Ivancevic_Applied-Diff-Geom

444 Applied Differential Geometry: A Modern IntroductionKilling tensor Q ab there is an associated Killing tensor K ab given byK ab = Q ab − qg ab , (3.248)which is defined only up to the addition of a constant multiple of the inversemetric tensor g ab .Some authors define a conformal Killing tensor as a trace–free tensorP ab satisfying P (ab;c) = p (a g bc) . Note that there is no contradiction betweenthe two definitions: if P ab is a trace–free conformal Killing tensor then forany scalar field λ, P ab + λg ab is a conformal Killing tensor and converselyif Q ab is a conformal Killing tensor, its trace–free part Q ab − 1 n Qgab is atrace–free Killing tensor [Rani et. al. (2003)].Killing tensor–fields are of importance owing to their connection withquadratic first integrals of the geodesic equations: if p a is tangent to anaffinely parameterized geodesic (i.e., p a ;b pb = 0) it is easy to see thatK ab p a p b is constant along the geodesic. For conformal Killing tensorsQ ab p a p b is constant along null geodesics and here, only the trace–free partof Q ab contributes to the constants of motion. Both Killing tensors andconformal Killing tensors are also of importance in connection with the separabilityof the Hamiltonian–Jacobi equations [Conway and Hopf (1964)](as well as other PDEs).A Killing tensor is said to be reducible if it can be written as a constantlinear combination of the metric and symmetrized products of Killingvectors,K ab = a 0 g ab + a IJ ξ I(a ξ |J|b) , (3.249)where ξ I for I = 1 . . . N are the Killing vectors admitted by the manifold(M, g) and a 0 and a IJ for 1 ≤ I ≤ J ≤ N are constants. Generally one isinterested only in Killing tensors which are not reducible since the quadraticconstant of motion associated with a reducible Killing tensor is a constantlinear combination of p a p a and of pairwise products of the linear constantsof motion ξ Ia p a [Rani et. al. (2003)].More generally, any linear differential operator on a Riemannian manifold(M, g) may be written in the form [Eastwood (1991); Eastwood (2002)]D = V bc···d ∇ b ∇ c · · · ∇ d + lower order terms,where V bc···d is symmetric in its indices, and ∇ a = ∂/∂x a (differentiation incoordinates). This tensor is called the symbol of D. We shall write φ (ab···c)for the symmetric part of φ ab···c .