Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 321As I p (T ′ cΩ) is positive definite for p ∈ (−∞, 0) ∪ (1, ∞) and negativedefinite for p ∈ (0, 1), it is sufficient to study the behavior of I p restrictedto T ⊥ c Ω. Since X⊥ċ and the curve c is a geodesic it followsHence, for all X ∈ T ⊥ c Ω, we have1pv p−2 I p(X) =∫ 10(g ċ, ∇X )= 0.dt[ ∥∥∥∥∇Xdt∥2− R 2 (X, ċ, X, ċ)]dt = I(X).3.11.3 Application: Finsler–Lagrangian Field TheoryIn this subsection we present generalized Finsler–Lagrangian field theory.The geometrical background of this theory relies on the notion of generalizedLagrangian space, GL n = (M, g ij (x k , y k )), which is a real nD manifold Mwith local coordinates {x i }, (i = 1, ..., n) and a symmetric fundamentalmetric tensor–field g ij = g ij (x k , y k ) of rank n and constant signature on T[Miron et. al. (1988); Miron and Anastasiei (1994)].From physical point of view, the fundamental metric tensor representsa unified gravitational field on T M, which consists of one external(x)−gravitational field spanned by points {x i }, and the one internal(y)−gravitational field spanned by directions {y i } and equipped with somemicroscopic character of the space–time structure.The field theory developed on a generalized Lagrangian space GL n relieson a fixed a priori nonlinear connection Γ = (Nj i (x, y)) on the tangent bundleT M. This plays the role of mapping operator of the internal (y)−fieldonto the external (x)−field, and prescribes the interaction between (x)−and (y)−fields. From geometrical point of view, the nonlinear connectionallows the construction of the adapted bases [Miron et. al. (1988);Miron and Anastasiei (1994)]{ δδx i =∂∂x i − N j i}∂∂y j , ∂∂y i ⊂ X (T M),{dx i , δy i = dy i + N i jdx j } ⊂ X ∗ (T M).As to the spatial structure, the most important thing is to determinethe Cartan canonical connection CΓ = (L i jk , Ci jk ) with respect to g ij, which