Let T(M) be the tangent bundle over a Finslerian manifold M of n-dimension endowed with the Cartan connection ∇. One makes T(M) into a 2n dimensional affinely connected manifold by assigning a connection ∇cto T(M). The cross-section$$mathfrak{B}$$of a vector field V defined in M reveals in T(M) an n-dimensional submanifold and its geometry is developed by means of the affine subspace theory and of the affine collineations in the base Finsler manif
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