Ivancevic_Applied-Diff-Geom

502 Applied Differential Geometry: A Modern Introduction(1) time–like, if f < 0;(2) nonspacelike or causal, if f ≤ 0;(3) null or lightlike, if f = 0;(4) space–like, if f > 0.Let ∇ be the Levi–Civita connection of (M, g). Using the semi–Riemannian version of the covariant derivative operator (3.157), we getthe prolongation∇dtẋ = ∇ ẋX (4.18)of the differential system (4.17) or of any perturbation of the system (4.17)get adding to the second member X a parallel vector–field Y with respect tothe covariant derivative ∇. The prolongation by derivation represents thegeneral dynamics of the flow. The vector–field Y can be used to illustratea progression from stable to unstable flows, or converse.The vector–field X, the metric g, and the connection ∇ determine theexternal (1, 1)−tensor–fieldF = ∇X − g −1 ⊗ g(∇X), F j i = ∇ j X i − g ih g kj ∇ h X k ,(with i, j, h, k = 1, ..., n), which characterizes the helicity of vector–field Xand its flow.First we write the differential system (4.18) in the equivalent form∇dtẋ = g−1 ⊗ g(∇X) (ẋ) + F (ẋ) . (4.19)Successively we modify the differential system (4.19) as follows [Udriste(2000)]:∇dtẋ = g−1 ⊗ g(∇X)(X) + F (ẋ) , (4.20)∇dtẋ = g−1 ⊗ g(∇X) (ẋ) + F (X), (4.21)∇dtẋ = g−1 ⊗ g(∇X)(X) + F (X). (4.22)Obviously, the second–order systems (4.20), (4.21), (4.22) are prolongationsof the first–order system (4.17). Each of them is connected either to thedynamics of the field X or to the dynamics of a particle which is sensitiveto the vector–field X. Sinceg −1 ⊗ g(∇X)(X) = grad f,