Introductory Works in Finsler Geometry

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An Introduction to Finsler Geometry
by Juan Carlos Álvarez Paiva and Carlos E. Durán

This short book is based on the notes for seven lectures given by the authors at the "Escuela Venezolana de Matemáticas" in the summer of 1998. Special emphasis is placed on geometric concepts, the construction of interesting examples, and the geometric interpretation of the differential invariants of Finsler surfaces. This book is not representative of the great body of work in Finsler geometry in the past 50 years and readers looking for tensor calculations will be much dissapointed. On the other hand, convex geometers, global-Riemannian geometers, and lovers of synthetic, geometric constructions are likely to find it interesting.


Crofton formulas and Gelfand transforms
by J.C. Álvarez Paiva and E. Fernandes.

The term integral geometry has come to describe two different fields of research: one, geometrical, based on the works of Blaschke, Chern, and Santaló, and another, analytical, based on the works of Radon, John, Helgason, and Gelfand. In this paper the authors bridge the gap by showing that classical integral-geometric formulas such as those of Crofton, Cauchy, and Chern can be easily and systematically obtained through the study of Radon-type transforms on double fibrations. The methods are also used to extend these formulas to non-homogeneous settings where group-theoretic techniques are no longer useful. To illustrate this point, all Finsler metrics on projective space such that hyperplanes are area-minimizing are constructed and the theory of Crofton densities developed by Busemann, Pogorelov, Gelfand, and Smirnov is substantially simplified.

Some Problems in Finsler Geometry
New Version

by Juan Carlos Álvarez Paiva

From the introduction:
... the present paper includes thirty one simply-stated open problems, as well as a survey of the more elementary and geometric chapters of Finsler geometry. It presents a detailed discussion of the notion of volume and area on Finsler manifolds with a strong bias towards the so-called Holmes-Thompson definition which, because of its symplectic nature, is easier to work with than the Hausdorff measure the other highlights of the paper are its presentation of Hilbert's fourth problem and its elementary approach to the differential invariants of Finsler surfaces. [...]

In view of the often-made criticisms of Finsler geometry — very few concrete and interesting examples, very few non-Riemannian theorems of real geometric content, and too many subindices — I have tried to include as many concrete examples, simply-stated results, and geometric constructions as possible. In this way, many of the jewels, so to speak, of Finsler geometry find their way in to the following pages.

This new version contains a more careful treatment of of areas in Finsler and Minkowski spaces, the geometry of unit spheres in normed spaces, and the symplectic equivalence of Finsler manifolds. I have removed some problems and I have added others which seem more interesting.