0 Extrema of p - energy functional on a Finsler manifold

In §1 the authors present the basic properties of the scalar product along a curve on a Finsler manifold. In §2 they investigate the variational formulae for the p-energy functional (p ∈ R − {0}). This concept generalises the notions of length (p = 1) and energy (p = 2) of a curve. §3 analyses the extrema of p-energy when the Finsler space has constant curvature. (F3) the fundamental tensor g ij (x, y) = 1 2 ∂ 2 F 2 ∂y i ∂y j is positive definite. (F4) F is C ∞ at every point (x, y) ∈ T M with y = 0 and continuous at every (x, 0) ∈ T M. Then, the absolute Finsler energy is F 2 (x, y) = g ij (x, y)y i y j. Let c : [a, b] → M be a C ∞ regular curve on M. For any two vector fields X(t) = X i (t) ∂ ∂x i c(t) , Y (t) = Y i (t) ∂ ∂x i c(t) along the curve c, we introduce [1], [6] the scalar product g(X, Y)(c(t)) = g ij (c(t), ˙ c(t))X i (t)Y j (t) along the curve c. Remarks. i) If X = Y , then we obtain X = g(X, X). ii) The vector fields X and Y are orthogonal along the curve c and we write X⊥Y iff g(X, Y) = 0.