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In [[Riemannian geometry]], the '''geodesic curvature''' <math>k_g</math> of a curve <math>\gamma</math> measures how far the curve is from being a [[geodesic]]. In a given manifold <math>\bar{M}</math>, the '''geodesic curvature''' is just the usual '''curvature''' of <math>\gamma</math> (see below), but when <math>\gamma</math> is restricted to lie on a submanifold <math>M</math> of <math>\bar{M}</math> (e.g. for [[Curvature#Curves on surfaces|curves on surfaces]]), geodesic curvature refers to the curvature of <math>\gamma</math> in <math>M</math> and it is different in general from the curvature of <math>\gamma</math> in the ambient manifold <math>\bar{M}</math>. The (ambient) curvature <math>k</math> of <math>\gamma</math> depends on two factors: the curvature of the submanifold <math>M</math> in the direction of <math>\gamma</math> (the [[normal curvature]] <math>k_n</math>), which depends only from the direction of the curve, and the curvature of <math>\gamma</math> seen in <math>M</math> (the geodesic curvature <math>k_g</math>), which is a second order quantity. The relation between these is <math>k = \sqrt{k_g^2+k_n^2}</math>. In particular geodesics on <math>M</math> have zero geodesic curvature (they are "straight"), so that <math>k=k_n</math>, which explains why they appear to be curved in ambient space whenever the submanifold is.

==Definition==
Consider a curve <math>\gamma</math> in a manifold <math>\bar{M}</math>, parametrized by [[arclength]], with unit tangent vector <math>T=d\gamma/ds</math>. Its curvature is the norm of the [[Covariant derivative#Derivative along curve|covariant derivative]] of <math>T</math>: <math>k = \|DT/ds \|</math>. If <math>\gamma</math> lies on <math>M</math>, the '''geodesic curvature''' is the norm of the projection of the covariant derivative <math>DT/ds</math> on the tangent space to the submanifold. Conversely the '''normal curvature''' is the norm of the projection of <math>DT/ds</math> on the normal bundle to the submanifold at the point considered.

If the ambient manifold is the euclidean space <math>\mathbb{R}^n</math>, then the covariant derivative <math>DT/ds</math> is just the usual derivative <math>dT/ds</math>.

==Example==
Let <math>M</math> be the unit sphere <math>S^2</math> in three dimensional Euclidean space. The normal curvature of <math>S^2</math> is identically 1, independently of the direction considered. Great circles have curvature <math>k=1</math>, so they have zero geodesic curvature, and are therefore geodesics. Smaller circles of radius <math>r</math> will have curvature <math>1/r</math> and geodesic curvature <math>k_g = \sqrt{1-r^2}/r</math>.

==Some results involving geodesic curvature==

*The geodesic curvature is no other than the usual curvature of the curve when computed intrinsically in the submanifold <math>M</math>. It does not depend on the way the submanifold <math>M</math> sits in <math>\bar{M}</math>.

* Geodesics of <math>M</math> have zero geodesic curvature, which is equivalent to saying that <math>DT/ds</math> is orthogonal to the tangent space to <math>M</math>.

*On the other hand the normal curvature depends strongly on how the submanifold lies in the ambient space, but marginally on the curve: <math>k_n</math> only depends on the point on the submanifold and the direction <math>T</math>, but not on <math>DT/ds</math>.

*In general Riemannian geometry, the derivative is computed using the [[Levi-Civita connection]] <math>\bar{\nabla}</math> of the ambient manifold: <math>DT/ds = \bar{\nabla}_T T</math>. It splits into a tangent part and a normal part to the submanifold: <math>\bar{\nabla}_T T = \nabla_T T + (\bar{\nabla}_T T)^\perp</math>. The tangent part is the usual derivative <math>\nabla_T T</math> in <math>M</math> (it is a particular case of Gauss equation in the [[Gauss-Codazzi equations]]), while the normal part is <math>\mathrm{I\!I}(T,T)</math>, where <math>\mathrm{I\!I}</math> denotes the [[second fundamental form]].

*The [[Gauss–Bonnet theorem]].

==See also==
* [[Curvature]]
* [[Darboux frame]]
* [[Gauss–Codazzi equations]]

== References ==
*{{citation | last = do Carmo|first =Manfredo P. | title=Differential Geometry of Curves and Surfaces | publisher=Prentice-Hall | year=1976 | isbn = 0-13-212589-7}}
* {{citation|first=Heinrich|last=Guggenheimer|author-link=Heinrich Guggenheimer|title=Differential Geometry|year=1977|publisher=Dover|chapter=Surfaces|isbn=0-486-63433-7}}.
* {{springer|id=G/g044070|title=Geodesic curvature|first=Yu.S.|last=Slobodyan|year=2001}}.

==External links==
* {{Mathworld|urlname=GeodesicCurvature|title=Geodesic curvature}}

[[Category:Geodesic (mathematics)]]
[[Category:Manifolds]]

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