derivative is computed using the Levi-Civita connection   of the ambient manifold:  . It splits into a tangent part and a normal part to the submanifold:  . The tangent part is the usual derivative   in   (it is a particular case of Gauss equation in the Gauss-Codazzi equations), while the normal part is  , where   denotes the second fundamental form.
  • The Gauss–Bonnet theorem.

See also

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References

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  • 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.
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