On the Finsler group and an almost symplectic structure on a tangent bundle (Q1108593)

In a previous paper [Finsler geometry, Proc. Rom.-Jap. Colloq., Braşov, Iaşi, Bucureşti/Rom. 1984, 91-106 (1985; Zbl 0602.53043)] the author has found a Lie group F(n) which is called the Finsler group and has investigated tangent bundles T(M) admitting an F(n)-structure in the sense of the theory of G-structures. An F(n)- structure which is defined on T(M) as a reduction of the standard tangent structure has been called an almost Finsler structure. In the present paper the author first studies almost Finsler structures without the assignment of a nonlinear connection, and finds a necessary and sufficient condition for T(M) to admit an almost Finsler structure, which is expressed in terms of some quantities in the base manifold M. Based on the fact that the Finsler group F(n) is a subgroup of the symplectic group, and consequently T(M) admits an almost symplectic structure if it admits an almost Finsler structure, the author introduces a special 2-form on T(M), which plays an important role in the development of the theory of almost Finsler structures. In {\S} 2 the author is concerned with this 2-form and deals with the case where the 2- form is closed or has an integrating factor. Finally, {\S} 3 is devoted to the consideration on almost Hamiltonian vectors with respect to the almost symplectic structure derived from the almost Finsler structure. In the case of Finsler manifolds, Hamiltonian vectors and Hamiltonian systems are treated and Hamiltonian functions are shown concretely.