Geodesic curvature
Encyclopedia
In Riemannian geometry
, the geodesic curvature
of a curve lying on a submanifold
of the ambient space measures how far the curve is from being a geodesic
. (For instance it applies to curves on surfaces.) The notion of geodesic curvature allows to distinguish the part of the curvature in ambient space that is due to the submanifold (the normal curvature
) and the one that comes from the curve itself. The curvature 
of the curve is related to these two by 
. In particular geodesics have no geodesic curvature (they are straight), and that is their definition, so that 
, which explains why they appear to be curved in ambient space whenever the submanifold is.
in ambient manifold 
, parametrized by arclength 
, with unit tangent vector 
. The geodesic curvature is the norm of the projection of the derivative 
on the tangent plane 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.
be the unit sphere 
in three dimensional Euclidean space. The normal curvature of 
is identically 1. Great circles have curvature 
, which implies zero geodesic curvature, thus they are geodesics. Smaller circles of radius 
will have curvature 
and geodesic curvature 
.
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
, the geodesic curvature
of a curve lying on a submanifoldSubmanifold
In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required...
of the ambient space measures how far the curve is from being a geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
. (For instance it applies to curves on surfaces.) The notion of geodesic curvature allows to distinguish the part of the curvature in ambient space that is due to the submanifold (the normal curvature
) and the one that comes from the curve itself. The curvature of the curve is related to these two by . In particular geodesics have no geodesic curvature (they are straight), and that is their definition, so that , which explains why they appear to be curved in ambient space whenever the submanifold is.Definition
Consider a curve lying on a submanifold in ambient manifold , parametrized by arclength , with unit tangent vector . The geodesic curvature is the norm of the projection of the derivative on the tangent plane 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.Example
Let be the unit sphere in three dimensional Euclidean space. The normal curvature of is identically 1. Great circles have curvature , which implies zero geodesic curvature, thus they are geodesics. Smaller circles of radius will have curvature and geodesic curvature .Some results involving geodesic curvature
- The geodesic curvature is no 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 
.
- On the contrary 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 will be computed using the Levi-Civita connectionLevi-Civita connectionIn Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
of the ambient manifold. It splits into a tangent part and a normal part to the submanifold: 
, and the tangent part is then the usual derivative 
in 
(see Gauss equation in the Gauss-Codazzi equations).
- The Gauss–Bonnet theoremGauss–Bonnet theoremThe Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry to their topology...
.