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
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.
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 , ,  and  for another connections of Finsler Geometry
, and ,  and  for the interesting application of Finsler Geometry
A thorough study of Finsler geometry
and Clifford algebras has been undertaken by Vacaru  where Clifford/spinor structures were defined with respect to Nonlinear connections associated with certain nonholonomic modifications of Riemann-Cartan gravity.