Geodesic curvature
In Riemannian geometry, the geodesic curvature [math]\displaystyle{ k_g }[/math] of a curve [math]\displaystyle{ \gamma }[/math] 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 [math]\displaystyle{ \bar{M} }[/math], the geodesic curvature is just the usual curvature of [math]\displaystyle{ \gamma }[/math] (see below). However, when the curve [math]\displaystyle{ \gamma }[/math] is restricted to lie on a submanifold [math]\displaystyle{ M }[/math] of [math]\displaystyle{ \bar{M} }[/math] (e.g. for curves on surfaces), geodesic curvature refers to the curvature of [math]\displaystyle{ \gamma }[/math] in [math]\displaystyle{ M }[/math] and it is different in general from the curvature of [math]\displaystyle{ \gamma }[/math] in the ambient manifold [math]\displaystyle{ \bar{M} }[/math]. The (ambient) curvature [math]\displaystyle{ k }[/math] of [math]\displaystyle{ \gamma }[/math] depends on two factors: the curvature of the submanifold [math]\displaystyle{ M }[/math] in the direction of [math]\displaystyle{ \gamma }[/math] (the normal curvature [math]\displaystyle{ k_n }[/math]), which depends only on the direction of the curve, and the curvature of [math]\displaystyle{ \gamma }[/math] seen in [math]\displaystyle{ M }[/math] (the geodesic curvature [math]\displaystyle{ k_g }[/math]), which is a second order quantity. The relation between these is [math]\displaystyle{ k = \sqrt{k_g^2+k_n^2} }[/math]. In particular geodesics on [math]\displaystyle{ M }[/math] have zero geodesic curvature (they are "straight"), so that [math]\displaystyle{ k=k_n }[/math], which explains why they appear to be curved in ambient space whenever the submanifold is.
Definition
Consider a curve [math]\displaystyle{ \gamma }[/math] in a manifold [math]\displaystyle{ \bar{M} }[/math], parametrized by arclength, with unit tangent vector [math]\displaystyle{ T=d\gamma/ds }[/math]. Its curvature is the norm of the covariant derivative of [math]\displaystyle{ T }[/math]: [math]\displaystyle{ k = \|DT/ds \| }[/math]. If [math]\displaystyle{ \gamma }[/math] lies on [math]\displaystyle{ M }[/math], the geodesic curvature is the norm of the projection of the covariant derivative [math]\displaystyle{ DT/ds }[/math] on the tangent space to the submanifold. Conversely the normal curvature is the norm of the projection of [math]\displaystyle{ DT/ds }[/math] on the normal bundle to the submanifold at the point considered.
If the ambient manifold is the euclidean space [math]\displaystyle{ \mathbb{R}^n }[/math], then the covariant derivative [math]\displaystyle{ DT/ds }[/math] is just the usual derivative [math]\displaystyle{ dT/ds }[/math].
If [math]\displaystyle{ \gamma }[/math] is unit-speed, i.e. [math]\displaystyle{ \|\gamma'(s)\|=1 }[/math], and [math]\displaystyle{ N }[/math] designates the unit normal field of [math]\displaystyle{ M }[/math] along [math]\displaystyle{ \gamma }[/math], the geodesic curvature is given by
- [math]\displaystyle{ 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.
Example
Let [math]\displaystyle{ M }[/math] be the unit sphere [math]\displaystyle{ S^2 }[/math] in three-dimensional Euclidean space. The normal curvature of [math]\displaystyle{ S^2 }[/math] is identically 1, independently of the direction considered. Great circles have curvature [math]\displaystyle{ k=1 }[/math], so they have zero geodesic curvature, and are therefore geodesics. Smaller circles of radius [math]\displaystyle{ r }[/math] will have curvature [math]\displaystyle{ 1/r }[/math] and geodesic curvature [math]\displaystyle{ 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]\displaystyle{ M }[/math]. It does not depend on the way the submanifold [math]\displaystyle{ M }[/math] sits in [math]\displaystyle{ \bar{M} }[/math].
- Geodesics of [math]\displaystyle{ M }[/math] have zero geodesic curvature, which is equivalent to saying that [math]\displaystyle{ DT/ds }[/math] is orthogonal to the tangent space to [math]\displaystyle{ 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]\displaystyle{ k_n }[/math] only depends on the point on the submanifold and the direction [math]\displaystyle{ T }[/math], but not on [math]\displaystyle{ DT/ds }[/math].
- In general Riemannian geometry, the derivative is computed using the Levi-Civita connection [math]\displaystyle{ \bar{\nabla} }[/math] of the ambient manifold: [math]\displaystyle{ DT/ds = \bar{\nabla}_T T }[/math]. It splits into a tangent part and a normal part to the submanifold: [math]\displaystyle{ \bar{\nabla}_T T = \nabla_T T + (\bar{\nabla}_T T)^\perp }[/math]. The tangent part is the usual derivative [math]\displaystyle{ \nabla_T T }[/math] in [math]\displaystyle{ M }[/math] (it is a particular case of Gauss equation in the Gauss-Codazzi equations), while the normal part is [math]\displaystyle{ \mathrm{I\!I}(T,T) }[/math], where [math]\displaystyle{ \mathrm{I\!I} }[/math] 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.
- Hazewinkel, Michiel, ed. (2001), "Geodesic curvature", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=G/g044070.
External links
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