Improved version which takes into account the comments of the refree. In particular, we extend to compact simply connected Riemannian orbifolds the result of Gromoll-Meyer
In this paper, we try to generalize to the case of compact Riemannian orbifolds $Q$ some classical results about the existence of closed geodesics of positive length on compact Riemannian manifolds $M$. We shall also consider the problem of the existence of infinitely many geometrically distinct closed geodesics.
In the classical case the solution of those problems involve the consideration of the homotopy groups of $M$ and the homology properties of the free loop space on $M$(Morse theory). Those notions have their analogue in the case of orbifolds (see [7]). The main part of this paper will be to recall those notions and to show how the classical techniques can be adapted to the case of orbifolds.
In this note, we prove the existence of a closed geodesic of positive length on any compact developable orbifold of dimension 3, 5, or 7. The argument uses the stratification of the singular locus, and reduces the problem of existence of a closed geodesic on a compact developable orbifold to the case of even dimensional orbifolds with zero-dimensional singular locus and orbifold fundamental group infinite torsion and of odd exponent.
Manifolds all of whose geodesics are closed have been studied a lot, but there are only few examples known. The situation is different if one allows in addition for orbifold singularities. We show, nevertheless, that the abundance of new examples is restricted to even dimensions. As one key ingredient we provide a characterization of orientable manifolds among orientable orbifolds in terms of characteristic classes.
In this paper we analyze and classify the totally geodesic subspaces of finite volume quaternionic hyperbolic orbifolds and their generalizations, locally symmetric orbifolds arising from irreducible lattices in Lie groups of the form $(\mathbf{Sp}_{2n}(\mathbb{R}))^q \times \prod_{i=1}^r \mathbf{Sp}(p_i,n-p_i) \times (\mathbf{Sp}_{2n}(\mathbb{C}))^s$. We give criteria for when the totally geodesic subspaces of such an orbifold determine its commensurability class. We give a parametrization of the commensurability classes of finite volume quaternionic hyperbolic orbifolds in terms of arithmeti…
We show that the geodesic period spectrum of a Riemannian 2-orbifold all of whose geodesics are closed depends, up to a constant, only on its orbifold topology and compute it. In the manifold case we recover the fact proved by Gromoll, Grove and Pries that all prime geodesics have the same length. In the appendix we partly strengthen our result in terms of conjugacy of contact forms and explain how to deduce rigidity on the real projective plane based on a systolic inequality due to Pu. (We do not use a Lusternik-Schnirelmann type theorem on the existence of at least three simple closed geodes…
Let $Q$ be a closed manifold admitting a locally-free action of a compact Lie group $G$. In this paper we study the properties of geodesic flows on $Q$ given by Riemannian metrics which are invariant by such an action. In particular, we will be interested in the existence of geodesics which are closed up to the action of some element in the group $G$, since they project to closed magnetic geodesics on the quotient orbifold $Q/G$.
Using the theory of geodesics on surfaces of revolution, we introduce the period function. We use this as our main tool in showing that any two-dimensional orbifold of revolution homeomorphic to S^2 must contain an infinite number of geometrically distinct closed geodesics. Since any such orbifold of revolution can be regarded as a topological two-sphere with metric singularities, we will have extended Bangert's theorem on the existence of infinitely many closed geodesics on any smooth Riemannian two-sphere. In addition, we give an example of a two-sphere cone-manifold of revolution which poss…
Using the Selberg trace formula, we show that for a hyperbolic 2-orbifold, the spectrum of the Laplacian acting on functions determines, and is determined by, the following data: the volume; the total length of the mirror boundary; the number of conepoints of each order, counting a mirror corner as half a conepoint; and the number of primitive closed geodesics of each length and orientability class, counting a geodesic running along the boundary as half orientation-preserving and half orientation-reversing, and discounting imprimitive geodesics appropriately. This implies that Laplace-isospect…
In this paper, we construct families of nonisometric hyperbolic orbifolds that contain the same isometry classes of nonflat totally geodesic subspaces. The main tool is a variant of the well-known Sunada method for constructing length-isospectral Riemannian manifolds that handles totally geodesic submanifolds of multiple codimensions simultaneously.
We determine the extent to which the collection of $Γ$-Euler-Satake characteristics classify closed 2-orbifolds. In particular, we show that the closed, connected, effective, orientable 2-orbifolds are classified by the collection of $Γ$-Euler-Satake characteristics corresponding to free or free abelian $Γ$ and are not classified by those corresponding to any finite collection of finitely generated discrete groups. Similarly, we show that such a classification is not possible for non-orientable 2-orbifolds and any collection of $Γ$, nor for noneffective 2-orbifolds. As a corollary, we generate…
We show that closed arithmetic hyperbolic n-dimensional orbifolds with larger and larger volumes give rise to triangulations of the underlying spaces whose 1-skeletons are harder and harder to embed nicely in Euclidean space. To show this we generalize an inequality of Gromov and Guth to hyperbolic n-orbifolds and find nearly optimal geodesic triangulations of arithmetic hyperbolic n-orbifolds.
In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first $m$ geodesic lengths. In the present paper, we show that this phenomenon is surprisingly common in the arithmetic set…
We determine several necessary and sufficient conditions for a closed almost-complex orbifold $Q$ with cyclic local groups to admit a nonvanishing vector field. These conditions are stated separately in terms of the orbifold Euler-Satake characteristics of $Q$ and its sectors, the Euler characteristics of the underlying topological spaces of $Q$ and its sectors, and in terms of the orbifold Euler class $e_{orb}(Q)$ in Chen-Ruan orbifold cohomology $H_{orb}^\ast (Q; \R)$.
We extend to orbifolds classical results on quantum ergodicity due to Shnirelman, Colin de Verdière and Zelditch, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow of p implies quantum ergodicity for the operator P. We also prove ergodicity of the geodesic flow on a compact Riemannian orbifold of negative sectional curvature.
Topology of singular foliations of closed 1-forms on orbifolds
Instituto de Matemática e Estatística, Universidade Federal Fluminense（联邦弗拉明戈大学数学与统计研究所）；Instituto de Matemática e Estatística, Universidade de São Paulo（圣保罗大学数学与统计研究所）
In this paper by using Teichmuller theory of a sphere with four holes/orbifold points, we obtain a system of flat coordinates on the general affine cubic surface having a D_4 singularity at the origin. We show that the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere coincides with the Etingof-Ginzburg Poisson bracket on the affine D_4 cubic. We prove that this bracket is the image under the Riemann-Hilbert map of the Poisson Lie bracket on the direct sum of three copies of sl_2. We realise the action of the mapping class group by the action of the braid group on the…
We introduce a complete obstruction to the existence of nonvanishing vector fields on a closed orbifold $Q$. Motivated by the inertia orbifold, the space of multi-sectors, and the generalized orbifold Euler characteristics, we construct for each finitely generated group $Γ$ an orbifold called the space of $Γ$-sectors of $Q$. The obstruction occurs as the Euler-Satake characteristics of the $Γ$-sectors for an appropriate choice of $Γ$; in the case that $Q$ is oriented, this obstruction is expressed as a cohomology class, the $Γ$-Euler-Satake class. We also acquire a complete obstruction in the …
We provide an algebraic description of the Teichmüller space and moduli space of flat metrics on a closed manifold or orbifold and study its boundary, which consists of (isometry classes of) flat orbifolds to which the original object may collapse. It is also shown that every closed flat orbifold can be obtained by collapsing closed flat manifolds, and the collapsed limits of closed flat 3-manifolds are classified.
We obtained a complete classification of simple closed geodesics on regular tetrahedra in Lobachevsky space. Also, we evaluated the number of simple closed geodesics of length not greater than $L$ and found the asymptotic of this number as $L$ goes to infinity.
In this survey results on the behavior of simple closed geodesics on regular tetrahedra in three-dimensional spaces of constant curvature are presented.
We present some results concerning the Morse Theory of the energy function on the free loop space of the three sphere for metrics all of whose geodesics are closed. We also explain how these results relate to the Berger Conjecture in dimension three.
We prove the existence of Alexandrov embedded closed magnetic geodesics on closed hyperbolic surfaces. Closed magnetic geodesics correspond to closed curves with prescribed geodesic curvature.
If all prime closed geodesics on $(S^n,F)$ with an irreversible Finsler metric $F$ are irrationally elliptic, there exist either exactly $2\left[\frac{n+1}{2}\right]$ or infinitely many distinct closed geodesics. As an application, we show the existence of three distinct closed geodesics on bumpy Finsler $(S^3,F)$ if any prime closed geodesic has non-zero Morse index.
Given a (possibly singular) Riemannian foliation $\mathcal{F}$ with closed leaves on a compact manifold $M$ with an adapted metric, we investigate the wave trace invariants for the basic Laplacian about a non-zero period. We compare them to the wave invariants of the underlying Riemannian orbifold that exists when the leaves in the regular region are identified to points, equipped with the metric that is transverse to the leaves of the foliation. Recalling that the basic Laplacian differs from the underlying orbifold Laplacian by a term that is the mean curvature vector field associated to the…
In this paper, we show a local energy convexity of $W^{1,2}$ maps into $CAT(K)$ spaces. This energy convexity allows us to extend Colding and Minicozzi's width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of the Birkhoff-Lyusternik theorem on the existence of non-trivial closed geodesics in the Alexandrov setting.
In this article, we give multiple situations when having one or two geometrically distinct closed geodesics on a complete Riemannian cylinder $M\simeq S^1\times\mathbb{R}$ or a complete Riemannian plane $M\simeq\mathbb{R}^2$ leads to having infinitely many geometrically distinct closed geodesics. In particular, we prove that any complete cylinder with isolated closed geodesics has zero, one or infinitely many homologically visible closed geodesics; this answers a question of Alberto Abbondandolo.
Let $M$ be a compact simply connected manifold satisfying $H^*(M;\mathbf{Q})\cong T_{d,n+1}(x)$ for integers $d\ge 2$ and $n\ge 1$. If all prime closed geodesics on $(M,F)$ with an irreversible bumpy Finsler metric $F$ are elliptic, either there exist exactly $\frac{dn(n+1)}{2}$ (when $d\ge 2$ is even) or $(d+1)$ (when $d\ge 3$ is odd) distinct closed geodesics, or there exist infinitely many distinct closed geodesics.
In this paper, we prove that on every Finsler $n$-sphere $(S^n, F)$ for $n\ge 6$ with reversibility $λ$ and flag curvature $K$ satisfying $(\fracλ{λ+1})^2<K\le 1$, either there exist infinitely many prime closed geodesics or there exist $[\frac{n}{2}]-2$ closed geodesics possessing irrational average indices. If in addition the metric is bumpy, then there exist $n-3$ closed geodesics possessing irrational average indices provided the number of closed geodesics is finite.
In this paper, we construct Delaunay type constant mean curvature surfaces along a nondegenerate closed geodesic in a 3-dimensional Riemannian manifold.
In this paper we show that the geodesic flow of a Finsler metric is Anosov if and only if there exists a $C^2$ open neighborhood of Finsler metrics all of whose closed geodesics are hyperbolic. For surfaces this result holds also for Riemannian metrics. This follows from a recent result of Contreras and Mazzucchelli. Furthermore, geodesic flows of Riemannian or Finsler metrics on surfaces are $C^2$ stably ergodic if and only if they are Anosov.
We prove that for every $\Q$-homological Finsler 3-sphere $(M,F)$ with a bumpy and irreversible metric $F$, either there exist two non-hyperbolic prime closed geodesics, or there exist at least three prime closed geodesics.
A short survey on the type numbers of closed geodesics, on applications of the Morse theory to proving the existence of closed geodesics and on the recent progress in applying variational methods to the periodic problem for Finsler and magnetic geodesics
Counting Salem numbers arising from arithmetic hyperbolic orbifolds
University of Minnesota（明尼苏达大学）；Universidade Federal Fluminense（弗鲁米嫩塞联邦大学）；Centro de Investigación en Matemáticas AC（数学研究中心AC）；Universiteit Utrecht（乌得勒支大学）
We show the existence of at least two geometrically distinct closed geodesics on an n-dimensional sphere with a bumpy and non-reversible Finsler metric for n>2.
In this paper we study 1/k geodesics, those closed geodesics that minimize on all subintervals of length $L/k$, where $L$ is the length of the geodesic. We develop new techniques to study the minimizing properties of these curves on doubled polygons, and demonstrate a sequence of doubled polygons whose closed geodesics exhibit unbounded minimizing properties. We also compute the length of the shortest closed geodesic on doubled odd-gons and show that this length approaches 4 times the diameter.
We prove that a riemannian metric on the 2-sphere or the projective plane can be C2-approximated by a smooth metric whose geodesic flow has an elliptic closed geodesic.
We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.
We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.
We obtain asymptotics of sequences of the holomorphic sections of the pluricanonical bundles on ball quotients associated to closed geodesics. A nonvanishing result follows.
Metrics on Lie groupoids and differentiable stacks have been introduced recently, extending the Riemannian geometry of manifolds and orbifolds to more general singular spaces. Here we continue that theory, studying stacky curves on Riemannian stacks, measuring their length using stacky metrics, and introducing stacky geodesics. Our main results show that the length of stacky curves measure distances on the orbit space, characterize stacky geodesics as locally minimizing curves, and establish a stacky version of Hopf-Rinow Theorem. We include a concise overview that bypasses nonessential techni…
We introduce the notion of spectral flow along a periodic semi-Riemannian geodesic, as a suitable substitute of the Morse index in the Riemannian case. We study the growth of the spectral flow along a closed geodesic under iteration, determining its asymptotic behavior.
In this paper we survey on some recent results on Riemannian orbifolds and singular Riemannian foliations and combine them to conclude the existence of closed geodesics in the leaf space of some classes of singular Riemannian foliations (s.r.f.), namely s.r.f. that admit sections or have no horizontal conjugate points. We also investigate the shortening process with respect to Riemannian foliations.
In this paper, we prove that on every Finsler manifold $(M,\,F)$ with reversibility $λ$ and flag curvature $K$ satisfying $\left(\fracλ{λ+1}\right)^2<K\le 1$, there exist $[\frac{\dim M+1}{2}]$ closed geodesics. If the number of closed geodesics is finite, then there exist $[\frac{\dim M}{2}]$ non-hyperbolic closed geodesics. Moreover, there are 3 closed geodesics on $(M,\,F)$ satisfying the above pinching condition when $\dim M=3$.
This article explores closed geodesics on hyperbolic surfaces. We show that, for sufficiently large $k$, the shortest closed geodesics with at least $k$ self-intersections, taken among all hyperbolic surfaces, all lie on an ideal pair of pants and have length $2\arccosh(2k+1)$.
