Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 447symmetries because, even though their existence is ensured by integrability,they are not easily recognizable [Clementi and Pettini (2002)].Let us now extend what has been presented so far about Killing vector–fields, trying to generalize the form of the conserved quantity along ageodesic flow from J = X i ẋ i to J = K j1j 2...j rẋ j1 ẋ j2 . . . ẋ jr , with K j1j 2...j ra tensor of rank r. Thus, we look for the conditions that entail∂ s (K j1j 2...j rẋ j1 ẋ j2 . . . ẋ jr ) = ẋ j ∇ j (K j1j 2...j rẋ j1 ẋ j2 . . . ẋ jr ) = 0. (3.258)In order to work out from this equation a condition for the existence ofa suitable tensor K j1j 2...j r, which is called a Killing tensor–field, let usfirst consider the 2r rank tensor K j1j 2...j r ẋ i1 ẋ i2 . . . ẋ ir and its covariantderivative along a geodesic [Clementi and Pettini (2002)]ẋ j ∇ j (K j1j 2...j r ẋ i1 ẋ i2 . . . ẋ ir ) = ẋ i1 ẋ i2 . . . ẋ ir ẋ j ∇ j K j1j 2...j r, (3.259)where we have again used ẋ j ∇ j ẋ i k= 0 along a geodesic, and a standardcovariant differentiation formula (see 3.10.1 above). Now, by contraction onthe indices i k and j k the 2r−rank tensor in (3.259) gives a new expressionfor (3.258), which reads∂ s (K j1j 2...j r ẋ j1 ẋ j2 . . . ẋ jr ) = ẋ j1 ẋ j2 . . . ẋ jr ẋ j ∇ (j K j1j 2...j r), (3.260)where ∇ (j K j1j 2...j r) = ∇ j K j1j 2...j r+ ∇ j1 K jj2...j r+ · · · + ∇ jr K j1j 2...j r−1j.The vanishing of (3.260), entailing the conservation of K j1j 2...j r ẋ j1 ẋ j2 . . . ẋ jralong a geodesic flow, is therefore guaranteed by the existence of a tensor–field fulfilling the conditions [Clementi and Pettini (2002)]∇ (j K j1j 2...j r) = 0. (3.261)These equations generalize (3.255) and give the definition of a Killingtensor–field on a Riemannian biodynamical manifold (M, g). These N r+1equations in (N + r − 1)!/r!(N − 1)! unknown independent components ofthe Killing tensor constitute an overdetermined system of equations. Thus,a‘priori, we can expect that the existence of Killing tensor–fields has to berather exceptional.If a Killing tensor–field exists on a Riemannian manifold, then the scalarK j1j 2...j r ˙q j1 ˙q j2 . . . ˙q jris a constant of motion for the geodesic flow on the same manifold. Withthe only difference of a more tedious combinatorics, also in this case it turns