Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 445Now, a conformal Killing tensor on (M, g) is a symmetric trace–freetensor–field, with s indices, satisfyingor, equivalently,the trace–free part of ∇ (a V bc···d) = 0, (3.250)for some tensor–field T c···d or, equivalently,∇ (a V bc···d) = g (ab T c···d) , (3.251)∇ (a V bc···d) =sn+2s−2 g(ab ∇ e V c···d)e , (3.252)where ∇ a = g ab ∇ b (the standard convention of raising and lowering indiceswith the metric tensor g ab ). When s = 1, these equations define a conformalKilling vector.M. Eastwood proved the following Theorem: any symmetry D of theLaplacian ∆ = ∇ a ∇ a on a Riemannian manifold (M, g) is canonicallyequivalent to one whose symbol is a conformal Killing tensor [Eastwood(1991); Eastwood (2002)].3.15.3 Application: Killing Vector and Tensor Fields inMechanicsRecall from subsection 3.15 above, that on a Riemannian manifold (M, g)with the system’s kinetic energy metric tensor g = (g ij ), for any pair ofvectors V and T , the following relation holds 9∂ s 〈V, T 〉 = 〈∇ s V, T 〉 + 〈V, ∇ s T 〉, (3.253)where 〈V, T 〉 = g ij V i T j . If the curve γ(s) is a geodesic, for a generic vectorX we havewhere∂ s 〈X, ˙γ〉 = 〈∇ s X, ˙γ〉 + 〈X, ∇ s ˙γ〉 = 〈∇ s X, ˙γ〉 ≡ 〈∇ ˙γ X, ˙γ〉, (3.254)so that in components it reads(∇ ˙γ X) i = ∂ s x l ∂ x lX i + Γ i jk∂ s x j X k ,∂ s (X i ẋ i ) = ẋ i ∇ i (X j ẋ j ).9 In this subsection, the overdot denotes the derivative upon the arc–length parameters, namely () ˙ ≡ ∂ s ≡ d/ds, while ∇ s is the covariant derivative along a curve γ(s).