Просмотр кода страницы Английская Википедия:Geodesic curvature

{{Английская Википедия/Панель перехода}}
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]]. For example, for [[Curvature of curves on surfaces|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]] <math>\bar{M}</math>, the '''geodesic curvature''' is just the usual '''curvature''' of <math>\gamma</math> (see below). However, when the curve <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 on 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>.

If <math>\gamma</math> is unit-speed, i.e. <math>\|\gamma'(s)\|=1</math>, and <math>N</math> designates the unit normal field of <math>M</math> along <math>\gamma</math>, the geodesic curvature is given by
:<math>
  k_g
  = \gamma''(s) \cdot 
  \Big( N( \gamma(s)) \times \gamma'(s) \Big)
  = \left[ \frac{\mathrm{d}^2 \gamma(s)}{\mathrm{d}s^2} ,
  N(\gamma(s)) , \frac{\mathrm{d}\gamma(s)}{\mathrm{d}s} \right]\,,
</math>
where the square brackets denote the scalar [[Triple_product|triple product]].

==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 = \frac{\sqrt{1-r^2}}{r}</math>.

==Some results involving geodesic curvature==

*The geodesic curvature is none 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. |authorlink=Manfredo do Carmo | 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]]
{{Навигационная таблица/Портал/Английская Википедия}}
[[Категория:Английская Википедия]]
[[Категория:Википедия]]
[[Категория:Статья из Википедии]]
[[Категория:Статья из Английской Википедии]]