Anton Deitmar, Mark Pavey
A prime geodesic theorem is proven for singular geodesics in quotients of SL(4). This is a case where regularity assumptions of previous papers fail. As a consequence, the analysis becomes much more involved. For applications in number theory (class number asymptotics) it is, however, necessary to consider this case, too.
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A prime geodesic theorem for singular geodesics in a locally symmetric space is proved. As an application, an asymptotic formula for units in number fields is given.
The prime geodesic theorem for regular geodesics in a higher rank locally symmetric space is proved. An application to class numbers is given. The proof relies on a Lefschetz formula that is based on work of Andreas Juhl.
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 of H. Duan, Y. Long, H.B. Rademacher, W. Wang and others on the existence of two prime closed geodesics to the equivariant situation, we purpose the question if a closed Finsler manifold has only one orbit of prime closed geodesic if and only if it is a compact rank-one Riemannian symmetric space. In this paper, we study this problem in homogeneous Finsler g…
In this paper, we consider a Finsler sphere $(M,F)=(S^n,F)$ with the dimension $n>1$ and the flag curvature $K\equiv 1$. The action of the connected isometry group $G=I_o(M,F)$ on $M$, together with the action of $T=S^1$ shifting the parameter $t\in \mathbb{R}/\mathbb{Z}$ of the closed curve $c(t)$, define an action of $\hat{G}=G\times T$ on the free loop space $λM$ of $M$. In particular, for each closed geodesic, we have a $\hat{G}$-orbit of closed geodesics. We assume the Finsler sphere $(M,F)$ described above has only finite orbits of prime closed geodesics. Our main theorem claims, if the …
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.
We prove prime geodesic theorems counting primitive closed geodesics on a compact hyperbolic 3-manifold with length and holonomy in prescribed intervals, which are allowed to shrink. Our results imply effective equidistribution of holonomy and have both the rate of shrinking and the strength of the error term fully symmetric in length and holonomy.
In this paper, we prove that for every Finsler $n$-sphere $(S^n, F)$ for $n\ge 3$ 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 exists one elliptic closed geodesic whose linearized Poincaré map has at least one eigenvalue which is of the form $\exp(πi μ)$ with an irrational $μ$. Furthermore, there always exist three prime closed geodesics on any $(S^3, F)$ satisfying the above pinching condition.
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.
In this paper we study the Finsler sphere $(S^n,F)$ with $n>1$, which has constant flag curvature $K\equiv 1$ and only finite prime closed geodesics. In this case, the connected isometry group $I_0(S^n,F)$ must be a torus which dimension satisfies $0<\dim I(S^n,F) \leq[\frac{n+1}{2}]$. We will prove that the number of geometrically distinct reversible closed geodesics on $(S^n,F)$ is at least $\dim I(S^n,F)$. When $\dim I_0(S^n,F)=[\frac{n+1}{2}]$, the equality happens, and there are exactly $2[\frac{n+1}{2}]$ prime closed geodesics, which verifies Anosov conjecture in this special case.
The main goal of the paper is to prove the sandwich theorem for geodesic convex functions in a complete Riemannian manifold. Then by using this theorem we have proved an inequality in a manifold with bounded sectional curvature. Finally, we have shown that the gradient of a convex function is orthogonal to the tangent vector at some point of any geodesic.
A central limit theorem for prime geodesics on random surfaces of large genus
该研究证明了大亏格Weil–Petersson随机闭双曲曲面上，范数在特定区间内的素测地线的归一化加权计数满足渐近高斯分布，拓展了素测地线计数的相关结论。
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。
Prime scattering geodesic theorem
本文研究模曲面上散射测地线的分布与逗留时间，首次建立了散射测地线计数与素因子位于算术级数中的正整数之间的联系。
We study the asymptotics of the number N(t) of geometrically distinct closed geodesics of a Riemannian or Finsler metric on a connected sum of two compact manifolds of dimension at least three with non-trivial fundamental groups and apply this result to the prime decomposition of a three-manifold. In particular we show that the function N(t) grows at least like the prime numbers on a compact 3-manifold with infinite fundamental group. It follows that a generic Riemannian metric on a compact 3-manifold has infinitely many geometrically distinct closed geodesics. We also consider the case of a c…
A Partial $ε$-Regularity Theorem for Integral Stationary Geodesic Networks in $\mathbb{R}^n$
研究\(\mathbb{R}^n\)中整数重数平稳测地线网络序列，通过证明部分\(\epsilon\)-正则性定理，得出其收敛到重数为\(k\)直线的结论。
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.
In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the support function of the surface, and a natural potential function in hyperbolic space and hemisphere.
We show that any two non-conjugate points on a forward or backward complete connected Finsler manifold can be joined by infinitely many geodesics which are not covered by finitely many closed ones, provided that the Betti numbers of the based loop space grow unbounded.
Multiple closed geodesics on Finsler $3$-dimensional sphere
本文在所有素闭测地线Morse指数非零的条件下，证明了Anosov关于bumpy Finsler $S^3$至少存在四条素闭测地线的猜想。
Geodesic Connectedness on Statistical Manifolds with Divisible Cubic Forms
研究统计流形上具有可除三次形式的仿射连接的测地连通性，建立类似Hopf-Rinow定理的结论，并推导出统计流形的Cartan-Hadamard型定理。
Prime geodesic theorem and closed geodesics for large genus
本文研究大亏格双曲曲面模空间中素数测地线定理的误差上界及其期望值，并利用所得结果证明长度显著小于√g的闭测地线大多简单且不可分离、长度显著大于√g的闭测地线大多非简单，从而证实Lipnowski-Wright猜想。
Geometric transformation theorem, fundamental groups and monotone of numbers of almost Euclidean factors of geodesic balls
研究具有非负Ricci曲率的开流形，利用与广义Reifenberg条件相关的变换定理，在测地球几乎欧几里得因子数量单调时，证明其基本群有限生成且几乎阿贝尔，还给出无单调性时变换定理不成立的例子。
We prove two injectivity theorems for the geodesic ray transform on two-dimensional, complete, simply connected Riemannian manifolds with non-positive Gaussian curvature, also known as Cartan-Hadamard manifolds. The first theorem is concerned with bounded non-positive curvature and the second with decaying non-positive curvature.
The two main topics of this text are as follows: Firstly, three modifications of the theorem of Beltrami will be presented for diffeomorphisms between Riemannian manifolds and a space form which preserve the geodesic circles, the geodesic hyperspheres, or the minimal surfaces, respectively. Secondly, it is defined what it means for an infinitesimal deformation of a metric to preserve the geodesics up to first order, and a corresponding infinitesimal version of Beltrami's theorem is given.
In this paper, we first generalize the common index jump theorem for symplectic matrix paths proved in 2002 by Long and Zhu in [LoZ], and get an enhanced version of it. As its applications, we further prove that for a compact simply-connected manifold $(M,F)$ with a bumpy, irreversible Finsler metric $F$ and $H^*(M;{\bf Q})\cong T_{d,n+1}(x)$ for some even integer $d\ge 2$ and integer $n\ge 1$, there exist at least $\frac{dn(n+1)}{2}$ distinct non-hyperbolic closed geodesics with odd Morse indices, provided the number of distinct prime closed geodesics is finite and every prime closed geodesic…
We give new proofs of two implications in the Donaldson--Uhlenbeck--Yau theorem. Our proofs are based on geodesic rays of Hermitian metrics, inspired by recent work on the Yau--Tian--Donaldson conjecture.
We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical comparison theorems for the sub-Riemannian distance may be obtained as a limit of horizontal and vertical comparison theorems for the Riemannian distances approximations.
In this paper, we prove that for every Finsler $4$-dimensional sphere $(S^4,F)$ with reversibility $λ$ and flag curvature $K$ satisfying $\frac{25}{9}\left(\fracλ{1+λ}\right)^2<K\le 1$ with $λ<\frac{3}{2}$, either there exist at least four prime closed geodesics, or there exist exactly three prime non-hyperbolic closed geodesics and at least two of them are irrationally elliptic.
The main result of the paper is Egorov's theorem for transversally elliptic operators on compact foliated manifolds. This theorem is applied to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.
We use a result of J. Mather on the existence of connecting orbits for compositions of monotone twist maps of the cylinder to prove the existence of connecting geodesics on the unit tangent bundle $ST^2$ of the 2-torus in regions without invariant tori.
Let $(M,g)$ be a simple Riemannian manifold. Under the assumption that the metric $g$ is real-analytic, it is shown that if the geodesic ray transform of a function $f\in L^{2}(M)$ vanishes on an appropriate open set of geodesics, then $f=0$ on the set of points lying on these geodesics. The approach is based on a microlocal version of unique continuation of analytic functions.
Let $(M,g)$ be a simple Riemannian manifold with boundary and consider the geodesic ray transform of symmetric 2-tensor fields. Let the integral of $f$ along maximal geodesics vanish on an appropriate open subset of the space of geodesics in $M$. Under the assumption that the metric $g$ is real-analytic, it is shown that there exists a vector field $v$ satisfying $f=dv$ on the set of points lying on these geodesics and $v=0$ on the intersection of this set with the boundary $\PD M$ of the manifold $M$. Using this result, a Helgason's type of a support theorem for the geodesic ray transform is …
In this article we construct L--A representations of geodesic flows on quadrics and of billiard problems within ellipsoids in the pseudo--Euclidean spaces. A geometric interpretation of the integrability analogous to the classical Chasles theorem for symmetric ellipsoids is given. We also consider a generalization of the billiard within arbitrary quadric allowing virtual billiard reflections.
Bumpy Metrics Theorem for Geodesic Nets
本文研究平稳测地网的非退化性问题，证明光滑流形上的Baire泛型黎曼度量下，所有连通嵌入平稳测地网均非退化，推广了闭测地线的相关结果。
Comparison and Rigidity Theorems for geodesic curvatures in two dimensional Alexandrov spaces
本文研究了二维Alexandrov空间边界的测地曲率，证明了比较和刚性定理，扩展了已知的曲率上界和下界结果。
The Deformation Space of Geodesic Triangulations and Generalized Tutte's Embedding Theorem
本文证明了负曲率闭曲面上测地三角剖分形变空间的可缩性，通过推广Tutte嵌入定理解决了Connelly等人1983年提出的双曲曲面公开问题。
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…
Superdiffusive central limit theorems for geodesic flows on nonpositively curved surfaces
本文研究了非正曲率表面测地流的超扩散中心极限定理，证明了非标准中心极限定理和弱不变性原理，并改进了稳定/不稳定叶的正则性结果。
We extend the Besicovitch-Federer projection theorem to transversal families of mappings. As an application we show that on a certain class of Riemann surfaces with constant negative curvature and with boundary, there exist natural 2-dimensional measures invariant under the geodesic flow having 2-dimensional supports such that their projections to the base manifold are 2-dimensional but the supports of the projections are Lebesgue negligible. In particular, the union of complete geodesics has Hausdorff dimension 2 and is Lebesgue negligible.
Perturbed geodesics are trajectories of particles moving on a semi-Riemannian manifold in the presence of a potential. Our purpose here is to extend to perturbed geodesics on semi-Riemannian manifolds the well known Morse Index Theorem. When the metric is indefinite, the Morse index of the energy functional becomes infinite and hence, in order to obtain a meaningful statement, we substitute the Morse index by its relative form, given by the spectral flow of an associated family of index forms. We also introduce a new counting for conjugate points, which need not to be isolated in this context,…
Some Rigidity Theorem for Anosov Geodesic Flows
研究完备无共轭点流形上的 Anosov 测地流，证明其收缩常数下界及有限体积情形的曲率刚性，并用于推出双 Lipschitz 与 $C^1$ 共轭结果。
The computation of the index of the Hessian of the action functional in semi-Riemannian geometry at geodesics with two variable endpoints is reduced to the case of a fixed final endpoint. Using this observation, we give an elementary proof of the Morse Index Theorem for Riemannian geodesics with two variable endpoints, in the spirit of the original Morse's proof. This approach reduces substantially the effort required in the proofs of the Theorem given in previous articles on the subject. Exactly the same argument works also in the case of timelike geodesics between two submanifolds of a Loren…
Klingenberg's theorem on Anosov geodesic flows
本文填补了Klingenberg关于具有Anosov测地流的闭黎曼流形无共轭点的定理原Morse理论证明中的关键漏洞，给出该定理的完整Morse理论证明。
We prove Wadsley's theorem for foliations by closed non-lightlike geodesics. As an application we show that every pseudo-Riemannian and non-Riemannian 2-mainfold, all of whose time- or spacelike geodesics are closed, is diffeomorphic to $S^1\times \R$. Further we show that every pseudo-Riemannian 2-manifold with index 1 contains a non-closed timelike or spacelike geodesic.
We extend two celebrated theorems on closed geodesics of Riemannian 2-spheres to the larger class of reversible Finsler 2-spheres: Lusternik-Schnirelmann's theorem asserting the existence of three simple closed geodesics, and Bangert-Franks-Hingston's theorem asserting the existence of infinitely many closed geodesics. In order to prove the first theorem, we employ the generalization of Grayson's curve shortening flow developed by Angenent-Oaks.
Quantitative versions of Pansu Asymptotic Theorem and of Mitchell Tangent Theorem
本文量化研究了测地黎曼群向其度量极限收敛的速度，并改进了现有文献中的相关界。
We study bifurcation from a branch of trivial solutions of semilinear elliptic Dirichlet boundary value problems on a geodesic ball, whose radius is used as the bifurcation parameter. In the proof of our main theorem we obtain in addition a special case of an index theorem due to S. Smale.
We use the Gauss-Bonnet theorem and the triangle comparison theorems of Rauch and Toponogov to show that on compact Riemann surfaces of negative curvature period integrals of eigenfunctions $e_λ$ over geodesics go to zero at the rate of $O((\logλ)^{-1/2})$ if $λ$ are their frequencies. As discussed in \cite{CSPer}, no such result is possible in the constant curvature case if the curvature is $\ge0$. Notwithstanding, we also show that these bounds for period integrals are valid provided that integrals of the curvature over all geodesic balls of radius $r\le 1$ are pinched from above by $-δr^N$ …
The Hopf--Rinow Theorem and Mañé's Critical Value for Magnetic Geodesics on Half Lie-Groups
本文研究半李群上的右不变磁系统，定义了弱恰当右不变磁场万有覆叠的Mañé临界值，证明能量高于该值时磁流与Finsler测地流一致且Hopf--Rinow定理成立，将相关结果推广到无限维磁测地流情形。
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