Dear Reader,
The papers showcased in these pages are deemed to be of general interest to geometers. When this is not obvious from the title or abstract of paper itself, we have ventured to write a short commentary to put its content in perspective. Many important works in Finsler geometry do not explicitly mention the word "Finsler". This is specially true of works on variational calculus (elliptic parametric integrands, Tonelli lagrangians), but it is also a consequence of the opposition between the metric approach of H. Busemann and the line-element approach of E. Cartan. We have made a special effort to bring these papers to the attention of researchers interested in Finsler geometry.
The quotient girth of normed spaces, and an extension of Schäffer's dual girth conjecture to Grassmannians
Author: Dmitry Faifman
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Consider a star-shaped hypersurface $S$ in a normed space $(V,\|\cdot\|)$. Given a point $x$ on $S$, define $D_x S$ to be image of the projection the unit ball of $V$ onto $T_x S$ along the direction of $x$. Perform this construction at every point of the hypersurface and consider the union of $D_x S$ $(x \in S)$ as the unit disc bundle of a Finsler metric on $S$. Equivalently, define a Finsler metric $F: TS \longrightarrow [0, \infty)$ by setting $F(x,v)$ to be the distance from the origin to the line $v + tx$ $(t \in R)$.
Faifman presents this delightful construction as dual (in the sense that projections and intersections are dual operations in convex geometry) to the usual way of inducing a Finsler metric on a hypersurface in a normed space. Specially interesting is the case where this construction is used to define a Finsler metric on the unit sphere of the normed space. Note that in dimension greater than two this is never the standard metric induced from the embedding unless the normed space is Euclidean. Nevertheless, Faifman shows that Alvarez-Paiva's solution of the girth conjecture admits the following variant:
Theorem (Faifman). The infimum of the lengths of all closed curves on the unit sphere of $V$ that are symmetric about the origin equals the infimum of the lengths of all closed curves on the unit sphere of the dual normed space $V^*$ that are symmetric about the origin.
Moreover, this variant admits a generalization to Grasmannian manifolds whereas the original formulation does not.
Local monotonicity of Riemannian and Finsler volume with respect to boundary distances
Author: Sergei Ivanov
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This short paper represents the state of the art in boundary monotonicity for Riemannian and Finsler metrics and is probably one of the most remarkable applications of Finsler geometry to Riemannian geometry.
Contact geometry and isosystolic inequalities
Authors: Juan-Carlos Alvarez-Paiva, Florent Balacheff
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A smooth deformation $(M,F_s)$ of a closed Finsler manifold $(M,F_0)$ is said to be isosystolic if the length of the shortest closed geodesic of $(M,F_s)$ remains constant along the deformation. The Finsler manifold $(M,F_0)$ is said to be extremal if for every smooth isosystolic deformation the derivative of the function $s \mapsto {\rm vol}(M, F_s)$ vanishes at $s = 0$.
Among many other things, Alvarez-Paiva and Balacheff show that a smooth Finsler manifold is extremal if and only if all of its geodesics are closed and of the same (prime) length.
A natural Finsler—Laplace operator
Author: Thomas Barthelmé
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Good Finsler analogues of Riemannian constructions are hard to find, so it comes as a pleasant surprise that there is a definition of the Laplace operator for Finsler manifolds that is both insightful and simple. Indeed, it can even be made even simpler that in Barthelmé's paper if we use the notion of fiber integration (see pages 61-63 of Bott and Tu's Differential forms in Algebraic Topology).
Let $\pi : S^*M \rightarrow M$ be the unit co-sphere bundle of an n-dimensional orientable Finsler manifold, let $\Omega$ denote the Liouville volume form on $S^*M$, let $X$ be the Hamiltonian (or Reeb) vector field that defines the geodesic flow, and let $\pi_*$ be the fiber integration map that takes $(2n-1)$-forms in $S^*M$ to $n$-forms on $M$. If $f : M \rightarrow R$ is twice-differentiable, define the laplacian of $f$ by the formula
(1)In the left-hand side of this formula, the function $f$ is considered as a function on the unit cotangent bundle (i.e., $f$ and $f \circ \pi$ are identified) so that the Lie derivative ${\cal L }_X f \Omega$ makes sense. The definition of the Laplacian is local, so the reader should not worry about the assumption of orientability that appears in the definition above.
Barthelmé studies this Laplacian and shows that it is a self-adjoint, elliptic differential operator of order two. He also studies the Laplacian and the spectral geometry of the Katok spheres.
