Mini-workshop: Aspects of Finsler Geometry

Titles and abstracts

Lecture 1: The origins of Finsler geometry in the calculus of variations

Abstract: In Riemann's famous 1854 address "On the hypotheses that lie at the foundations of geometry", he proposed that our notions of geometry were based on optimization principles, i.e., that what we perceive as 'straight lines' are actually just the curves that minimize some functional on curves in space and that 'geometry' is just the study of the appropriate functional.

It took some time before this notion in this generality was taken seriously, with the fundamental work being done in the early 20th century by Finsler, Cartan, and Chern. By that time, so-called 'Riemannian geometry' (a very special case of what Riemann originally had in mind) had undergone extensive development and much effort was put into forcing the general 'Finsler' case to fit the pattern of the special case of 'Riemannian' geometry, with somewhat unsatisfactory results.

In this talk, I will outline an approach to Finsler geometry that starts from the point of view that the central importance of the subject is to uncover the properties of the solutions to the calculus of variations problem that underlies Finsler geometry and will explain how this leads to a more natural understanding of Finsler geometry than one often encounters in its expositions.

Lecture 2: Finsler manifolds of constant flag curvature

Abstract: Continuing in the natural development of Finsler geometry, we consider the Finslerian analog of the problem that Riemann solved for Riemannian geometry in his 1854 lecture: Finding and describing the examples of constant curvature. In the more general case, the curvature, which is the main invariant that governs the behavior of geodesics, is a more subtle concept than in the Riemannian case, and the classification of the examples of constant curvature is, correspondingly more delicate. On the other hand, it turns out that the greater richness of the family of Finsler manifolds of constant flag curvature means that it has close connections with other kinds of geometry, such as complex geometry and manifolds endowed with other interesting structures. I will describe what is known about this classification problem and some of these connections, especially with complex and Kahler geometry, in this lecture.

Lecture 3: Recent developments in special holonomy manifolds (DG Seminar)

Abstract: One of the big classification results in differential geometry from the past century has been the classification of the possible holonomies of affine manifolds, with the major first step having been taken by Marcel Berger in his 1954 thesis. However, Berger's classification was only partial, and, in the past 20 years, an extensive research effort has been expended to complete this classification and extend it in a number of ways. In this talk, after recounting the major parts of the history of the subject, I will discuss some of the recent results and surprising new examples discovered as a by-product of research into Finsler geometry. If time permits, I will also discuss some of the open problems in the subject.

Lecture 4: The geometry of rolling surfaces and non-holonomic mechanics (Colloquium)

Abstract: In mechanics, the system of a sphere rolling over a plane without slipping or twisting is a fundamental example of what is called a non-holonomic mechanical system, the study of which belongs to the subject of control theory. The more general case of one surface rolling over another without slipping or twisting is, similarly, of great interest for both practical and theoretical reasons. In this talk, which is intended for a general mathematical audience (i.e., no familiarity with control theory or differential geometry will be assumed), I will describe some of the basic features of this problem, a bit of its history, and some of the surprising developments that its study reveals, such as the unexpected appearance of the exceptional group G_2.