Recent findings in works by the PI, Co-PI and Walsh, however, indicate that in infinite dimensions there exist alternative characterisations of real Jordan algebras in terms of the

Finsler geometry of cones and their associated order structure.

So, people try to establish a modified gravitational theory described by

Finsler geometry.

Finsler geometry is the most general differential geometry, which regards Riemannian geometry as its special case, and it is just Riemannian geometry without quadratic restriction [22].

Up to now,

Finsler geometry has developed rapidly in its global and analytic aspects.

In 1980 Professor Radu Miron initiates at the University of Brasov The First National Seminar on

Finsler Geometry, which was held every two years ever since, and where Prof.

In this context,

Finsler geometry is a very strong tool than well-known Riemann geometry for modeling physical phenomena that are genuinely asymmetric and/or nonisotropic [23-26].

Today, as per differential geometry and topology, both Riemannian and non-Riemannian geometry (such as

Finsler geometry) can be used in General Relativity to understand better its geometric-folitional structure (such as Riemannian sub-manifolds and singular spaces) as well as its extensions (most ontologically and epistemologically unique, though, would be General Relativity's orthometric extensions--not just any extension--as I have alluded to elsewhere).

Projective

Finsler geometry studies equivalent Finsler metrics on the same manifold with the same geodesics as points [1].

Bejancu:

Finsler Geometry and Applications, Ellis Horwood Ltd., 1990.

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.

An important special case is when [F.sup.2] = [g.sub.ij] (x) [dx.sup.i][dx.sup.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].

These proceedings of the September 2005 symposium reflect the respect participants felt for their late colleague Makato Matsumoto and his work in

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.