Exponential map of a weak Riemannian Hilbert manifold

Einstein-Hilbert Action and on the Gauss-Bonnet Theorem for Riemannian Noncommutative Tori

We show the non-positivity of the Einstein-Hilbert action for conformal flat Riemannian metrics. The action vanishes only when the metric is constant flat. This recovers an earlier result of Fathizadeh-Khalkhali in the setting of spectral triples on noncommutative four-torus. Furthermore, computations of the gradient flow and the scalar curvature of this space based on modular operator are given. We also show the Gauss-Bonnet theorem for a parametrized class of non-diagonal metrics on noncommutative two-torus.

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。

具有Sobolev正则黎曼度量的流形的无穷小希尔伯特性：通过一致下容量界

Infinitesimal Hilbertianity for manifolds with Sobolev regular Riemannian metrics through uniform lower capacity bounds

本文证明具有Sobolev正则黎曼度量的二维流形上的度量测度空间是无穷小希尔伯特的，并通过建立一致下容量界实现。

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.

无限维希尔伯特空间上等价高斯测度的Fisher-Rao几何

Fisher-Rao geometry of equivalent Gaussian measures on infinite-dimensional Hilbert spaces

本文在无限维希尔伯特空间上显式描述了等价中心高斯测度的Fisher-Rao黎曼度量，证明有限维高斯密度相关几何量可自然推广，并建立了与正定单位化希尔伯特-施密特算子黎曼几何的联系。

The Einstein-Hilbert type action on almost $k$-product manifolds

A Riemannian manifold endowed with $k>2$ orthogonal complementary distributions (called here a Riemannian almost $k$-product structure) appears in such topics as multiply warped products, the webs composed of several foliations, and proper Dupin hypersurfaces of real space-forms. In the paper, we consider the mixed scalar curvature of such structure for $k>2$, derive Euler-Lagrange equations for the Einstein-Hilbert type action with respect to adapted variations of metric, and present them in a nice form of Einstein equation.

Universal infinitesimal Hilbertianity of sub-Riemannian manifolds

We prove that sub-Riemannian manifolds are infinitesimally Hilbertian (i.e., the associated Sobolev space is Hilbert) when equipped with an arbitrary Radon measure. The result follows from an embedding of metric derivations into the space of square-integrable sections of the horizontal bundle, which we obtain on all weighted sub-Finsler manifolds. As an intermediate tool, of independent interest, we show that any sub-Finsler distance can be monotonically approximated from below by Finsler ones. All the results are obtained in the general setting of possibly rank-varying structures.

Hilbert-Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvature is $-\infty$.

The co-Riemannian Structure of Smooth Loop Spaces

We construct a natural co-Riemannian structure on the manifold of smooth loops in a Riemannian manifold. We show that the smooth loop space of a string manifold is a per-Hilbert-Schmidt locally equivalent co-spin manifold and thus admits a Dirac operator.

F-爱因斯坦-希尔伯特泛函的第二变分

Second Variation of F-Einstein-Hilbert Functional

本文推导了黎曼流形上广义爱因斯坦-希尔伯特泛函的第二变分公式，扩展了稳定爱因斯坦流形的定义，并给出相关性质。

希尔伯特H型群中的Michor-Mumford猜想

The Michor-Mumford conjecture in Hilbertian H-type groups

本文引入无限维希尔伯特H型群并研究其弱左不变黎曼度量，证明测地距离消失与截面曲率局部无界性共存，验证Michor-Mumford猜想，还揭示退化测地距离与无限维子黎曼结构的关联及Levi-Civita导数的不存在性。

Some Results on Infinite Dimensional Riemannian Geometry

In this paper we will investigate the global properties of complete Hilbert manifolds with upper and lower bounded sectional curvature. We shall prove the Focal Index Lemma that we will allow us to extend some classical results of finite dimensional Riemannian geometry such as Rauch and Berger Theorems and the Topogonov Theorem in the class of manifolds on which the Hopf-Rinow Theorem holds.

The Completion of the Manifold of Riemannian Metrics with Respect to its $L^2$ Metric

This is the author's Ph.D. thesis, submitted to the University of Leipzig. It deals with the $L^2$ Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold. The main body of the thesis is a description of the completion manifold of metrics with respect to the $L^2$ metric. The primary motivation for studying this problem comes from Teichmueller theory, where similar considerations lead to a completion of the well-known Weil-Petersson metric. We give an application of the main theorem to the completions of Teichmueller space with respect …

Deformations and Hilbert's Fourth Problem

In this paper we study a class of Finsler metrics defined by a Riemannian metric and an 1-form. We classify those of projectively flat in dimension $n\geq3$ by a special class of deformations. The results show that the projective flatness of such kind of Finsler metrics always arises from that of some Riemannian metric.

常负曲率子流形：希尔伯特定理的推广

Submanifolds of Constant Negative Curvature: A Generalization of Hilbert's Theorem

本文通过广义高斯-博内公式证明n维双曲空间无法嵌入到2n-1维欧几里得空间，推广了希尔伯特定理。

关于指数次椭圆调和映射的稳定性

On stability of exponentially subelliptic harmonic maps

该文研究次黎曼流形到黎曼流形的指数次椭圆调和映射稳定性，推导两类变分公式，证明非正曲率目标流形下的稳定性，得到球面目标流形下的不稳定性结果。

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.

Exponential convergence rate of the harmonic heat flow

We consider the harmonic heat flow for maps from a compact Riemannian manifold into a Riemannian manifold that is complete and of non-positive curvature. We prove that if the harmonic heat flow converges to a limiting harmonic map that is a non-degenerate critical point of the energy functional, then the rate of convergence is exponential (in the $L^2$ norm).

Quantization for an elliptic equation with critical exponential growth on compact Riemannian surface without boundary

In this paper, using blow-up analysis, we prove a quantization result for an elliptic equation with critical exponential growth on compact Riemannian surface without boundary. Similar results for Euclidean space were obtained by Adimurthi-Struwe \cite{Adi-Stru}, Druet \cite{Druet}, Lamm-Robert-Struwe \cite{L-R-S}, Martinazzi \cite{Mart}, Martinazzi-Struwe \cite{Mar-Stru}, and Struwe \cite{Struwe} respectively.

关于希尔伯特流形上完备性的共形灵活性与极小化测地线

On Conformal Flexibility of Completeness and Minimizing Geodesics on Hilbert Manifolds

研究希尔伯特流形上完备性相关问题，通过共形变形得到度量和测地线完备且有长度极小化测地线的代表，同时建立了度量完备性的局部灵活性现象，即能在保持测地线完备性的同时破坏度量完备性。

Conjugate points and Maslov index in locally symmetric semi-Riemannian manifolds

We study the singularities of the exponential map in semi Riemannian locally symmetric manifolds. Conjugate points along geodesics depend only on real negative eigenvalues of the curvature tensor, and their contribution to the Maslov index of the geodesic is computed explicitly. We prove that degeneracy of conjugate points, which is a phenomenon that can only occur in semi-Riemannian geometry, is caused in the locally symmetric case by the lack of diagonalizability of the curvature tensor. The case of Lie groups endowed with a bi-invariant metric is studied in some detail, and conditions are g…

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.

Analytic and Reidemeister torsion for representations in finite type Hilbert modules

For a closed Riemannian manifold we extend the definition of analytic and Reidemeister torsion associated to an orthogonal representation of fundamental group on a Hilbert module of finite type over a finite von Neumann algebra. If the representation is of determinant class we prove, generalizing the Cheeger-Müller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, the L2 analytic and Reidemeister torsions are equal.

Trudinger-Moser inequalities on complete noncompact Riemannian manifolds

Though Trudinger-Moser inequalities on compact Riemannian manifolds or Euclidean space are well understood, we know little about them on complete noncompact Riemannian manifolds. In this paper, we established respectively necessary condition and sufficient condition under which Trudinger-Moser inequalities hold on complete noncompact Riemannian manifolds. For applications of such inequalities, we obtained existence results for some quasilinear equations with nonlinearity of exponential growth.

Non-Embedding Theorems of Nilpotent Lie groups and Sub-Riemannian Manifolds

We prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the RCD(0,N), with N > 1. In fact, we can prove that a subRiemannian manifold whose generic degree of nonholonomy is not smaller than 2 can not be biLipschitzly embedded in any Banach space with the Radon-Nikodym property. We also get that every regular sub-Riemannian manifold do not satisfy the CD(K,N) with N > 1. We also prove that the subRiemannian manifold is infinitesimally Hilbert space.

Absence of super-exponentially decaying eigenfunctions on Riemannian manifolds with pinched negative curvature

Let (X,g) be a metrically complete, simply connected Riemannian manifold with bounded geometry and pinched negative curvature, i.e. there are constants a>b>0 such that -a^2<K<-b^2 for all sectional curvatures K. Here bounded geometry is used in the sense that all covariant derivatives of the Riemannian curvature tensor are bounded and the injectivity radius is uniformly bounded below by a positive constant. We show that there are no superexponentially decaying eigenfunctions of the Laplacian of g. We also show the analogous conclusion for other geometric operators, and prove a theorem with the…

常数曲率设定下的希尔伯特第四问题

Hilbert's fourth problem in the constant curvature setting

本文研究了常数旗曲率下希尔伯特第四问题的全局几何结构，推导了距离公式并建立了流形的分类与唯一性定理。

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。

(In)equality distance patterns and embeddability into Hilbert spaces

We show that compact Riemannian manifolds, regarded as metric spaces with their global geodesic distance, cannot contain a number of rigid structures such as (a) arbitrarily large regular simplices or (b) arbitrarily long sequences of points equidistant from pairs of points preceding them in the sequence. All of this provides evidence that Riemannian metric spaces admit what we term loose embeddings into finite-dimensional Euclidean spaces: continuous maps that preserve both equality as well as inequality. We also prove a local-to-global principle for Riemannian-metric-space loose embeddabilit…

Concentration of eigenfunctions of the Laplacian on a closed Riemannian manifold

We study concentration phenomena of eigenfunctions of the Laplacian on closed Riemannian manifolds. We prove that the volume measure of a closed manifold concentrates around nodal sets of eigenfunctions exponentially. Applying the method of Colding and Minicozzi we also prove restricted exponential concentration inequalities and restricted Sogge-type $L_p$ moment estimates of eigenfunctions.

Riemannian Holonomy Groups of Statistical Manifolds

Normal distribution manifolds play essential roles in the theory of information geometry, so do holonomy groups in classification of Riemannian manifolds. After some necessary preliminaries on information geometry and holonomy groups, it is presented that the corresponding Riemannian holonomy group of the $d$-dimensional normal distribution is $SO\left(\frac{d\left(d+3\right)}{2}\right)$, for all $d\in\mathbb{N}$. As a generalization on exponential family, a list of holonomy groups follows.

The global geometry of Riemannian manifolds with commuting curvature operators

We give manifolds in both the Riemannian and in the higher signature settings whose Riemann curvature operators commute, i.e. which satisfy R(a,b)R(c,d)=R(c,d)R(a,b) for all tangent vectors. These manifolds have global geometric phenomena which are quite different for higher signature manifolds than they are for Riemannian manifolds. Our focus is on global properties; questions of geodesic completeness and the behaviour of the exponential map are investigated.

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.

Finding Geodesics on Surfaces Using Taylor Expansion of Exponential Map

Our aim in this paper is to construct a numerical algorithm using Taylor expansion of exponential map to find geodesic joining two points on a 2-dimensional surface for which a Riemannian metric is defined

Regge's Einstein-Hilbert Functional on the Double Tetrahedron

The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, constant scalar curvature metrics, and the Yamabe problem on the double tetrahedron, with some reference to the possibilities on a general piecewise flat manifold. The main tool is analysis of Regge's Einstein-Hilbert functional, a piecewise flat analogue of the Einstein-Hilbert (or to…

Remarks on the spectrum of the Dirac operator of pseudo-Riemannian spin manifolds

We study the spectrum of the Dirac operator $D$ on pseudo-Riemannian spin manifolds of signature $(p,q)$, considered as an unbounded operator in the Hilbert space $L^2_ξ(S)$. The definition of $L^2_ξ(S)$ involves the choice of a $p$-dimensional time-like subbundle $ξ\subset TM$. We establish a sufficient criterion for the spectra of $D$ induced by two maximal time-like subbundles $ξ_1,ξ_2\subset TM$ to be equal. If the base manifold $M$ is compact, the spectrum does not depend on $ξ$ at all. We then proceed by explicitely computing the full spectrum of $D$ for $\mathbb R^{p,q}$, the flat torus…

Riemannian Geometry

These notes on Riemannian geometry use the bases bundle and frame bundle, as in Geometry of Manifolds, to express the geometric structures. It has more problems and omits the background material. It starts with the definition of Riemannian and semi-Riemannian structures on manifolds. Affine connections, geodesics, torsion and curvature, the exponential map, and the Riemannian connection follow quickly. The Taylor series for of the metric in normal coordinates is an unusual feature. The last third covers, first and second variation of energy, completeness, cut points, the Hadamard-Cartan theore…

A remark on vanishing geodesic distances in infinite dimensions

We observe that a vanishing geodesic distance arising from a weak Riemannian metric in a Hilbert manifold can be constructed.

On the mixed scalar curvature of almost multi-product manifolds

A pseudo-Riemannian manifold endowed with $k>2$ orthogonal complementary distributions (called a Riemannian almost multi-product structure) appears in such topics as multiply warped products, the webs composed of several foliations, Dupin hypersurfaces and in stu\-dies of the curvature and Einstein equations. In this article, we consider the following two problems on the mixed scalar curvature of a Riemannian almost multi-product manifold with a linear connection: a) integral formulas and applications to splitting of manifolds, b) variation formulas and applications to the mixed Einstein-Hilbe…

非连续黎曼度量流形的微分结构

The infinitesimal structure of manifolds with non-continuous Riemannian metrics

本文通过构造例子研究非连续黎曼度量空间无法成为微分Hilbert manifold或近似黎曼流形的问题，核心方法是利用局部可积的度量和弱导数。

Stochastic Characterization of Harmonic maps on Riemannian polyhedra

We have completely rewritten the paper, and corrected the proofs. We construct an exponential map at any point in the (n-1)-skeleton minus the (n-2)-skeleton of an n-dimensional Riemannian polyhedron. We have added allover the extra-assumption that the exponential map is totally geodesic at points in the (n-1)-skeleton minus the (n-2)-skeleton. This condition is automatically realized for instance if the dimension is one, or if the metric is flat.

On the $L^2-$Poincaré duality for incomplete riemannian manifolds: a general construction with applications

Let $(M,g)$ be an open, oriented and incomplete riemannian manifold of dimension $m$. Under some general conditions we show that it is possible to build a Hilbert complex $(L^2Ω^i(M,g),d_{\mathfrak{M},i})$ such that its cohomology groups, labeled with $H^i_{2,\mathfrak{M}}(M,g)$, satisfy the following properties: \begin{itemize} \item $H^i_{2,\mathfrak{M}}(M,g)=ker(d_{max,i})/\im(d_{min,i})$ \item $H^i_{2,\mathfrak{M}}(M,g)\cong H^{m-i}_{2,\mathfrak{M}}(M,g)$ (Poincaré duality holds) \end{itemize} Finally in the rest of the paper we study some properties of this complex with particular attenti…

The exponential map of the group of area-preserving diffeomorphisms of a surface with boundary

We prove that the Riemannian exponential map of the right-invariant $L^2$ metric on the group of volume-preserving diffeomorphisms of a two-dimensional manifold with a nonempty boundary is a nonlinear Fredholm map of index zero.

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.

On the biharmonic heat equation on complete Riemannian manifolds

We study entire solutions of the biharmonic heat equation on complete Riemannian manifolds without boundary. We provide exponential decay estimates for the biharmonic heat kernel under assumptions on the lower bound of Ricci curvature and noncollapsing of unit balls. And we prove a uniqueness criteria for the Cauchy problem. As corollaries we prove the conservation law for the biharmonic heat kernel and a uniform L-infinite estimate for entire solutions starting with bounded initial data.

Cut time in sub-Riemannian problem on Engel group

The left-invariant sub-Riemannian problem on the Engel group is considered. The problem gives the nilpotent approximation to generic nonholonomic systems in four-dimensional space with two-dimensional control, for instance to a system which describes motion of mobile robot with a trailer. The global optimality of extremal trajectories is studied via geometric control theory. The global diffeomorphic structure of the exponential mapping is described. As a consequence, the cut time is proved to be equal to the first Maxwell time corresponding to discrete symmetries of the exponential mapping.

关于不完整黎曼流形上 $L^2$-调和形式的上同调

On the cohomology of $L^2$-harmonic forms of an incomplete Riemannian manifold

本文将紧致流形上调和形式的上同调概念扩展到Hilbert复形，应用于不完整黎曼流形，特别是光滑分层Thom-Mather空间。

Short geodesics losing optimality in contact sub-Riemannian manifolds and stability of the 5-dimensional caustic

We study the sub-Riemannian exponential for contact distributions on manifolds of dimension greater or equal to 5. We compute an approximation of the sub-Riemannian Hamiltonian flow and show that the conjugate time can have multiplicity 2 in this case. We obtain an approximation of the first conjugate locus for small radii and introduce a geometric invariant to show that the metric for contact distributions typically exhibits an original behavior, different from the classical 3-dimensional case. We apply these methods to the case of 5-dimensional contact manifolds. We provide a stability analy…

流体力学中黎曼指数映射的奇异集

Singular Sets of Riemannian Exponential Maps in Hydrodynamics

该研究通过谱分析证明二维无粘Euler方程等的拉格朗日解映射奇异集为高斯零集，明确相关算子的Schatten-von Neumann类，并指出三维对应方程中解映射不具备Fredholm性质是本质性的。

Any sub-Riemannian Metric has Points of Smoothness

We prove the result stated in the title; it is equivalent to the existence of a regular point of the sub-Riemannian exponential mapping. We also prove that the metric is analytic on an open everywhere dense subset in the case of a complete real-analytic sub-Riemannian manifold.

The Riemannian manifold of all Riemannian metrics

The space of all Riemannian metrics on a smooth second countable finite dimensional manifold is itself a smooth manifold modeled on the space of symmetric (0,2)-tensor fields with compact support. It carries a canonical Riemannian metric which is invariant under the action of the diffeomorphism group. We determine its geodesics, exponential mapping, curvature, and Jacobi fields in a very explicit manner.

On the Exponential Stability of Projected Primal-Dual Dynamics on a Riemannian Manifold

Equivalence of convex optimization, saddle-point problems, and variational inequalities is a well-established concept. The variational inequality (VI) is a static problem which is studied under dynamical settings using a framework called the projected dynamical system, whose stationary points coincide with the static solutions of the associated VI. VI has rich properties concerning the monotonicity of its vector-valued map and the uniqueness of its solution, which can be extended to convex optimization and saddle-point problems. Moreover, these properties also extend to the representative proj…

Topological and Geometric Obstructions on Einstein-Hilbert-Palatini Theories

In this article we introduce $A$-valued Einstein-Hilbert-Palatini functional ($A$-EHP) over a n-manifold $M$, where $A$ is an arbitrary graded algebra, as a generalization of the functional arising in the study of the first order formulation of gravity. We show that if $A$ is weak $(k,s)$-solvable, then $A$-EHP is non-null only if $n<k+s+3$. We prove that essentially all algebras modeling classical geometries (except semi-Riemannian geometries with specific signatures) satisfy this condition for $k=1$ and $s=2$, including Hitchin's generalized complex geometry, Pantilie's generalized quaternio…

Semi - Riemannian Geometry with Nonholonomic Constraints

In the present article the geometry of semi-Riemannian manifolds with nonholonomic constraints is studied. These manifolds can be considered as analogues to the sub-Riemannian manifolds, where the positively definite metric is substituted by a nondegenerate metric. To study properties of the exponential map the Christoffel symbols and other differential operators were introduced. We study solutions of the Hamiltonian system and their projections into the underlying manifold. The explicit formulae were found for a specific example of a semi-Riemannian manifold with nonholonomic constraints.