In
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 that varies smoothly from point to poin ...
, the geodesic curvature
of a curve
measures how far the curve is from being a
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
. 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
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, 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 In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any non-umbilic point of a s ...
), 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
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a different ...
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
In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves th ...
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
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundamen ...
.
*The
Gauss–Bonnet theorem
In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology.
In the simplest application, the case of a t ...
.
See also
*
Curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
*
Darboux frame In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any non-umbilic point of a s ...
*
Gauss–Codazzi equations
In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi Formulas) are fundamental formulas which link together the induced ...
References
*
* .
* .
External links
* {{Mathworld, urlname=GeodesicCurvature, title=Geodesic curvature
Geodesic (mathematics)
Manifolds