multiplication loops of locally compact topological translation planes; Lie groups which are the groups topologically generated by all left and right translations of topological loops; the inverse problem of the calculus of variations for second order ordinary differential equations: existence of variational multipliers, in particular, of multipliers satisfying the Finsler homogeneity conditions, and Riemannian and Finsler metrizability; metric structures associated with Lagrangians and Finsler functions variational structures in

Finsler geometry and applications in physics (general relativity, Feynmam integral); Hamiltonian structures for homogeneous Lagrangians.

Bejancu:

Finsler Geometry and Applications, Ellis Horwood Ltd.

Linfan Mao [4, 5] showed that SG are generalizations of Pseudo-Manifold Geometries, which in their turn are generalizations of

Finsler Geometry, and which in its turn is a generalization of Riemann Geometry.

Matsumoto, Foundations of

Finsler geometry and Special Finsler spaces, Kaiseisha press, Otsu, Saikawa 1986.

j] Historical developments have conferred the name Riemannian geometry to this case while the general case, Riemannian geometry without the quadratic restriction, has been known as

Finsler geometry [24].

Contributors address such topics as two curvature-driven problems in Riemann-

Finsler geometry, curvature properties of certain metrics, a connectiveness principle in positively curved Finsler manifolds, Riemann-Finsler surfaces,

Finsler geometry in the tangent bundle, and topics in Finsler-inspired differential geometry such as perturbations of constant connection Wagner spaces, path geometries of almost-Grassmann structures, Ehresmann connections in relation to metrics and good metric derivatives and dynamical systems of the Lagrangian and Hamiltonian mechanical systems.

All of those materials have established the pseudo-manifold geometry and combinatorially

Finsler geometry or Riemannian geometry.

Our primary aim is to develop a new foundational world-geometry based on the intuitive notion of a novel, fully naturalized kind of

Finsler geometry, which extensively mimics the Eulerian description of the mechanics of continuous media with special emphasis on the world-velocity field, in the sense that the whole space-time continuum itself is taken to be globally dynamic on both microscopic and macroscopic scales.

Finsler geometry is the most natural generalization of Riemannian geometry.

Let us pass to a new

Finsler geometry on the base of the space of non-degenerated polynumbers [P.

which is equivalent to the torsion-free condition of the Chern-Rund connection in natural coordinates (see [3], [7], [8] and [9] for another connections of

Finsler Geometry, and [2], [3] and [8] for the interesting application of

Finsler Geometry).

A thorough study of

Finsler geometry and Clifford algebras has been undertaken by Vacaru [81] where Clifford/spinor structures were defined with respect to Nonlinear connections associated with certain nonholonomic modifications of Riemann-Cartan gravity.