Geodesic curvature
In Riemannian geometry, the geodesic curvature of a curve
measures how far the curve is from being a geodesic. In a given manifold
, the geodesic curvature is just the usual curvature of
(see below), but when
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 from 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
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
.
Example
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
- 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
References
- 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), "Geodesic curvature", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.