Cornelia Vizman
Geodesics and curvature of semidirect product groups with right invariant metrics are determined. In the special case of an isometric semidirect product, the curvature is shown to be the sum of the curvature of the two groups. A series of examples, like the magnetic extension of a group, are then considered.
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On the ambient space of a Lie group with a left invariant metric that is isometric and isomorphic to a semidirect product $\mathbb{R}^2\rtimes_A\mathbb{R}$, we consider a domain $Ω\subseteq \mathbb{R}^2\rtimes_A\{0\}$ and vertical $π$-graphs over $Ω$ and study the partial differential equation a function $u:Ω\rightarrow \mathbb{R}$ must satisfy in order to have prescribed mean curvature $H$. Using techniques from quasilinear elliptic equations we prove that if a $π-$graph has non-negative mean curvature, then it satisfy some uniform height estimates that depend on $Ω$ and on a parameter $α$, g…
The Landau-Lifshitz equation is derived as the reduction of a geodesic flow on the group of maps into the rotation group. Passing the symmetries of spatial isotropy to the reduced space is an example of semidirect product reduction by stages.
Minimal hypersurfaces in $\mathbb{S}^5$ with constant scalar curvature and zero Gauss curvature are totally geodesic
本文证明五维球面中常数量曲率且零高斯曲率的闭极小超曲面必为全测地超曲面。
Geodesics in Randers spaces of constant curvature are classified.
In this paper, we construct Delaunay type constant mean curvature surfaces along a nondegenerate closed geodesic in a 3-dimensional Riemannian manifold.
Considering Riemannian submersions, we find necessary and sufficient conditions for when sub-Riemannian normal geodesics project to curves of constant first geodesic curvature or constant first and vanishing second geodesic curvatures. We describe a canonical extension of the sub-Riemannian metric and study geometric properties of the obtained Riemannian manifold. This work contains several examples illustrating the results.
We construct real analytic flat Moebius strips of arbitrary isotopy types, whose centerlines are geodesics or lines of curvature.
The geodesic total curvature of rectifiable spherical curves is analyzed. We extend to the case of high dimension spheres the explicit formula that holds true for curves supported into the 2-sphere. For this purpose, we take advantage of some new integral-geometric formulas concerning both the Euclidean and geodesic total curvature of spherical curves.
We compute curvatures of a three-manifold formed by a Weil-Petersson geodesic in Teichmuller space.
We prove the existence of immersed closed curves of constant geodesic curvature in an arbitrary Riemannian 2-sphere for almost every prescribed curvature. To achieve this, we develop a min-max scheme for a weighted length functional.
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 notion of geodesic curvature as the first corrective term in the Taylor expansion of the distance between two close points of the curve.
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…
We consider equitorsion second type almost geodesic mappings of a non-symmetric affine connection space in this article. Using different computational methods, we obtained some invariants of these mappings. Last generalized Thomas projective parameter and Weyl projective tensor as invariants of a second type almost geodesic mapping of a non-symmetric affine connection space are further generalized here.
It is well-known that if a curve is a geodesic line of the tangent (sphere) bundle with Sasaki metric of a locally symmetric Riemannian manifold then the projected curve has all its geodesic curvatures constant. In this paper we consider the case of tangent (sphere) bundle over the real, complex and quaternionic space form and give a unified proof of the following property: all geodesic curvatures of projected curve are zero starting from k_3,k_6 and k_{10} for the real, complex and quaternionic space formes respectively.
We will develop simple relations between the arc-lengths of a pair of geodesics that share common end-points. The two geodesics differ only by the requirement that one is constrained to lie in a subspace of the parent manifold. We will present two applications of our results. In the first example we explore the convergence of Gaussian curvature estimates on a simple triangular mesh. The second example demonstrates an improved error estimate for the area of a Schwarz lantern.
A Finsler space is said to be geodesically reversible if each oriented geodesic can be reparametrized as a geodesic with the reverse orientation. A reversible Finsler space is geodesically reversible, but the converse need not be true. In this note, building on recent work of LeBrun and Mason, it is shown that a geodesically reversible Finsler metric of constant flag curvature on the 2-sphere is necessarily projectively flat. As a corollary, using a previous result of the author, it is shown that a reversible Finsler metric of constant flag curvature on the 2-sphere is necessarily a Riemannian…
In this paper, we generalize the classification of geodesic orbit spheres from Riemannian geometry to Finsler geometry. Then we further prove if a geodesic orbit Finsler sphere has constant flag curvature, it must be Randers. It provides an alternative proof for the classification of invariant Finsler metrics with $K\equiv1$ on homogeneous spheres other than $Sp(n)/Sp(n-1)$.
We prove that on a closed surface, for any $c>0$, our min-max theory for prescribing mean curvature produces a solution given by a curve of constant geodesic curvature $c$ which is almost embedded, except for finitely many points, at which the solution is a stationary junction with integer density. Moreover, each smooth segment has multiplicity one. The key is a classification of blowups which is new even for $c=0$.
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 $8 π(n-1)$.
Geodesic boundary of constant mean curvature surfaces in $\mathbb{H}^2\times \mathbb{R}$
将极小曲面测地边界结果推广至常平均曲率 $0\le H\le 1/2$ 的曲面，研究其在 $\mathbb{H}^2\times \mathbb{R}$ 中的边界性质。
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.
On the boundedness of the geodesic curvature measure of a regular curve passing through a cross cap singularity
该研究针对内蕴交叉帽奇点相关几何问题，证明了穿过该奇点的正则曲线测地曲率测度的有界性，推广了已有Gauss-Bonnet型公式至带边界曲面情形，并给出奇点处测地曲率有界的条件。
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.
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.
We prove some rigidity results on geodesic orbit Finsler spaces with non-positive curvature. In particular, we show that a geodesic Finsler space with strictly negative flag curvature must be a non-compact Riemannian symmetric space of rank one.
Logarithmic spirals on surfaces of constant Gaussian curvature
本研究针对常高斯曲率曲面上的对数螺线，计算其测地曲率，证明测地曲率渐近行为与曲面曲率无关，且固定距离处测地曲率随高斯曲率连续可微，丰富了曲面几何的相关结论。
I show that if a geodesic space has curvature bounded below locally in the sense of Alexandrov then its completion has the same lower curvature bound globally.
Min-max construction of two capillary embedded geodesics on Riemannian $2$-disks
本研究针对边界严格凸的黎曼2-圆盘，通过极小极大方法证明了满足特定条件时存在两条指定接触角的毛细管嵌入测地线，验证了总测地曲率条件的最优性，并给出了一般度量下不同Morse指数测地线的存在性结果。
A Constrained Mean Curvature Flow On Capillary Hypersurface Supported On Totally Geodesic Plane
针对双曲空间中支撑于全测地超平面的毛细超曲面，本文提出新的 Minkowski 型公式并构造保体积约束平均曲率流，证明其长时间存在、一致收敛到 θ-全脐帽，并验证该帽在固定体积下极小化能量。
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 …
Non-canonical variations of Riemannian submersions with totally geodesic fibers
研究通过保持给定黎曼子浸入的纤维全纯性和纤维限制度量不变，但改变水平分布，探讨水平和垂直方向的截面曲率变化，提出使乘积空间的截面曲率正的条件，并分析从K-接触结构获得弱接触度量结构的条件。
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。
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。
We prove that a homogeneous Finsler sphere with constant flag curvature $K\equiv1$ and a prime closed geodesic of length $2π$ must be Riemannian. This observation provides the evidence for the non-existence of homogeneous Bryant spheres. It also helps us propose an alternative approach proving that a geodesic orbit Finsler sphere with $K\equiv1$ must be Randers. Then we discuss the behavior of geodesics on a homogeneous Finsler sphere with $K\equiv1$. We prove that many geodesic properties for homogeneous Randers spheres with $K\equiv1$ can be generalized to the non-Randers case.
In this paper, we study homogeneous geodesics in homogeneous Finsler spaces. We first give a simple criterion that characterizes geodesic vectors. We show that the geodesics on a Lie group, relative to a bi-invariant Finsler metric, are the cosets of the one-parameter subgroups. The existence of infinitely many homogeneous geodesics on compact semi-simple Lie group is established. We introduce the notion of naturally reductive homogeneous Finsler space. As a special case, we study homogeneous geodesics in homogeneous Randers spaces. Finally, we study some curvature properties of homogeneous ge…
We begin a systematic study of these spaces, initially following along the lines of Eberlein's comprehensive study of the Riemannian case. In particular, we integrate the geodesic equation, discuss the structure of the isometry group, and make a study of lattices and periodic geodesics. Some major differences from the Riemannian theory appear. There are many flat groups (versus none), including Heisenberg groups. While still a semidirect product, the isometry group can be strictly larger than the obvious analogue. Everything is illustrated with explicit examples. We introduce the notion of pH-…
We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. Furthermore, we give a structural result concerning expanding semi-algebraic solitons on complex Lie groups. It turns out that the restriction of the soliton metric to the nilradical is also an expanding algebraic soliton and we explain how to construct expanding solitons on comp…
In this paper we prove that a properly embedded constant mean curvature surface in $\mathbb{H}^2\times\mathbb{R}$ which has finite topology and stays at a finite distance from a vertical geodesic line is invariant by rotation around a vertical geodesic line.
We prove transverse Weitzenböck identities for the horizontal Laplacians of a totally geodesic foliation. As a consequence, we obtain nullity theorems for the de Rham cohomology assuming only the positivity of curvature quantities transverse to the leaves. Those curvature quantities appear in the adiabatic limit of the canonical variation of the metric.
In this paper, we study the interaction between the geodesic orbit (g.o.~in short) property and certain flag curvature conditions. A Finsler manifold is called g.o.~if each constant speed geodesic is the orbit of a one-parameter subgroup. Besides the non-negatively curved condition, we also concern the (FP) condition for the flag curvature, i.e., in any flag we can find a flag pole, such that the flag curvature is positive. The main theorem we will prove is the following. If a g.o.~Finsler space $(M,F)$ has non-negative flag curvature and satisfies the (FP) condition, then $M$ must be compact.…
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.
In this survey results on the behavior of simple closed geodesics on regular tetrahedra in three-dimensional spaces of constant curvature are presented.
Invariant constant mean curvature tubes around a horizontal geodesic in $\mathbb{E}(κ,τ)$-spaces
本文研究齐次3-流形$\mathbb{E}(κ,τ)$中水平测地线周围的不变常平均曲率管道，证明其在特定参数条件下叶化空间，并揭示Daniel姊妹对应保持该管道族，且在环面情形计算共形类型与等周轮廓。
It is known that the so-called rotation minimizing (RM) frames allow for a simple and elegant characterization of geodesic spherical curves in Euclidean, hyperbolic, and spherical spaces through a certain linear equation involving the coefficients that dictate the RM frame motion (da Silva, da Silva in Mediterr J Math 15:70, 2018). Here, we shall prove the converse, i.e., we show that if all geodesic spherical curves on a Riemannian manifold are characterized by a certain linear equation, then all the geodesic spheres with a sufficiently small radius are totally umbilical and, consequently, th…
We propose a new two-component geodesic equation with the unusual property that the underlying space has constant positive curvature. In the special case of one space dimension, the equation reduces to the two-component Hunter-Saxton equation.
We give a universal upper bound for the total curvature of minimizing geodesic on a convex surface in the Euclidean space.
We construct examples of compact and one-ended constant mean curvature surfaces with large mean curvature in Riemannian manifolds with axial symmetry by gluing together small spheres positioned end-to-end along a geodesic. Such surfaces cannot exist in Euclidean space, but we show that the gradient of the ambient scalar curvature acts as a `friction term' which permits the usual analytic gluing construction to be carried out.
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。
In this paper we show that on a complete Riemannian manifold of negative curvature and dimension $n>1$ every two points which realize a local maximum for the distance function are connected by at least $2n+1$ geometrically distinct geodesic segments (i.e. length minimizing). Using a similar method, we obtain that in the case of non-positive curvature, for every two points with the same property as above the number of connecting distinct geodesic segments is at least $n+1$.
In this paper we discuss the stability of geodesic spheres in $\mathbb{S}^{n+1}$ under constrained curvature flows. We prove that under some standard assumptions on the speed and weight functions, the spheres are stable under perturbations that preserve a volume type quantity. This extends results by Escher and Simonett, 1998, and the author, 2015, to a Riemannian manifold setting.
An expansion is developed for the Weil-Petersson Riemann curvature tensor in the thin region of the Teichmüller and moduli spaces. The tensor is evaluated on the gradients of geodesic-lengths for disjoint geodesics. A precise lower bound for sectional curvature in terms of the surface systole is presented. The curvature tensor expansion is applied to establish continuity properties at the frontier strata of the augmented Teichmüller space. The curvature tensor has the asymptotic product structure already observed for the metric and covariant derivative. The product structure is combined with t…
Combinatorial p-th Calabi Flows for Total Geodesic Curvatures in hyperbolic background geometry
本文研究双曲背景几何下带边界曲面上具锥奇点的广义圆填充，利用变分原理证明其存在性与刚性，并首次引入组合p次Calabi流以求解预设全测地曲率与离散高斯曲率的圆填充。
Equidistribution of divergent geodesics in negative curvature
本文研究非紧有限体积负曲率流形及几何有限树商空间中的发散测地线，证明其随复杂度增大分别趋向最大熵测度与 Bowen–Margulis 测度等分布。
We study conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundary in supercritical regimes. Exploiting a variational argument, we derive a general existence result for surfaces with at least two boundary components. This seems to be the first result in this setting. Moreover, we allow to have conical singularities with both positive and negative orders, that is cone angles both less and grater than $2π$.
This paper is concerned with the existence of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures. Being more specific, given nonnegative smooth functions $K: \overline{\mathbb{D}} \to \mathbb{R}$ and $h: \partial \mathbb{D} \to \mathbb{R}$, we consider the problem of finding a conformal metric realizing $K$ and $h$ as Gaussian and geodesic curvatures, respectively. This is the natural analogue of the classical Nirenberg problem posed on the disk. As we shall see, both curvatures play a role in the existence of solutions. Indeed we are able to give existence results …
In a Riemannian manifold, it is well known that the scalar curvature at a point can be recovered from the volumes (areas) of small geodesic balls (spheres). We show the scalar curvature is likewise determined by the relative capacities of concentric small geodesic balls. This result has motivation from general relativity (as a complement to a previous study by the author of the capacity of large balls in an asymptotically flat manifold) and from weak definitions of nonnegative scalar curvature. It also motivates a conjecture (inspired by the famous volume conjecture of Gray and Vanhecke), rega…
Conformal Metrics on the Disk with Prescribed Negative Gaussian Curvature and Boundary Geodesic Curvature
本文研究了通过共形变换给定圆盘负高斯曲率和边界测地曲率的问题，通过变分方法和发散分析得出存在性结果。
An upper bound for the mean Lorentz force of magnetic flows without conjugate points in terms of the geodesic curvature of horocycles
本文研究亏格至少二的闭黎曼曲面上无共轭点的磁流，证明在 Mañé 临界值之上，洛伦兹力积分受水平圆测地曲率约束，并由此推出基于面积和高斯曲率的显式上界。
We introduce a notion of geodesic curvature $k_ζ$ for a smooth horizontal curve $ζ$ in a three-dimensional contact sub-Riemannian manifold, measuring how much a horizontal curve is far from being a geodesic. We show that the geodesic curvature appears as the first corrective term in the Taylor expansion of the sub-Riemannian distance between two points on a unit speed horizontal curve $$ d_{SR}^2( ζ(t),ζ(t+ε))=ε^2-\frac{k_ζ^2(t)}{720} ε^6 +o(ε^{6}). $$ The sub-Riemannian distance is not smooth on the diagonal, hence the result contains the existence of such an asymptotics. This can be seen as …
In this paper, we prove a scalar curvature rigidity result for geodesic balls in S^n. This result contrasts sharply with the recent counterexamples to Min-Oo's conjecture for the hemisphere (cf. [5]).
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