Higher order geometries



The higher order geometrical structures appear in a natural way as plrongations of Riemannian and Finslerian structures to the Jet bundle. The attempts of A. Kawaguchi and J. K. Singe to define Higher Order Finsler structures were not sustantiated because the conditions imposed on the matrical function impliy that the fundamental tensor is singular.
   Higher Order Geometries were introduced by Prof. Dr. R. Miron (Al. I. Cuza University, Iasi, Romania). In a joint work we have defined the notion of homogeneity on the fibres of the Jet bundle. The fundamental tensor of such a structure is not singular anymore.

Recommended bibliography:
1. R. Miron, The Geometry of Higher Order Lagrange Spaces. Applications in Mechanics and Physics, Kluwer Acad. Publ, FTP, Vol. 82, 1997.
2. R. Miron,
The Geometry of Higher Order Finsler Spaces,  Hadronic Press, USA, 1998.
3 . R. Miron, D. Hrimiuc, H. Shimada, S. V. Sabau, The Geometry of Lagrange and Hamilton Spaces, Kluwer Acad. Publ, FTP, Vol. 118, 2001.


Finsler geometry



Finsler geometry is "the geometry of families of convex sets in R^n, parametrized by a manifold"(H. Busseman). It is also sometimes described as being "just the Riemannian geometry without the quadratic restriction" (S. S. Chern).


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