The exponential map of a $C^{1,1}$-metric

Exponential map and normal form for cornered asymptotically hyperbolic metrics

This paper considers asymptotically hyperbolic manifolds with a finite boundary intersecting the usual infinite boundary -- cornered asymptotically hyperbolic manifolds -- and proves a theorem of Cartan-Hadamard type near infinity for the normal exponential map on the finite boundary. As a main application, a normal form for such manifolds at the corner is then constructed, analogous to the normal form for usual asymptotically hyperbolic manifolds and suited to studying geometry at the corner. The normal form is at the same time a submanifold normal form near the finite boundary and an asympto…

The exponential map for time-varying vector fields

The exponential map that characterises the flows of vector fields is the key in understanding the basic structural attributes of control systems in geometric control theory. However, this map does not exists due to the lack of completeness of flows for general vector fields. An appropriate substitute is devised for the exponential map, not by trying to force flows to be globally defined by any compact assumptions on the manifold, but by categorical development of spaces of vector fields and flows, thus allowing for systematic localisation of such spaces. That is to say, we give a presheaf cons…

次黎曼指数映射的正则性与连续性性质

Regularity and Continuity properties of the sub-Riemannian exponential map

本文通过次黎曼Jacobi场和Jacobi曲线的Maslov指标，证明了次黎曼指数映射的Warner正则性与连续性性质，并由此推出三维Heisenberg群的指数映射在共轭向量附近非单射。

Exponential map of a weak Riemannian Hilbert manifold

We prove the Focal Index Lemma and the Rauch and Berger comparison theorems on a weak Riemannian Hilbert manifold with a smooth Levi-Civita connection and we apply these results to the free loop spaces of a compact manifold with the L^2 metrics

The Taylor series related to the differential of the exponential map

In this paper we study the Taylor series of an operator-valued function related to the differential of the exponential map. For a smooth manifold $\mathcal{M}$ with a torsion-free affine connection the operator $\mathcal{E}_p(v)$ acting on the space $T_p\mathcal{M}$ is defined to be the composition of the differential of the exponential map at $v\in T_p\mathcal{M}$ with parallel transport to $p$ along the geodesic. The Taylor series of $\mathcal{E}_p$ as a function of $v$ is found explicitly in terms of the curvature tensor and its high order covariant derivatives at $p$.

一类埃尔米特-杨-米尔斯度量的极限行为，II：指数逼近

Limiting Behavior of a Class of Hermitian Yang--Mills Metrics, II: Exponential Approximation

本文是相关研究续篇，针对两个一维复环面乘积上稳定秩二全纯向量丛的近似埃尔米特-杨-米尔斯度量，证明大凯勒极限下该近似度量对精确度量具有任意阶指数精度的渐近描述，克服了全局C^0估计等解析难点。

The Taylor Expansion of the Exponential Map and Geometric Applications

In this work we consider the Taylor expansion of the exponential map of a submanifold immersed in R^n up to order three, in order to introduce the concepts of lateral and frontal deviation. We compute the directions of extreme lateral and frontal deviation for surfaces in R^3. Also we compute, by using the Taylor expansion, the directions of high contact with hyperspheres of a surface immersed in R^4 and the asymptotic directions of a surface immersed in R^5.

Normal forms for the sub-Riemannian exponential map of $\mathbb{G}_α$, $\mathrm{SU}(2)$, and $\mathrm{SL}(2)$

The goal of this paper is to use singularity theory to find normal forms near the critical points of the sub-Riemannian exponential map. Three cases are studied: the $α$-Grushin plane with fold singularities, and the special unitary group $\mathrm{SU}(2)$ and special linear group $\mathrm{SL}(2)$ with fold and saddle-like singularities. They serve as examples of different sub-Riemannian structures and the techniques presented can be applied to other contexts. The paper also includes a discussion of the implications of this approach, as well as open problems.

曲线在指数型映射下的提升原理

A Lifting principle of curves under exponential-type maps

本文提出曲线在指数型映射下的提升原理，克服经典障碍，通过局部提升得到全局提升，为半黎曼几何中的基础定理提供新证明。

On analysis of the exponential map of volume-preserving diffeomorphism group on closed orientable surfaces through the vorticity

We study the exponential map of group of volume-preserving diffeomorphisms on closed orientable surfaces via the vorticity formulation of the incompressible Euler equation. We present an alternative, fluid dynamical proof of the theorem of Ebin--Misiołek--Preston: the exponential is a nonlinear Fredholm mapping of index zero. We extend Shnirelman's rigidity result for the exponential map from 2-dimensional flat torus to arbitrary orientable closed surfaces. That is, we prove that the exponential map is Fredholm quasiregular.

The Cartan exponential map in semi-simple compact Lie groups

In this paper we introduce a new type of exponential map in semi-simple compact Lie groups, which is related to the sub-Riemannian geometry generated by the orthogonal complement of a Cartan subalgebra in a similar way to how the group exponential map is related to the Riemannian geometry.

Path-lifting properties of the exponential map with applications to geodesics

We revisit certain path-lifting and path-continuation properties of abstract maps as described in the work of F. Browder and R. Rheindboldt in 1950-1960s, and apply their elegant theory to exponential maps. We obtain thereby a number of novel results of existence and multiplicity of geodesics joining any two points of a connected affine manifold, as well as ${\it causal}$ geodesics connecting any two causally related points on a Lorentzian manifold. These results include a generalization of the well-known Hadamard-Cartan theorem of Riemannian geometry to the affine manifold context, as well as…

Exponential gaps in the length spectrum

We present a separation property for the gaps in the length spectrum of a compact Riemannian manifold with negative curvature. In arbitrary small neighborhoods of the metric for some suitable topology, we show that there are negatively curved metrics with a length spectrum exponentially separated from below. This property was previously known to be false generically.

On the Singularities of the Exponential Map in Infinite Dimensional Riemannian Manifolds

Using symplectic techniques and spectral analysis of smooth paths of self-adjoint operators, we characterize the set of conjugate instants along a geodesic in an infinite dimensional Riemannian Hilbert manifold.

通过SPD几何在非参数指数统计流形上的可处理伪度量

A Tractable Pseudo-Metric on Non-Parametric Exponential Statistical Manifolds via SPD Geometry

研究非参数统计流形上概率分布距离难计算问题，提出两阶段框架，先投影到有限维参数指数族，再嵌入对称正定矩阵流形，得到可计算伪度量及仿射不变检验统计量，应用于两样本检验，无需连续调整参数。

Local non-injectivity of the exponential map at critical points in sub-Riemannian geometry

We prove that the sub-Riemannian exponential map is not injective in any neighbourhood of certain critical points. Namely that it does not behave like the injective map of reals given by $f(x) = x^3$ near its critical point $x = 0$. As a consequence, we characterise conjugate points in ideal sub-Riemannian manifolds in terms of the metric structure of the space. The proof uses the Hilbert invariant integral of the associated variational problem.

Cut and conjugate points of the exponential map, with applications

The goal of this thesis is to study the singularities of the exponential map of Riemannian and Finsler manifolds (a concept related to caustics and catastrophes), and the object known as the cut locus (aka ridge, medial axis or skeleton), to improve existing results about its structure, to look at it in new ways, and to derive applications to the Ambrose conjecture and the Hamilton-Jacobi equations.

Formal exponential map for graded manifolds

We introduce, for every $\mathbb{Z}$-graded manifold, a formal exponential map defined in a purely algebraic way and study its properties. As an application, we give a simple new construction of a Fedosov type resolution of the algebra of smooth functions of $\mathbb{Z}$-graded manifolds and we extend the Emmrich--Weinstein theorem to the context of $\mathbb{Z}$-graded manifolds.

Exponential map and $L_\infty$ algebra associated to a Lie pair

In this note, we unveil homotopy-rich algebraic structures generated by the Atiyah classes relative to a Lie pair $(L,A)$ of algebroids. In particular, we prove that the quotient $L/A$ of such a pair admits an essentially canonical homotopy module structure over the Lie algebroid $A$, which we call Kapranov module.

The Exponential Map for Hopf Algebras

We give an analogue of the classical exponential map on Lie groups for Hopf $*$-algebras with differential calculus. The major difference with the classical case is the interpretation of the value of the exponential map, classically an element of the Lie group. We give interpretations as states on the Hopf algebra, elements of a Hilbert $C^{*} $-bimodule of $\frac{1}{2}$ densities and elements of the dual Hopf algebra. We give examples for complex valued functions on the groups $S_{3}$ and $\mathbb{Z}$, Woronowicz's matrix quantum group $\mathbb{C}_{q}[SU_2] $ and the Sweedler-Taft algebra.

秩为2情形下 Hitchin 度量与半平坦度量的比较

Comparison of the Hitchin metric and the semi-flat metric in the rank two case

本文研究秩2、次数0且谱曲线光滑的 Higgs 丛模空间中，Hitchin 度量与半平坦度量沿大参数曲线的指数衰减率改进问题。

A bug's eye view: the Riemannian exponential map on polyhedral surfaces

We explore the perspective of a bug living on the two-dimensional surface of a polyhedron. Images of various kinds of effects like lensing and cloaking are shown via color pictures of three viewpoints: the first person perspective of the bug, a map of the bug's viewpoint, and a look at the bug on the embedded polyhedron from a three-dimensional exterior viewer. The pictures were constructed by computing the exponential map of a polyhedron by cutting and rotating faces into the tangent plane of the bug.

Chentsov's theorem for exponential families

Chentsov's theorem characterizes the Fisher information metric on statistical models as essentially the only Riemannian metric that is invariant under sufficient statistics. This implies that each statistical model is naturally equipped with a geometry, so Chentsov's theorem explains why many statistical properties can be described in geometric terms. However, despite being one of the foundational theorems of statistics, Chentsov's theorem has only been proved previously in very restricted settings or under relatively strong regularity and invariance assumptions. We therefore prove a version o…

The exponential map at a cuspidal singularity

We study spaces with a cuspidal (or horn-like) singularity embedded in a smooth Riemannian manifold and analyze the geodesics in these spaces which start at the singularity. This provides a basis for understanding the intrinsic geometry of such spaces near the singularity. We show that these geodesics combine to naturally define an exponential map based at the singularity, but that the behavior of this map can deviate strongly from the behavior of the exponential map based at a smooth point or at a conical singularity: While it is always surjective near the singularity, it may be discontinuous…

Formal exponentials and linearisations of QP-manifolds

We define formal exponential maps for any graded manifold as maps from the formal tangent bundle (that we also define) into the graded manifold. We show that each such map uniquely determines and is determined by its associated Grothendieck connection, which is shown to be flat, and to furnish a resolution of the ring of functions. We then show how a recent construction involving the data of a connection on the tangent bundle recovers a large class of formal exponentials in our definition. As an application, we use a formal exponential map to linearise a QP-manifold at a point. This gives the …

SE(3)上的指数映射和Cayley映射综述：与广义泊松方程和柔体多体系统的李群积分相关

Review of the Exponential and Cayley Map on SE(3) as relevant for Lie Group Integration of the Generalized Poisson Equation and Flexible Multibody Systems

本文总结了SE(3)上指数和Cayley映射的闭式关系及证明，用于李群积分方案，并提出了一种计算效率更高的隐式generalized-alpha方案。