Related papers: The Length of a Shortest Geodesic Loop

In this paper, we investigate left-invariant geodesic orbit metrics on connected simple Lie groups, where the metrics are formed by the structures of generalized flag manifolds. We prove that all these left-invariant geodesic orbit metrics…

We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal, is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by…

In this paper we prove asymptotic estimates for closed geodesic loops on compact surfaces with no conjugate points. These generalize the classical counting results of Huber and Margulis and sector theorems for surfaces of strictly negative…

When a closed Finsler manifold admits continuous isometric actions, estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics. To generalize the works…

We consider any Finsler metric on a closed, orientable surface of genus greater than one. H. M. Morse proved that we can associate an asymptotic direction to minimal rays in the universal cover (in the Poincar\'e disc: a point on the unit…

A recently developed tool allows for a description of spacetime as a manifold with a Lorentz-invariant (lower) limit length built-in. This is accomplished in terms of geometric quantities depending on two spacetime events (bitensors) and…

A Riemannian metric is of constant curvature if and only if it is locally projectively flat. There are infinitely many locally projectively flat Finsler metrics of constant curvature, that are special solutions to the Hilbert's Fourth…

Let (M.F) be a complete Finsler manifold and P be a minimal and compact submanifold of M. Ric_k(x), x in M is a differential invariant that interpolates between the flag curvature and the Ricci curvature. We prove that if on any geodesic…

We derive the geodesic equation for relatively K\"ahler metrics on fibrations and prove that any two such metrics with fibrewise constant scalar curvature are joined by a unique smooth geodesic. We then show convexity of the log-norm…

In this paper we study the notion of geodesic curvature of smooth horizontal curves parametrized by arc lenght in the Heisenberg group, that is the simplest sub-Riemannian structure. Our goal is to give a metric interpretation of this…

In some recent papers, the relations existing between the metric properties of Randers spaces and the conformal geometry of stationary Lorentzian manifolds were discovered and investigated. In this note, we focus on the equality between the…

We consider the geodesic flow of reversible Finsler metrics on the 2-sphere and the 2-torus, whose geodesic flow has vanishing topological entropy. Following a construction of A. Katok, we discuss examples of Finsler metrics on both…

We prove that a homogeneous Finsler sphere with constant flag curvature $K\equiv1$ and a prime closed geodesic of length $2\pi$ must be Riemannian. This observation provides the evidence for the non-existence of homogeneous Bryant spheres.…

We present a viscosity approach to the min-max construction of closed geodesics on compact Riemannian manifolds of arbitrary dimension. We also construct counter-examples in dimension $1$ and $2$ to the $\varepsilon$-regularity in the…

It is known that the shortest non-simple closed geodesic on an orientable hyperbolic 2-orbifold passes through an orbifold point of the orbifold. This raises questions about minimal length non-simple closed geodesics disjoint from the…

A long-standing open problem in systolic geometry asks whether a Riemannian metric on the real projective space whose volume equals that of the canonical metric, but is not isometric to it, must necessarily carry a periodic geodesic of…

We find obstructions to the existence of Einstein metrics of non-negative sectional curvature on a smooth closed simply connected manifold of any dimension. The results are achieved by combining the classical Morse theory of the loop space…

Let $V$ be a separable Hilbert space, possibly infinite dimensional. Let $\St(p,V)$ be the Stiefel manifold of orthonormal frames of $p$ vectors in $V$, and let $\Gr(p,V)$ be the Grassmann manifold of $p$ dimensional subspaces of $V$. We…

In his famous 1981 paper, Lempert proved that given a point in a strongly convex domain the complex geodesics (i.e., the extremal disks) for the Kobayashi metric passing through that point provide a very useful fibration of the domain. In…

A conformal metric ${\rm d}s^{2}$ with finitely many conical singularities of constant Gaussian curvature $K=1$ on a compact Riemann surface is referred to as a spherical conical metric. When the associated monodromy group of ${\rm d}s^{2}$…