News

Welcome to the Finsler Geometry Newsletter.

The aim of the Newsletter is to promote the interaction between researchers in convex, integral, metric, and symplectic geometry by providing them with a quick, accessible medium for communicating ideas, announcements, examples, counter-examples, and remarks.

Criticisms and comments should be addressed to the webmaster

The webmaster: Juan Carlos Alvarez

New postings

The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations by Yan Yan Li and Louis Nirenberg.

Minimalité de sous-variétés totalement géodésiques en géométrie finslerienne by Gautier Berck.

Antinorms and Radon curves by Horst Martini and Konrad J. Swanepoel

About Finsler Geometry

Finsler manifolds, manifolds whose tangent spaces carry a norm that varies smoothly with the base point, were born prematurely in 1854 together with the Riemannian counterparts in Riemann's ground-breaking Habilitationsvortrag. I say prematurely because in 1854 Minkowski's work on normed spaces and convex bodies was still forty three years away, and thus not even the infinitesimal geometry on which Finsler manifolds are based was understood at the time. Apparently, Riemann did not know what to make of these 'more general class' of manifolds whose element of arc-length does not originate from a scalar product and, fatefully, put in a bad word for them:

Investigation of this more general class would actually require no essential different principles, but it would be rather time-consuming and throw relatively little new light on the study of Space, especially since the results cannot be expressed geometrically

Given the awe in which we rightfully regard Riemann's achievements and uncanny geometrical intuition, it is tempting to take the above quotation out of historical context and to dismiss Finsler geometry altogether. But, if we think of the great advances in convex geometry, the calculus of variations, integral geometry, the theory of metric spaces, and symplectic geometry that have taken place since 1854, then we may be moved to reassess Riemann's statement and to consider applying these new tools to develop the subject in a way that Riemann could not have foreseen.