By Pfeffer, Riemannian

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11). d. 1. 1. 2. 3) on TM, we obtain Consequently, the tensorial equations Wjk = 0, h ing. 3. 12), the Lie brackets U— ,7-r give an horizontal vector fields \_dx3 ox"J if and only if &ik = 0. The previous property allows to say that Rljk is the curvature tensor field of the nonlinear connection N. 14) is the torsion of the nonlinear connection N. 3. have The almost complex structure W is integrable if and only if we # V = 0, fjk = 0. 16) Proof. , we deduce dxl oyJ 5 M d \ _ , t = Rl 6 { d —~ + t- \ dyi dy J oyl Now it follows that Afw = 0 «=* {R}^ = fjk = 0}.

1) £(c) =/>(*,§),«. ,: Remark. Starting from the property that L(x,y) = F2(x,y) is energy function of the regular Lagrangian F2, we will prove in the next chapter the following two properties. Theorem A. Along with the integral curves of the Euler-Lagrange Q equations ffr' Et (F2) = 0, y* = — we have: at dt Theorem B. 2. 1. 1) can be expressed in the form where Gl is given by (4-4) 2 Proof. 3). 4) give the integral curve of the spray The vector field S is called canonical spray of the space. 2.

The number L(c) does not depend by the changing of coordinates on TM and, by means of 1-homogeneity of the fundamental function F, L(c) does not depend on the parametrization of the curve c. So L(c) depends on c, only. We can fix a canonical parameter on the curve c, given by the arclength of c. Indeed, the function s = s(t), t € [0,1], given by is derivable, having the derivative: The Geometry of Hamilton & Lagrange Spaces 38 So the function s = s(t), t € [0,1], is invertible. Let t — t(s) be its inverse.