In this talk we present, in the Finslerian framework, two generalizations of the Myers theorem. These differ by conditions on the Ricci scalar as well as by the techniques of proof. The both extend some results from the Riemannian geometry published by Gregory J. Galloway around 1980.

In the first, the condition that the Ricci scalar is bounded below by a positive constant is replaced with a weaker one that allows negative values for it.

In the second it is assumed that the integral of the Ricci scalar on [0,∞) diverges to +∞ and one infers that the manifold is compact. Its hypothesis is weaker than of the first but it is not able to provide any information on the diameter of M. The proof of the first is standard while the proof of the second is based on a result regarding the solutions of a class of second order differential equations called of Jacobi type.

The relations which occur while increasing the order and dimension in the Grobner form are investigated from multilinear spectral algebra point of view.

We consider first-order, respectively second order differential dynamical systems, and we show that any such system is associated with one of these geometric structures, such that the system's trajectories are autoparallel curves with respect to the associated structure.

We then consider non-holonomic dynamical systems and we show that they are in a bijective correspondence with the above geometric structures.

[1] O. Constantinescu, M. Crasmareanu,

Extended Lagrange differential equations second order for fractional order system dynamics in matrix formformal form are introduced. Two theorems are formulated.

By use presented matrix method, as special case, the fractional order chain system dynamics is considered. Two body fractional order system dynamics in plane is considered.

In this talk, we discuss on CR-submanifolds in l.c.K.-manifolds considering the relation of $\alpha$ with the CR-distribution, and give some examples of CR-submanifolds.

In Riemannian spaces the square of the fundamental metric function is given by the quadratic form of the differential dx at each point x of the space whose coefficients are functions of the position x only, while in Finsler spaces the coefficients are generalized into functions of not only the position x but also the direction dx, and while in Higher Order spaces the coefficients are more generalized into functions of x, dx, dx^2 …

The speaker and author is interested in the research of the natural or necessary extension or generalization of various geometry but less than in many researches of the formal extension or generalization.

[1] B.Y, Chen,

[2] B. Kirik and F.Ö. Zengin,

[3] K. Matsumoto,

[4] K. Matsumoto and Z. Senturk,

[5] B. Sahin,

[6] B. Sahin,

[7] K. Yano,

[1] A. Mihai and I. Mihai,

[2] I. Mihai, R. Rosca and L. Verstraelen,

[1] S.L. Druta-Romaniuc, M.I. Munteanu,

[2] M.I. Munteanu, A.I. Nistor,

[3] S.L. Druta-Romaniuc, J. Inoguchi, M.I. Munteanu, A.I. Nistor,

[1] T. Nagano,

[2] T. Nagano,

[3] T. Nagano,

[4] T. Nagano,

[1] M. Anastasiei,

[2] M. Anastasiei and A. Sandovici,

[3] S. Lang,

This generalizations are related to the invariants which implies the curvature tensor and the Bianchi Tensor, as well as the transformations of the connections pairs which preserve the ω-conjugation, respectively the absolute ω-conjugation.