Geodesic curvature explained
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
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
kg
=\gamma''(s) ⋅ (N(\gamma(s)) x \gamma'(s))
=\left[
,
N(\gamma(s)),
\right],
where the square brackets denote the scalar
triple product.
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
.
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
.
of the ambient manifold:
. It splits into a tangent part and a normal part to the submanifold:
\bar{\nabla}TT=\nablaTT+(\bar{\nabla}TT)\perp
. 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.
See also
References