From flops to diffeomorphism groups

Higgs bundles and diffeomorphism groups

By studying the Higgs bundle equations with the gauge group replaced by the group of symplectic diffeomorphisms of the 2-sphere we encounter the notion of a folded hyperkaehler 4-manifold and conjecture the existence of a family of such metrics parametrised by an infinite-dimensional analogue of Teichmueller space.

Lie groups of Poisson diffeomorphisms

By considering suitable Poisson groupoids, we develop an approach to obtain Lie group structures on (subgroups of) the Poisson diffeomorphism groups of various classes of Poisson manifolds. As applications, we show that the Poisson diffeomorphism groups of (normal-crossing) log-symplectic, elliptic symplectic, scattering-symplectic and cosymplectic manifolds are regular infinite-dimensional Lie groups.

The Group of Contact Diffeomorphisms for Compact Contact Manifolds

For a compact contact manifold it is shown that the anisotropic Folland-Stein function spaces form an algebra. The notion of anisotropic regularity is extended to define the space of Folland-Stein contact diffeomorphisms, which is shown to be a topological group under composition and a smooth Hilbert manifold. These results are used in a subsequent paper to analyze the action of the group of contact diffeomorphisms on the space of CR structures on a compact, three dimensional manifold.

Bi-invariant metric on contact diffeomorphisms group

We show the existence of a weak bi-invariant symmetric nondegenerate 2-form on the contact diffeomorphisms group $\mathcal{D}_θ$ of a contact Riemannian manifold $(M,g,θ)$ and study its properties. We describe the Euler's equation on a Lie algebra of group $\mathcal{D}_θ$ and calculate the sectional curvature of $\mathcal{D}_θ$. In a case $\dim M =3$ connection between the bi-invariant metric on $\mathcal{D}_θ$ and the bi-invariant metric on volume-preserving diffeomorphisms group $\mathcal{D}_μ$ of $M^3$ is discover.

Finsler 2-manifolds whose holonomy group is the diffeomorphism group of the circle

In this paper we show that the topological closure of the holonomy group of a certain class of projectively flat Finsler 2-manifolds of constant curvature is maximal, that is isomorphic to the connected component of the diffeomorphism group of the circle. This class of 2-manifolds contains the standard Funk plane of constant negative curvature and the Bryant-Shen-spheres of constant positive curvature. The result provides the first examples describing completely infinite dimensional Finslerian holonomy structures.

Bi-invariant metric on symplectic diffeomorphisms group

We show the existence of a weak bi-invariant symmetric nondegenerate 2-form on the symplectic diffeomorphisms group $\mathcal{D}_ω$ of a symplectic Riemannian manifold $(M,g,ω)$ and study its properties. We describe the Euler's equation on a Lie algebra of group $\mathcal{D}_ω$ and calculate the sectional curvature of $\mathcal{D}_ω(T^n)$.

The action of the diffeomorphism group on the space of immersions

We study the action of the diffeomorphism group $\Diff(M)$ on the space of proper immersions $\Imm_{\text{prop}}(M,N)$ by composition from the right. We show that smooth transversal slices exist through each orbit, that the quotient space is Hausdorff and is stratified into smooth manifolds, one for each conjugacy class of isotropy groups.

Totally geodesic subgroups of diffeomorphisms

We determine the Riemannian manifolds for which the group of exact volume preserving diffeomorphisms is a totally geodesic subgroup of the group of volume preserving diffeomorphisms, considering right invariant $L^2$-metrics. The same is done for the subgroup of Hamiltonian diffeomorphisms as a subgroup of the group of symplectic diffeomorphisms in the Kähler case. These are special cases of totally geodesic subgroups of diffeomorphisms with Lie algebras big enough to detect the vanishing of a symmetric 2-tensor field.

A Lie group structure on the group of real analytic diffeomorphisms of a compact real analytic manifold with corners

We construct a smooth Lie group structure on the group of real analytic diffeomorphisms of a compact analytic manifold with corners. This generalises the known analogous results in the situation where the real analytic manifold has no corners. Additionally our approach uses a different construction.

Weighted diffeomorphism groups of Riemannian manifolds

In this paper, we define locally convex vector spaces of weighted vector fields and use them as model spaces for Lie groups of weighted diffeomorphisms on Riemannian manifolds. We prove an easy condition on the weights that ensures that these groups contain the compactly supported diffeomorphisms. We finally show that for the special case where the manifold is the euclidean space, these Lie groups coincide with the ones constructed in the author's earlier work ["Weighted diffeomorphism groups of Banach spaces and weighted mapping groups". In: Dissertationes Math. 484 (2012), p. 128. DOI: 10.40…

Geodesic Equations on Diffeomorphism Groups

We bring together those systems of hydrodynamical type that can be written as geodesic equations on diffeomorphism groups or on extensions of diffeomorphism groups with right invariant $L^2$ or $H^1$ metrics. We present their formal derivation starting from Euler's equation, the first order equation satisfied by the right logarithmic derivative of a geodesic in Lie groups with right invariant metrics.

Geometry of Image Registration: The Diffeomorphism Group and Momentum Maps

These lecture notes explain the geometry and discuss some of the analytical questions underlying image registration within the framework of large deformation diffeomorphic metric mapping (LDDMM) used in computational anatomy.

A manifold structure for the group of orbifold diffeomorphisms of a smooth orbifold

For a compact, smooth C^r orbifold (without boundary), we show that the topological structure of the orbifold diffeomorphism group is a Banach manifold for finite r \ge 1 and a Frechet manifold if r=infty. In each case, the local model is the separable Banach (Frechet) space of C^r (C^infty, resp.) orbisections of the tangent orbibundle.

$n$-transitivity of certain diffeomorphism groups

Some groups of real analytic diffeomorphism act n-transitively for each finite n.

Canonical Deformed Groups of Diffeomorphisms and Finite Parallel Transports in Riemannian Spaces

We show that finite parallel transports of vectors in Riemannian spaces, determined by the multiplication law in the deformed groups of diffeomorphisms, and sequences of infinitesimal parallel transports of vectors along geodesics are equivalent.

On the transitivity of the group of orbifold diffeomorphisms

Consider a connected manifold of dimension at least two and the group of compactly supported diffeomorphisms that are compactly supported isotopic to the identity. This group acts $n$-transitive: Any tuple of $n$ points can be moved to any other tuple of $n$ points by a compactly supported diffeomorphism that is compactly supported isotopic to the identity. An orbifold diffeomorphism group is typically not $n$-transitive: simple obstructions are given by isomorphism classes of isotropy groups of points. In this paper we investigate the transitivity properties of the group of compactly supporte…

Bi-invariant metric on volume-preserving diffeomorphisms group of a three-dimensional manifold

We show the existence of a weak bi-invariant symmetric nondegenerate 2-form on the volume-preserving diffeomorphism group of a three-dimensional manifold and study its properties. Despite the fact that the space $\mathcal{D}_μ(M^3)$ is infinite-dimensional, we succeed in defining the signature of the bi-invariant quadric form. It is equal to the $η$-invariant of the manifold $M^3$.

The diffeomorphism groups of the real line are pairwise bihomeomorphic

We prove that the group D^r(R) of C^r diffeomorphisms of the real line, endowed with the compact-open and Whitney C^r topologies, is bihomeomorphic to the group H(R) of homeomorphisms of the real line endowed with the compact-open and Whitney topologies. This implies that the diffeomorphism group D^r(R) endowed with the Whitney C^r topology is homeomorphic to the countable box-power of the separable Hilbert space.

Two-Point Boundary Value Problems on Diffeomorphism Groups

We consider a variety of geodesic equations on Sobolev diffeomorphism groups, including the equations of ideal hydrodynamics. We prove that solutions of the corresponding two-point boundary value problems are precisely as smooth as their boundary conditions. We further utilise this regularity property to construct continuously differentiable exponential maps in the Frechét setting.

The orientation-preserving diffeomorphism group of S^2 deforms to SO(3) smoothly

Smale proved that the orientation-preserving diffeomorphism group of S^2 has a continuous strong deformation retraction to SO(3). In this paper, we construct such a strong deformation retraction which is diffeologically smooth.

The Lie group of real analytic diffeomorphisms is not real analytic

We construct an infinite dimensional real analytic manifold structure for the space of real analytic mappings from a compact manifold to a locally convex manifold. Here a map is real analytic if it extends to a holomorphic map on some neighbourhood of the complexification of its domain. As is well known the construction turns the group of real analytic diffeomorphisms into a smooth locally convex Lie group. We prove then that the diffeomorphism group is regular in the sense of Milnor. In the inequivalent "convenient setting of calculus" the real analytic diffeomorphisms even form a real analyt…

The diffeomorphism group of a Lie foliation

We explicitly compute the diffeomorphism group of several types of linear foliations (with dense leaves) on the torus $T^n$, $n\geq 2$, namely codimension one foliations, flows, and the so-called non-quadratic foliations. We show in particular that non-quadratic foliations are rigid, in the sense that they do not admit transverse diffeomorphisms other than $\pm \id$ and translations. The computation is an application of a general formula that we prove for the diffeomorphism group of any Lie foliation with dense leaves on a compact manifold. Our results generalize those of P. Donato and P. Igle…

On Completeness of Groups of Diffeomorphisms

We study completeness properties of the Sobolev diffeomorphism groups $\mathcal D^s(M)$ endowed with strong right-invariant Riemannian metrics when the underlying manifold $M$ is $\mathbb R^d$ or compact without boundary. The main result is that for $s > \dim M/2 + 1$, the group $\mathcal D^s(M)$ is geodesically and metrically complete with a surjective exponential map. We then present the connection between the Sobolev diffeomorphism group and the large deformation matching framework in order to apply our results to diffeomorphic image matching.

Smooth perfectness for the group of diffeomorphisms

Given a result of Herman, we provide a new elementary proof of the fact that the connected component of the group of compactly supported diffeomorphisms is perfect and hence simple. Moreover, we show that every diffeomorphism $g$, which is sufficiently close to the identity, can be represented as a product of four commutators, $g=[h_1,k_1]\circ...\circ[h_4,k_4]$, where the factors $h_i$ and $k_i$ can be chosen to depend smoothly on $g$.

Non Jordan groups of diffeomorphisms and actions of compact Lie groups on manifolds

A recent preprint of Csikós, Pyber and Szabó (arXiv:1411.7524) proves that the diffeomorphism group of $T^2\times S^2$ is not Jordan. The purpose of this paper is to generalize the arguments of Csikós, Pyber and Szabó in order to obtain many other examples of compact manifolds whose diffeomorphism group fails to be Jordan. In particular we prove that for any $ε>0$ there exist manifolds admitting effective actions of arbitrarily large $p$-groups $Γ$ all of whose abelian subgroups have at most $|Γ|^ε$ elements. Finally, we also recover some results on nonexistence of effective actions of compact…

Families of vector fields which generate the group of diffeomorphisms

Given a compact manifold M, we prove that any bracket generating and invariant under multiplication on smooth functions family of vector fields on M generates the connected component of unit of the group Diff(M).

Tangent Lie algebra of a diffeomorphism group and application to holonomy theory

In this paper we introduce the notion of tangent space TG of a (not necessary smooth) subgroup G of the diffeomorphism group Diff(M) of a compact manifold M. We prove that TG is a Lie subalgebra of the Lie algebra of smooth vector fields on M. The construction can be generalized to subgroups of any (finite or infinite dimensional) Lie groups. The tangent Lie algebra TG introduced this way is a generalization of the classical Lie algebra in the smooth cases. As a working example, we discuss in detail the tangent structure of the holonomy group and fibered holonomy group of Finsler manifolds.

Amari-Chentsov connections and their geodesics on homogeneous spaces of diffeomorphism groups

We study the family of $α$-connections of Amari-Chentsov on the homogeneous space $\mathcal{D}(M)/\mathcal{D}_μ(M)$ of diffeomorphisms modulo volume-preserving diffeomorphims of a compact manifold $M$. We show that in some cases their geodesic equations yield completely integrable Hamiltonian systems.

Differentiable conjugacy for groups of area preserving circle diffeomorphisms

We study groups of circle diffeomorphisms whose action on the cylinder $\mathcal C=\mathbb S^1\times \mathbb S^1\setminus Δ$ preserves a volume form. We first show that such a group is topologically conjugate to a subgroup of $\rm{PSL}(2,\mathbb R)$, then discuss the existence of a differentiable conjugacy.

Shock waves for the Burgers equation and curvatures of diffeomorphism groups

We establish a simple relation between curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation corresponds to a focal point of the group of volume-preserving diffeomorphisms regarded as a submanifold of the full diffeomorphism group and, consequently, to a conjugate point along a geodesic in the Wasserstein space of densities. This establishes an intrinsic connection between ideal Euler hydrodynamics (via Arnold's approach), shock formation in the multidim…

Finite groups of diffeomorphisms are topologically determined by a vector field

In a previous work it is shown that every finite group $G$ of diffeomorphisms of a connected smooth manifold $M$ of dimension $\geq 2$ equals, up to quotient by the flow, the centralizer of the group of smooth automorphisms of a $G$-invariant complete vector field $X$ (shortly $X$ describes $G$). Here the foregoing result is extended to show that every finite group of diffeomorphisms of $M$ is described, within the group of all homeomorphisms of $M$, by a vector field. As a consequence, it is proved that a finite group of homeomorphisms of a compact connected topological $4$-manifold, whose ac…

Equivariant homotopy and deformations of diffeomorphisms

We present a way of constructing and deforming diffeomorphisms of manifolds endowed with a Lie group action. This is applied to the study of exotic diffeomorphisms and involutions of spheres and to the equivariant homotopy of Lie groups.

保体积微分同胚群的一个对偶对

A dual pair for the group of volume preserving diffeomorphisms

该研究利用光滑嵌入空间的余切丛构造保体积微分同胚群的辛对偶对，通过辛约化用增广子流形的非线性格拉斯曼流形描述其馀伴随轨道，并给出余维数1嵌入下子流形的约束条件。

The uniform perfectness of diffeomorphism groups of open manifolds

In this paper we study the uniform perfectness, boundedness and uniform simplicity of diffeomorphism groups of compact manifolds with boundary and open manifolds and obtain some upper bounds of their diameters with respect to commutator length, those with support in balls and conjugation-generated norm.

辛微分同胚群的几何

A translation of "The Geometry of the Group of Symplectic Diffeomorphisms"

本书介绍霍费尔度量，研究哈密顿微分同胚群的直径、测地线及单参数子群增长，并应用于动力学和遍历理论。

Asymptotic directions, Monge-Ampere equations and the geometry of diffeomorphism groups

In this note we obtain the characterization for asymptotic directions on various subgroups of the diffeomorphism group. We give a simple proof of non-existence of such directions for area-preserving diffeomorphisms of closed surfaces of non-zero curvature. Finally, we exhibit the common origin of the Monge-Ampere equations in 2D fluid dynamics and mass transport.

Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. II

The geodesic distance vanishes on the group of compactly supported diffeomorphisms of a Riemannian manifold $M$ of bounded geometry, for the right invariant weak Riemannian metric which is induced by the Sobolev metric $H^s$ of order $0\le s<\tfrac12$ on the Lie algebra $\mathfrak X_c(M)$ of vector fields with compact support.

A zoo of diffeomorphism groups on $\mathbb R^n$

We consider the groups $\operatorname{Diff}_{\mathcal B}(\mathbb R^n)$, $\operatorname{Diff}_{H^\infty}(\mathbb R^n)$, and $\operatorname{Diff}_{\mathcal S}(\mathbb R^n)$ of smooth diffeomorphisms on $\mathbb R^n$ which differ from the identity by a function which is in either $\mathcal B$ (bounded in all derivatives), $H^\infty = \bigcap_{k\ge 0}H^k$, or $\mathcal S$ (rapidly decreasing). We show that all these groups are smooth regular Lie groups.

Diffeomorphism Groups of Compact 4-manifolds are not always Jordan

We show that if $M$ is a compact smooth manifold diffeomorphic to the total space of an orientable $S^2$ bundle over the torus $T^2$, then its diffeomorphism group does not have the Jordan property, i.e., Diff$(M)$ contains a finite subgroup $G_n$ for any natural number $n$ such that every abelian subgroup of $G_n$ has index at leat $n$. This gives a counterexample to an old conjecture of Ghys.

Quasi-Isometric Embeddings into Diffeomorphism Groups

Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form. Assuming certain conditions on the fundamental group $π_1(M)$ we construct quasi-isometric embeddings of either free Abelian or direct products of non-Abelian free groups into the group of volume preserving diffeomorphisms of M equipped with the L^p metric induced by a Riemannian metric on M.

Bounded symplectic diffeomorphisms and split flux groups

We prove the bounded isometry conjecture of F. Lalonde and L. Polterovich for a special class of closed symplectic manifolds. As a byproduct, it is shown that the flux group of a product of these special symplectic manifold is isomorphic to the direct sum of the flux group of each symplectic manifold.

Diffeomorphisms of the circle and Brownian motions on an infinite-dimensional symplectic group

An embedding of the group $\Diff(S^{1})$ of orientation preserving diffeomorphims of the unit circle $S^1$ into an infinite-dimensional symplectic group, $\Sp(\infty)$, is studied. The authors prove that this embedding is not surjective. A Brownian motion is constructed on $\Sp(\infty)$. This study is motivated by recent work of H. Airault, S. Fang and P. Malliavin.

实微分同胚群的全纯逼近

Holomorphic Approximation for Real Diffeomorphism Groups

该研究利用密度性质，证明n≥2时ℝⁿ的微分同胚可在Whitney Cᵏ拓扑中被ℂⁿ自同构逼近，多数线性代数群的分裂实形式满足相关充分条件，且其保体积微分同胚也可被全纯逼近。

Higher analytic torsion and cohomology of diffeomorphism groups

We consider a closed odd-dimensional oriented manifold $M$ together with an acyclic flat hermitean vector bundle $\cF$. We form the trivial fibre bundle with fibre $M$ over the manifold of all Riemannian metrics on $M$. It has a natural flat connection and a vertical Riemannian metric. The higher analytic torsion form of Bismut/Lott associated to the situation is invariant with respect to the connected component of the identity of the diffeomorphism group of $M$. Using that the space of Riemannian metrics is contractible we define continuous cohomology classes of the diffeomorphism group and i…

流形边界上的可控性与微分同胚群

Controllability and diffeomorphism groups on manifolds with boundary

本文研究具有光滑边界的流形的微分同胚群，证明其与边界微分同胚群构成具有局部截面的短正合序列，并建立向量场族的可控性结果，推广了Agrachev和Caponigro的结论，表明该群由指数映射的像生成。

Normal families of functions and groups of pseudoconformal diffeomorphisms of quaternion and octonion variables

This paper is devoted to the specific class of pseudoconformal mappings of quaternion and octonion variables. Normal families of functions are defined and investigated. Four criteria of a family being normal are proven. Then groups of pseudoconformal diffeomorphisms of quaternion and octonion manifolds are investigated. It is proven, that they are finite dimensional Lie groups for compact manifolds. Their examples are given. Many charactersitic features are found in comparison with commutative geometry over $\bf R$ or $\bf C$.

Quasi-morphisms on surface diffeomorphism groups

We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its fragmentation norm is unbounded, answering a question of Burago--Ivanov--Polterovich. As a key tool we construct a hyperbolic graph on which these groups act, which is the analog of the curve graph for the mapping class group.

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.

通过微分同胚群上的梯度流进行形状分析

Shape analysis via gradient flows on diffeomorphism groups

本文研究图像配准问题中的Sobolev微分同胚梯度流，证明了该流的适定性，并利用动量映射给出梯度的几何描述。

Homotopy types of Diffeomorphism groups of noncompact 2-manifolds

Suppose M is a noncompact connected smooth 2-manifold without boundary and let D(M)_0 denote the identity component of the diffeomorphism group of M with the compact-open C^infty-topology. In this paper we investigate the topological type of D(M)_0 and show that D(M)_0 is a topological ell_2-manifold and it has the homotopy type of the circle if M is the plane, the open annulus or the open M"obius band, and it is contractible in all other cases. When M admits a volume form w, we also discuss the topological type of the group of w-preserving diffeomorphisms of M. To obtain these results we stud…

面积保持微分同胚环的中心扩张及其模糊球极限

Central extensions for loop groups of area-preserving diffeomorphisms and their fuzzy sphere limits

研究面积保持微分同胚环的中心扩张，并揭示其在大k极限下的模糊球极限性质。

微分同胚群与族的规范理论

Diffeomorphism groups and gauge theory for families

本文综述了族的规范理论，重点探讨了2021-2025年间发展出的用于4维流形微分同胚群的应用方法。

深度神经网络在微分同胚群上的最优形状重参数化

Deep neural networks on diffeomorphism groups for optimal shape reparameterization

本文提出一种基于初等微分同胚复合的算法，利用PyTorch实现，用于高效求解形状对齐中的最优重参数化问题，并证明了其通用逼近性质及Lipschitz常数界。

接触分布流形微分同胚群中的正路径

Positive paths in diffeomorphism groups of manifolds with a contact distribution

研究在接触分布流形微分同胚群中定义正路径的概念，并展示其在标准接触结构上的广泛应用。