Ivancevic_Applied-Diff-Geom

316 Applied Differential Geometry: A Modern Introduction3.11 Finsler Manifolds and Their ApplicationsRecall that Finsler geometry is such a generalization of Riemannian geometry,that is closely related to multivariable calculus of variations.3.11.1 Definition of a Finsler ManifoldLet M be a real, smooth, connected, finite–dimensional manifold. The pair(M, F ) is called a Finsler manifold iff there exists a fundamental functionF : T M → R, not necessary reversible (i.e., F (x, −y) need not be equal toF (x, y)), that satisfies the following set of axioms (see, e.g., [Udriste andNeagu (1999)]):F1 F (x, y) > 0 for all x ∈ M, y ≠ 0.F2 F (x, λy) = |λ|F (x, y) for all λ ∈ R, (x, y) ∈ T M.F3 the fundamental metric tensor g ij on M, given byg ij (x, y) = 1 ∂ 2 F 22 ∂y i ∂y j ,is positive definite.F4 F is smooth (C ∞ ) at every point (x, y) ∈ T M with y ≠ 0 and continuous(C 0 ) at every (x, 0) ∈ T M. Then, the absolute Finsler energyfunction is given byF 2 (x, y) = g ij (x, y)y i y j .Let c = c(t) : [a, b] → M be a∣smooth regular curve on∣M. For anytwo vector–fields X(t) = X i ∂(t) ∣c(t)∂xand Y (t) = Y i ∂(t) ∣c(t) i ∂xalong theicurve c = c(t), we introduce the scalar (inner) product [Chern (1996)]g(X, Y )(c) = g ij (c, ċ)X i Y jalong the curve c.In particular, if X = Y then we have ‖X‖ = √ g(X, X). The vector–fields X and Y are orthogonal along the curve c, denoted by X⊥Y , iffg(X, Y ) = 0.Let CΓ(N) = (L i jk , N j i, Ci jk) be the Cartan canonical N−linear metricconnection determined by the metric tensor g ij (x, y). The coefficients of