Hans-Bert Rademacher
We give a lower bound for the length of a non-trivial geodesic loop on a simply-connected and compact manifold of even dimension with a non-reversible Finsler metric of positive flag curvature. Harris and Paternain use this estimate in their recent paper [HP] to give a geometric characterization of dynamically convex Finsler metrics on the 2-sphere.
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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 show that the shortest closed geodesic on a 2-sphere with non-negative curvature has length bounded above by three times the diameter. We prove a new isoperimetric inequality for 2-spheres with pinched curvature; this allows us to improve our bound on the length of the shortest closed geodesic in the pinched curvature setting.
There are many equivalent definitions of Riemannian geodesics. They are naturally generalised to sub-Riemannian manifold, but become non-equivalent. We give a review of different definitions of geodesics of a sub-Riemannian manifold and interrelation between them. We recall three variational definitions of geodesics as (locally) shortest curves (Euler-Lagrange, Pontyagin and Hamilton) and three definitions of geodesics as straightest curves (d'Alembert , Levi-Civita-Schouten and Cartan-Tanaka ), used in nonholonomic mechanics and discuss their interrelations. We consider a big class of sub-Rie…
The authors found geodesics, shortest arcs, cut loci, and conjugate sets for left-invariant sub-Riemannian matric on the Lie group $SL(2)$, which is right-invariant relative to the Lie subgroup $SO(2)\subset SL(2)$ (in other words, for invariant sub-Riemannian metric on weakly symmetric space $(SL(2)\times SO(2))/SO(2)$).
Recently, F. Balacheff proved that the Calabi-Croke sphere made of two flat 1-unit-side equilateral triangles glued along their boundaries is a local extremum for the length of the shortest closed geodesic among the Riemannian spheres with conical singularities of fixed area. We give an alternative proof of this theorem, which does not make use of the uniformization theorem, and extend the result to Finsler metrics.
Let $M^n$ be a closed Riemannian manifold of dimension $n\geq 2$, with Ricci curvature $Ric \geq n-1$. We will show that any sphere of dimension $m$ in the space of closed loops on $M^n$ is homotopic to the sphere in the space of closed loops of length at most $8 πm$. It follows that the length of a shortest periodic geodesic on $M^n$ is bounded from above by $8 π(n-1)$.
In this paper, we show that for any closed 4-dimensional simply-connected Riemannian manifold $M$ with Ricci curvature $|Ric|\leq 3$, volume $vol(M)>v>0$, and diameter $diam(M)<D$, the length of a shortest closed geodesic is bounded by a function $F(v,D)$ which only depends on $v$ and $D$. The proofs of our result are based on a recent theorem of diffeomorphism finiteness of the manifolds satisfying the above conditions proven by J. Cheeger and A. Naber.
Shortest closed geodesics on three-dimensional isospectral manifolds
本文证明光滑闭三维流形的谱能给出其最短非平凡闭测地线长度的正下界，并由此推出等谱族在微分同胚意义下的光滑紧致性。
We establish sharp universal upper bounds on the length of the shortest closed geodesic on a punctured sphere with three or four ends endowed with a complete Riemannian metric of finite area. These sharp curvature-free upper bounds are expressed in terms of the area of the punctured sphere. In both cases, we describe the extremal metrics, which are modeled on the Calabi-Croke sphere or the tetrahedral sphere. We also extend these optimal inequalities for reversible and non-necessarily reversible Finsler metrics. In this setting, we obtain optimal bounds for spheres with a larger number of punc…
Geodesics and shortest arcs of some sub-Riemannian metrics on the Lie groups $SU(2)\times\mathbb{R}$ and $SO(3)\times\mathbb{R}$ with three-dimensional generating distributions
本研究针对李群 $SU(2)\times\mathbb{R}$ 与 $SO(3)\times\mathbb{R}$ 上带三维生成分布的左不变次黎曼度量，推导得到测地线、最短弧、割迹、第一共轭迹及任意元素间距离的结果。
In this paper we prove new upper bounds for the length of a shortest closed geodesic, denoted $l(M)$, on a complete, non-compact Riemannian surface $M$ of finite area $A$. We will show that $l(M) \leq 4\sqrt{2A}$ on a manifold with one end, thus improving the prior estimate of C. B. Croke, who first established that $l(M) \leq 31 \sqrt{A}$. Additionally, for a surface with at least two ends we show that $l(M) \leq 2\sqrt{2A}$, improving the prior estimate of Croke that $l(M) \leq (12+3\sqrt{2})\sqrt{A}$.
Geodesics and shortest arcs of some sub-Riemannian metrics on the Lie groups $SU(1,1)\times\mathbb{R}$ and $SO_0(2,1)\times\mathbb{R}$ with three-dimensional generating distributions
该研究针对李群$SU(1,1)\times\mathbb{R}$与$SO_0(2,1)\times\mathbb{R}$上带三维生成分布的左不变次黎曼度量，求解出测地线、最短弧、割迹及第一共轭迹，完善了相关几何结构的刻画。
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 $2\arccosh(2k+1)$.
The Shortest Geodesic Flower on a Manifold with Locally Convex Ends and Finite Volume
研究了具有局部凸端和有限体积的流形上存在非平凡测地网的性质，证明了其顶点数、边数和总长度的上界。
In every conformal class of Finsler (or Riemannian) metrics on a closed manifold there exists a residual subset of Finsler metrics, such that, with respect to the residual Finsler metrics, in any non-trivial homotopy class of free loops there is precisely one shortest geodesic loop.
Bounding Shortest Closed Geodesics with Diameter on compact 2-dimensional Orbifolds Homeomorphic to $S^2$
将长度有界扫除方法推广到紧致二维轨形，建立了最短闭轨形测地线长度与直径之间的不等式。
Shortest k-Geodesics on Hyperbolic Surfaces
本文研究双曲面上闭测地线的长度与自交数的关系，改进了现有估计，将最大自交数的渐进行界从512降至128。
Asymptotics of shortest filling closed multi-geodesics
本文研究了双曲面中闭合多测地线的渐进行为，证明了其长度与亏格和测地线长度的对数相关，并展示了在随机双曲面模型中的应用。
The authors find geodesics, shortest arcs, diameter, cut locus, and conjugate sets for left-invariant sub-Riemannian metric on the Lie group SO(3), under condition that the metric is right-invariant relative to the Lie subgroup $SO(2)\subset SO(3)$.
Shortest Geodesic Loops, Sectional Curvature, and Injectivity Radius of the Stiefel Manifold
本文确定了Stiefel流形在Hüper等人引入的单参数黎曼度量族下非平凡测地线环的最短长度，并结合截面曲率界限精确计算了该流形在广泛度量参数下的单射半径。
The author finds geodesics, shortest arcs, cut locus, and conjugate sets for left-invariant sub-Riemannian metric on the Lie group $SO_0(2,1)$ under the condition that the metric is right-invariant relative to the Lie subgroup $SO(2)\subset SO_0(2,1)$.
In this paper, we prove that on every Finsler $n$-sphere $(S^n, F)$ with reversibility $λ$ satisfying $F^2<(\frac{λ+1}λ)^2g_0$ and $l(S^n, F)\ge π(1+\frac{1}λ)$, there always exist at least $n$ prime closed geodesics without self-intersections, where $g_0$ is the standard Riemannian metric on $S^n$ with constant curvature 1 and $l(S^n, F)$ is the length of a shortest geodesic loop on $(S^n, F)$. We also study the stability of these closed geodesics.
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.
This note proves that any locally extremal non-self-conjugate geodesic loop in a Riemannian manifold is a closed geodesic. As a consequence, any complete and non-contractible Riemannian manifold with diverging injectivity radii along diverging sequences and without points conjugate to themselves, possesses a minimizing closed geodesic.
Length of closed geodesics on Riemannian manifolds with good covers
本文推广了先前结果，证明n维单连通黎曼流形若存在含N个元素的良好覆盖，其最短闭测地线长度可由仅依赖于体积V、直径D和N的函数界定。
Geodesic loops on tetrahedra in spaces of constant sectional curvature
本文研究了常截面曲率空间中四面体的测地环路，证明了球面空间中特定角度条件下简单测地环路的不存在性与存在性，以及双曲空间中正四面体上基于互质数对的简单测地环路的存在性。
Conditions for the existence of closed geodesics is a classic, much-studied subject in Riemannian geometry, with many beautiful results and powerful techniques. However, many of the techniques that work so well in that context are far less effective in Lorentzian geometry. In revisiting this problem here, we introduce the notion of timelike geodesic homotopy, a restriction to geodesics of the more standard timelike homotopy (also known as $t$-homotopy) of timelike loops on Lorentzian manifolds. This tool is combined with a local shortening/stretching of length argument to provide a number of n…
Geodesic loops and orthogonal geodesic chords without self-intersections
本文利用测地线段的扰动结果与正交测地弦的存在性定理，证明了在维数 $n\ge 3$ 的紧致流形和具有严格凸边界的 $n$-圆盘上，通用黎曼度量下的测地环路和正交测地弦无自相交。
We show that the volume of any Riemannian metric on a three sphere is bounded below by the length of the shortest closed curve that links its antipodal image. In particular, the volume is bounded below by the minimum of the length of the shortest closed geodesic and the minimal distance between antipodal points.
We study non-reversible Finsler metrics with constant flag curvature 1 on S^2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1-parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We also show, in any dimension, that the geodesic flow of a Finsler metrics with constant positive flag curvature is completely integrable. Finally, we give an example of a Finsler metric on~$S^2$ with positive flag curvature such that no two closed geodesics intersect and show that…
A consequence of the Cheeger-Gromoll splitting theorem states that for any open manifold $(M,x)$ of nonnegative Ricci curvature, if all the minimal geodesic loops at $x$ that represent elements of $π_1(M,x)$ are contained in a bounded ball, then $π_1(M,x)$ is virtually abelian. We generalize the above result: if these minimal representing geodesic loops of $π_1(M,x)$ escape from any bounded metric balls at a sublinear rate with respect to their lengths, then $π_1(M,x)$ is virtually abelian.
This paper is part of an undergraduate research project. We discuss the Heisenberg group H1, the three-dimensional space R3 equipped with one of two equivalent metrics, the Koranyi- and Carnot- Caratheodory metric. We show that the notion of length of curves for both metrics coincide, and that shortest curves, so-called geodesics, exist.
A well-known Lemma in Riemannian geometry by Klingenberg says that if $x_0$ is a minimum point of the distance function $d(p,\cdot)$ to $p$ in the cut locus $C_p$ of $p$, then either there is a minimal geodesic from $p$ to $x_0$ along which they are conjugate, or there is a geodesic loop at $p$ that smoothly goes through $x_0$. In this paper, we prove that: for any point $q$ and any local minimum point $x_0$ of $F_q(\cdot)=d(p,\cdot)+d(q,\cdot)$ in $C_p$, either $x_0$ is conjugate to $p$ along each minimal geodesic connecting them, or there is a geodesic from $p$ to $q$ passing through $x_0$. …
We study the shortest geodesics on flat cone spheres, i.e. flat metrics on the sphere with conical singularities. The length of the shortest geodesic between two singular points can be treated as a function on the moduli space of flat cone spheres with prescribed angle defects. We prove a recurrent relation on the distribution of this function with respect to Thurston's volume form on the moduli space.
This is a survey paper on Morse theory and the existence problem for closed geodesics. The free loop space plays a central role, since closed geodesics are critical points of the energy functional. As such, they can be analyzed through variational methods. The topics that we discuss include: Riemannian background, the Lyusternik-Fet theorem, the Lyusternik-Schnirelmann principle of subordinated classes, the Gromoll-Meyer theorem, Bott's iteration of the index formulas, homological computations using Morse theory, $SO(2)$- vs. $O(2)$-symmetries, Katok's examples and Finsler metrics, relations t…
Short Simple Geodesic Loops on a 2-Sphere
针对黎曼2-球面$M$，在经典定理及已有最短周期测地线长度界的基础上，证明$M$上任一点$p$处存在至少两条简单测地环路，长度分别以$8d$和$14d$为界，拓展了相关结果。
It is not known whether or not the lenth of the shortest periodic geodesic on a closed Riemannian manifold $M^n$ can be majorized by $c(n) vol^{ 1 \over n}$, or $\tilde{c}(n)d$, where $n$ is the dimension of $M^n$, $vol$ denotes the volume of $M^n$, and $d$ denotes its diameter. In this paper we will prove that for each $ε>0$ one can find such estimates for the length of a geodesic loop with with angle between $π-ε$ and $π$ with an explicit constant that depends both on $n$ and $ε$. That is, let $ε> 0$, and let $a = \lceil{ {1 \over {\sin ({ε\over 2})}}} \rceil+1 $. We will prove that there ex…
Let x and y be two (not necessarily distinct) points on a closed Riemannian manifold M of dimension n. According to a celebrated theorem by J.P. Serre there exist infinitely many geodesics between x and y. The length of the shortest of these geodesics is obviously less than the diameter of M. But what can be said about the length of the other geodesics? We conjecture that for every k there are k geodesics between x and y of length not exceeding kd, where d denotes the diameter of M.This conjecture is obviously true for round spheres and it is not difficult to prove it for all closed Riemannian…
This paper proves that in any closed Riemannian surface $M$ with diameter $d$, the length of the $k^\text{th}$-shortest geodesic between two given points $p$ and $q$ is at most $8kd$. This bound can be tightened further to $6kd$ if $p = q$. This improves prior estimates by A. Nabutovsky and R. Rotman.
Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. Bers' theorem gives a global bound on the length of the first $3g-3$ geodesics. We use the construction of Brooks and Makover of random Riemann surfaces to investigate the distribution of short ($< \log (g)$) geodesics on a random Riemann surfaces. We calculate the expected value of the shortest geodesic, and show that if one orders prime non-intersecting geodesics by length $γ_1\le γ_2\le ... \le γ_i ,...$, then for fixed $k$, if one allows the genus to go to infinity, the length of $γ_{k}$ is inde…
We show the Chas-Sullivan product (on the homology of the free loop space of a Riemannian manifold) is related to the Morse index of its closed geodesics. We construct related products in the cohomology of the free loop space and of the based loop space, and show they are nontrivial.
The action of the mapping class group of the thrice-punctured projective plane on its $\mathrm{GL}(2,\mathbb{C})$ character variety produces an algorithm for generating the simple length spectra of quasi-Fuchsian thrice-punctured projective planes. We apply this algorithm to quasi-Fuchsian representations of the corresponding fundamental group to prove: a sharp upper-bound for the length of its shortest geodesic, a McShane identity and the surprising result of non-polynomial growth for the number of simple closed geodesic lengths.
Infinitely many closed geodesic images on simply connected compact rank-one symmetric spaces
该研究证明，单连通紧致秩一对称空间（含球面、复射影空间等）上的任意光滑黎曼度量，均存在无穷多条几何不同的素闭测地线，其证明结合了长度滤子、环路旋转反转及受控切割等方法。
Counting geodesic loops on surfaces of genus at least 2 without conjugate points
本文在无共轭点的紧曲面上建立了闭测地环的渐近计数估计，并证明了更一般的扇形定理，推广了Huber、Margulis、Nicholls和Sharp等人在严格负曲率情形下的经典结果。
Large genus asymptotics for lengths of separating closed geodesics on random surfaces
本文研究了模空间上关于 Weil-Petersson 测度的随机双曲曲面几何量，证明了当亏格趋于无穷时分离 systole、半领子宽度和最短分离闭多重测地线长度的渐近行为，并讨论了相关极值量的渐近性质。
We prove an upper bound for the number of shortest closed geodesics in a closed hyperbolic manifold of any dimension in terms of its volume and systole, generalizing a theorem of Parlier for surfaces. We also obtain bounds on the number of primitive closed geodesics with length in a given interval that are uniform for all closed hyperbolic manifolds with bounded geometry. The proofs rely on the Selberg trace formula.
Mean curvature and closed geodesics in convex hypersurfaces
本文通过Birkhoff不变量给出了欧氏空间中凸超曲面的平均宽度下界，进而得到总平均曲率的尖锐下界，并刻画了球面为唯一使所有含最长弦的平面截面尽可能长的凸超曲面。
Extensions of the loop product and coproduct, the space of antipodal paths and resonances of closed geodesics
本文推广弦拓扑中的环积与环余积至对合确定的路径空间，并在偶数维球面上利用 Morse-Bott 方法完整计算，进而证明实射影空间闭测地线的共振定理。
For a Riemannian metric $g$ on the two-sphere, let $\ell_{\min}(g)$ be the length of the shortest closed geodesic and $\ell_{\max}(g)$ be the length of the longest simple closed geodesic. We prove that if the curvature of $g$ is positive and sufficiently pinched, then the sharp systolic inequalities \[ \ell_{\rm min}(g)^2 \leq π\ {\rm Area}(S^2,g) \leq \ell_{\max}(g)^2, \] hold, and each of these two inequalities is an equality if and only if the metric $g$ is Zoll. The first inequality answers positively a conjecture of Babenko and Balacheff. The proof combines arguments from Riemannian and s…
Loop space of a Kähler manifold
本文证明了 Kähler 流形的环路空间继承 Kähler 结构，且在自然度量下完备无界，并给出由各叶测地线拼接构造环路空间测地线的方法。
We study the topology of admissible-loop spaces on a step-two Carnot group G. We use a Morse-Bott theory argument to study the structure and the number of geodesics on G connecting the origin with a 'vertical' point (geodesics are critical points of the 'Energy' functional, defined on the loop space). These geodesics typically appear in families (critical manifolds). Letting the energy grow, we obtain an upper bound on the number of critical manifolds with energy bounded by s: this upper bound is polynomial in s of degree l (the corank of the distribution). Despite this evidence, we show that …
We enumerate a necessary condition for the existence of infinitely many geometrically distinct, non-constant, prime closed geodesics on an arbitrary closed Riemannian manifold $M$. That is, we show that any Riemannian metric on $M$ admits infinitely many prime closed geodesics such that the energy functional $E:ΛM\to\mathbb{R}$ has infinitely many non-degenerate critical points on the free loop space $ΛM$ of Sobolev class $H^1=W^{1,2}$. This result is obtained by invoking a handle decomposition of free loop space and using methods of cellular homology to study its topological invariants.
We investigate octonion product deformations coming from the parallelizable torsion of the 7-sphere $S^7$, obtaining a family of geometries from solutions of the Lagrangian formalism movement equations. This can be achieved by analyzing the spontaneous compactification $M_4\times S^7$, where $M_4$ is a Lorentzian $4$-dimensional manifold. Besides the usual Riemannian geometry and two others proposed by Cartan and Schouten, solutions in geometries with torsion and more general seven-dimensional spaces are obtained. Such formalism may by subsequently derived over the 7-sphere $S^7$, locally give…
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