On conformal invariance of isotropic geodesics
Maks A. Akivis, Vladislav V. Goldberg
我们考虑赋以伪共形结构的流形上的实迷向测地线及其在这类流形上的类光超曲面理论、洛仑兹型四维共形结构几何和爱因斯坦空间分类中的应用。
We consider real isotropic geodesics on manifolds endowed with a pseudoconformal structure and their applications to the theory of lightlike hypersurfaces on such manifolds, the geometry of four-dimensional conformal structures of Lorentzian type, and a classification of the Einstein spaces.
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On the three-dimensional homogeneous SO(2) - isotropic Riemannian manifolds
本文研究三维齐次SO（2）-迷向黎曼流形的一些性质。特别地，我们确定了旋转曲面的测地线、全测地线曲面、全脐曲面和测地线。
We study singularities of geodesics flows in two-dimensional generalized Finsler spaces (pseudo-Finsler spaces). Geodesics are defined as extremals of a certain auxiliary functional whose non-isotropic extremals coincide with extremals of the action functional. This allows to consider isotropic lines as (unparametrized) geodesics.
In the present paper, we try to investigate the Noether symmetries and Lie point symmetries of the Vaidya-Bonner geodesics. Classification of one-dimensional subalgebras of Lie point symmetries are considered. In fact, the collection of pairwise non-conjugate one-dimensional subalgebras that are called the optimal system of subalgebras is determined. Moreover, as illustrative examples, the symmetry analysis is implemented on two special cases of the system.
Jacobi osculating rank and isotropic geodesics on naturally reductive 3-manifolds
本文研究了自然约化齐次流形上测地线的Jacobi密切秩，并将这一理论应用于三维情形。在这里，每个非对称的单连通自然约化3-流形可以作为常曲率曲面上的主丛给出，使得它的水平分布的曲率也是常数。证明了除Hopf纤维为零外，所有测地线的Jacobi密切秩均为2。此外，我们确定了所有的迷向测地线和迷向切共轭轨迹。
Chern-Ricci invariance along G-geodesics
该研究在紧定向流形的度量空间中，证明G-测地线初始速度与微分同胚群轨道切空间G正交的性质沿测地线保持，且在复反不变变分和Nijenhuis张量为零的条件下陈-Ricci形式不变，为Kähler-Ricci孤立子流动态稳定性的切片定理提供支撑。
Loxodromes and geodesics on rotational surfaces in pseudo-isotropic space
本文首次研究了伪迷向空间 I_p^3 中的恒向线，定义了伪迷向角，并推导出旋转曲面上类空与类时恒向线及测地线的方程。
In this paper, we prove that lightlike geodesics of a pseudo-Finsler manifold and its focal points are preserved up to reparametrization by anisotropic conformal changes, using the Chern connection and the anisotropic calculus and the fact that geodesics are critical points of the energy functional and Jacobi fields, the kernel of its index form. This result has applications to the study of Finsler spacetimes.
In this article we investigate a type of totally geodesic map which has its image being a geodesic in an anisotropic Riemannian manifold. We consider its nonlinear stability among the family of wave maps. We first establish the factorization property and then formulate the stability problem into a PDE system in a specially constructed chart of geodesic normal coordinates. With a generalization of the hyperboloidal foliation, we establish the global existence result associate to small initial data for this PDE system, which leads to the geometric stability.
Finsler metrizabilities and geodesic invariance
该研究将芬斯勒可度量化问题转化为度量与角张量的测地线不变性问题，明确陀螺spray是具测地线不变角度量的最大类别，其关联标量函数不变性可提供spray的首次积分。
Invariance principle for Lifts of Geodesic Random Walks
本文研究 Riemannian 淹没下测地随机游动的提升，在速度分布适当条件下证明其收敛于水平 Brown 运动，并据此给出水平 Laplace 算子与 Laplace-Beltrami 算子几何恒等式的概率证明。
Isotropic embeddings of coadjoint orbits and magnetic geodesic flows
本文研究紧李群余伴随轨道的各向同性与 Lagrange 嵌入，将 SU(n) 的结果推广到 SO 和 Sp，并证明旗流形磁测地流与特殊经典自旋链系统等价。
Singularities of the geodesic flow on surfaces with pseudo-Riemannian metrics
我们考虑一个伪黎曼度量，改变签名沿表面上的平滑曲线，称为判别曲线。判别曲线将表面局部地分成黎曼域和洛伦兹域。我们研究的地方行为和性质的测地线在一个点上的判别式的各向同性方向是切线的判别曲线。
On uniqueness of free boundary minimal annuli in geodesic balls of $\mathbb{S}^3_+$ and $\mathbb{H}^3$
本文研究了球面S³₊和双曲空间H³中测地球内自由边界极小环的唯一性，证明其与临界旋转环全等。
Volume minimization and estimates for certain isotropic submanifolds in complex projective spaces
本文利用R的积分几何公式证明了如下结果。霍华德：考虑在$\mathbb{C}P^n$中的完全测地线$\mathbb{R}P^{2m}$。然后它使$\mathbb{C}P^n$中同一$\mathbb{Z}/2$同调类中的迷向子流形之间的体积最小化（但不是使这个$\mathbb{Z}/2$同调类中的所有子流形之间的体积最小化）。此外，全测地线$\mathbb{R}P^{2m-1}$在$\mathbb{C}P^n$的哈密顿变形类中使体积最小化。作为推论，我们将给出$\mathbb{C}P^n$中完全相交的拉格朗日子流形的体积估计。
Non-isotropic geometries are of interest to low-dimensional topologists, physicists and cosmologists. However, they are challenging to comprehend and visualize. We present novel methods of computing real-time native geodesic rendering of non-isotropic geometries. Our methods can be applied not only to visualization, but also are essential for potential applications in machine learning and video games.
Lightlike hypersurfaces on a four-dimensional manifold endowed with a pseudoconformal structure of signature (2, 2)
研究了四维流形$(M, c)$上具有伪共形结构$c = CO (2, 2)$的类光超曲面的几何性质.他们证明了类光超曲面$V \subset (M, c)$具有由共形不变的各向同性测地线和与这些测地线相切的两个各向同性分布构成的叶理，并且这两个分布是可积的当且仅当$V$是全脐的。利用类光超曲面$V \subset (M, c)$的奇点和奇异子流形，构造了与$V$内在联系的$V$的不变正规化.
We prove several versions of Driver's integration by parts formula for the horizontal Wiener measure on a totally geodesic Riemannian foliation and prove that the horizontal Wiener measure has a quasi-invariance property with respect to flows generated by suitable tangent processes.
We consider the projective Finsler metrizability problem: under what conditions the solutions of a given system of second-order ordinary differential equations (SODE) coincide with the geodesics of a Finsler metric, as oriented curves. SODEs with isotropic curvature have already been thoroughly studied in the literature and have proved to be projective Finsler metrizable. In this paper, we investigate the non-isotropic case and obtain new results by examining the integrability of the Rapcsák system extended with curvature conditions. We consider the $n$-dimensional generic case, where the eige…
Holomorphic vector bundles on Kähler manifolds and totally geodesic foliations on Euclidean open domains
本文建立了Kähler流形上整体生成的全纯向量丛的截面，关于非退化二次型的迷向截面和欧氏开域上的全测地叶理截面之间的关系.我们找到了全测地叶理起源于全纯向量丛的一个几何条件。对于余维为2的叶理，这种描述恢复了P.Baird和J.C.木材.起关键作用的泛对象是正交格拉斯曼。
Pseudo-Riemannian geodesic orbit nilmanifolds of signature $\boldsymbol{(n-3,3)}$
本文研究签名 $(n-3,3)$ 的伪黎曼测地轨道幂零流形，证明在导出代数上非退化时幂零步数限于 2 或 4，退化时给出双扩张约化，并用九维和八维例子说明结论的精确性。
Geodesic Lévy flights on Zoll surfaces
研究Zoll曲面上各向同性Lévy飞行首次捕获时间的渐近行为，证明首项普适而修正项由共轭点的局部奇异性决定。
Let $M$ be a complete Riemannian manifold and suppose $p\in M$. For each unit vector $v \in T_p M$, the $\textit{Jacobi operator}$, $\mathcal{J}_v: v^\perp \rightarrow v^\perp$ is the symmetric endomorphism, $\mathcal{J}_v(w) = R(w,v)v$. Then $p$ is an $\textit{isotropic point}$ if there exists a constant $κ_p \in \mathbf{R}$ such that $\mathcal{J}_v = κ_p \textit{ Id}_{v^\perp}$ for each unit vector $v \in T_pM$. If all points are isotropic, then $M$ is said to be isotropic; it is a classical result of Schur that isotropic manifolds of dimension at least 3 have constant sectional curvatures. …
Spray-Invariant Sets in Infinite-Dimensional Manifolds
研究无限维流形中喷雾不变集的几何性质，探讨其在不同正则性下的行为差异，扩展了对测地线保持性的理解。
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。
On Spherically Symmetric Sprays
本文研究了球对称喷射，即在正交变换下不变的喷射。通过建立其标准形式，证明其测地系数可表示为特定形式，并对具有各向同性曲率的投影平坦球对称喷射进行了完整分类，特别得到了零曲率喷射的显式形式。
By adapting a technique of Molchanov, we obtain the heat kernel asymptotics at the sub-Riemannian cut locus, when the cut points are reached by an $r$-dimensional parametric family of optimal geodesics. We apply these results to the bi-Heisenberg group, that is, a nilpotent left-invariant sub-Rieman\-nian structure on $\mathbb{R}^{5}$ depending on two real parameters $α_{1}$ and $α_{2}$. We develop some results about its geodesics and heat kernel associated to its sub-Laplacian and we illuminate some interesting geometric and analytic features appearing when one compares the isotropic ($α_{1}=…
Kundt Three Dimensional Left Invariant Spacetimes
本研究聚焦Kundt时空这一广义相对论中的重要对象，刻画其洛伦兹流形结构与相关性质，给出局部结构，并完成三维单连通幺模李群上所有左不变Kundt结构的分类。
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