Diffeomorphisms Research Articles

Controllability and diffeomorphism groups on manifolds with boundary

Abstract In this article we consider diffeomorphism groups of manifolds with smooth boundary. We show that the diffeomorphism groups of the manifold and its boundary fit into a short exact sequence which admits local sections. In other words, they form an infinite-dimensional fibre bundle. Manifolds with boundary are of interest in numerical analysis and with a view towards applications in machine learning we establish controllability results for families of vector fields. This generalises older results due to Agrachev and Caponigro in the boundary-less case. Our results show in particular that the diffeomorphism group of a manifold with smooth boundary is generated by the image of the exponential map.

Diffeomorphisms of Foliated Manifolds I

The set $Diff(M)$ of all diffeomorphisms of manifold $M$ onto itself is the group related to composition and inverse mapping. The group of diffeomorphisms of smooth manifolds is of great importance in differential geometry and analysis. It is known that the group $Diff(M)$ is topological group in compact open topology.In this paper we investigate the group $Diff_{F}(M)$ of diffeomorphisms foliated manifold $(M,F)$ with foliated compact open topology. In this paper we prove that if all leaves of the the foliation $F$ are closed subsets of $M$ then the foliated compact open topology of the group $Diff_{F}(M)$ coincides with compact open topology. In addition it is studied the question on the dimension of the group of isometries of foliated manifold is studied when foliation generated by riemannian submersion.

Symplectic Geometry of Teichmüller Spaces for Surfaces with Ideal Boundary

A hyperbolic 0-metric on a surface with boundary is a hyperbolic metric on its interior, exhibiting the boundary behavior of the standard metric on the Poincaré disk. Consider the infinite-dimensional Teichmüller spaces of hyperbolic 0-metrics on oriented surfaces with boundary, up to diffeomorphisms fixing the boundary and homotopic to the identity. We show that these spaces have natural symplectic structures, depending only on the choice of an invariant metric on sl(2,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {sl}(2,\\mathbb {R})$$\\end{document}. We prove that these Teichmüller spaces are Hamiltonian Virasoro spaces for the action of the universal cover of the group of diffeomorphisms of the boundary. We give an explicit formula for the Hill potential on the boundary defining the moment map. Furthermore, using Fenchel–Nielsen parameters we prove a Wolpert formula for the symplectic form, leading to global Darboux coordinates on the Teichmüller space.

Note on the group of vertical diffeomorphisms of a principal bundle & its relation to the Frölicher-Nijenhuis bracket

The group of vertical diffeomorphisms of a principal bundle forms the action Lie groupoid associated to the bundle. The former is generated by the group of maps with value in the structure group, which is also the group of bisections of the groupoid. The corresponding Lie algebra of general vertical vector fields is generated by maps with value in the Lie algebra of the structure group. The bracket on these maps, induced by the bracket of vertical vector fields, is an “extended” bracket on gauge parameters: it has been introduced heuristically in physics, notably in the study of asymptotic symmetries of gravity. Seeing the set of Lie algebra-valued maps as sections of the action Lie algebroid associated to the bundle, the extended bracket is understood to be a Lie algebroid bracket on those sections.Here, we highlight that this bracket can also be seen to arise from the Frölicher-Nijenhuis bracket of vector-valued differential forms. The benefit of this viewpoint is to insert this extended bracket within the general framework of derivations of forms on a bundle. Identities relating it to the usual operations of Cartan calculus — inner product, exterior and (Nijenhuis-) Lie derivative — are immediately read as special cases of general results. We also consider the generalised gauge transformations induced by vertical diffeomorphisms, and discuss their peculiar features. In particular, locally, and contrary to standard gauge transformations arising from vertical bundle automorphisms, they are distinguishable from local gluings when iterated. Yet, the gauge principle still holds.

Horizon phase spaces in general relativity

We derive a prescription for the phase space of general relativity on two intersecting null surfaces using the null initial value formulation. The phase space allows generic smooth initial data, and the corresponding boundary symmetry group is the semidirect product of the group of arbitrary diffeomorphisms of each null boundary which coincide at the corner, with a group of reparameterizations of the null generators. The phase space can be consistently extended by acting with half-sided boosts that generate Weyl shocks along the initial data surfaces. The extended phase space includes the relative boost angle between the null surfaces as part of the initial data.We then apply the Wald-Zoupas framework to compute gravitational charges and fluxes associated with the boundary symmetries. The non-uniqueness in the charges can be reduced to two free parameters by imposing covariance and invariance under rescalings of the null normals. We show that the Wald-Zoupas stationarity criterion cannot be used to eliminate the non-uniqueness. The different choices of parameters correspond to different choices of polarization on the phase space. We also derive the symmetry groups and charges for two subspaces of the phase space, the first obtained by fixing the direction of the normal vectors, and the second by fixing the direction and normalization of the normal vectors. The second symmetry group consists of Carrollian diffeomorphisms on the two boundaries.Finally we specialize to future event horizons by imposing the condition that the area element be non-decreasing and become constant at late times. For perturbations about stationary backgrounds we determine the independent dynamical degrees of freedom by solving the constraint equations along the horizons. We mod out by the degeneracy directions of the presymplectic form, and apply a similar procedure for weak non-degeneracies, to obtain the horizon edge modes and the Poisson structure. We show that the area operator of the black hole generates a shift in the relative boost angle under the Poisson bracket.

Characteristic cohomology II: Matrix singularities

AbstractFor a germ of a variety , a singularity of “type ” is given by a germ , which is transverse to in an appropriate sense, such that . In part I of this paper, we introduced for such singularities the Characteristic Cohomology for the Milnor fiber (for a hypersurface), and complement and link (for the general case). It captures the cohomology of inherited from and is given by subalgebras of the cohomology for for the Milnor fiber and complements, and is a subgroup for the cohomology of the link. We showed these cohomologies are functorial and invariant under diffeomorphism groups of equivalences for Milnor fibers and for complements and links. We also gave geometric criteria for detecting the nonvanishing of the characteristic cohomology.In this paper, we apply these methods in the case denotes any of the varieties of singular complex matrices, which may be either general, symmetric, or skew‐symmetric (with even). For these varieties, we have shown in another paper that their Milnor fibers and complements have compact “model submanifolds” for their homotopy types, which are classical symmetric spaces in the sense of Cartan. As a result, we first give the structure of the characteristic cohomology subalgebras for the Milnor fibers and complements as images of exterior algebras (or in one case a module on two generators over an exterior algebra). For links, the characteristic cohomology group is the image of a shifted upper truncated exterior algebra. In addition, we extend these results for the complement and link to the case of general complex matrices.Second, we then apply the geometric detection methods introduced in Part I to detect when specific characteristic cohomology classes for the Milnor fiber or complement are nonzero. We identify an exterior subalgebra on a specific set of generators and for the link that it contains an appropriate shifted upper truncated exterior subalgebra. The detection criterion involves a special type of “kite map germ of size ” based on a given flag of subspaces. The general criterion that detects such nonvanishing characteristic cohomology is then given in terms of the defining germ containing such a kite map germ of size . Furthermore, we use a restricted form of kite spaces to give a cohomological relation between the cohomology of local links and the global link for the varieties of singular matrices.

Correction to: Positivity for the curvature of the diffeomorphism group corresponding to the incompressible Euler equation with Coriolis force

Correction to: Positivity for the curvature of the diffeomorphism group corresponding to the incompressible Euler equation with Coriolis force

Stabilisation, scanning, and handle cancellation

In this note, we describe a family of arguments that link the homotopy type of (a) the diffeomorphism group of the disc D^{n} , (b) the space of co-dimension one embedded spheres in S^{n} , and (c) the homotopy type of the space of co-dimension two trivial knots in S^{n} . We also describe some natural extensions to these arguments. We begin with Cerf’s “upgraded” proof of Smale’s theorem, showing that the diffeomorphism group of S^{2} has the homotopy type of the isometry group. This entails a cancelling-handle construction, related to recently studied “scanning” maps of spaces of embeddings \operatorname{Emb}(D^{n-1}, S^{1}\times D^{n-1}) \to \Omega^{j} \operatorname{Emb}(D^{n-1-j}, S^{1} \times D^{n-1}) . We further give a Bott-style variation on Cerf’s construction and a related embedding calculus framework for these constructions. We use these arguments to prove that the monoid of Schönflies spheres \pi_{0} \operatorname{Emb}(S^{n-1}, S^{n}) is a group with respect to the connected-sum operation for all n \geq 2 . This last result is perhaps only interesting when n=4 , as when n \neq 4 , it follows from the resolution of the various generalised Schönflies problems.

The unbounded Lagrangian spectral norm and wrapped Floer cohomology

We investigate the question of whether the spectral metric on the orbit space of a fiber in the disk cotangent bundle of a closed manifold, under the action of the compactly supported Hamiltonian diffeomorphism group, is bounded. We utilize wrapped Floer cohomology to define the spectral invariant of an admissible Lagrangian submanifold within a Weinstein domain. We show that the pseudo-metric derived from this spectral invariant is a valid Ham-invariant metric. Furthermore, we establish that the spectral metric on the orbit space of an admissible Lagrangian is bounded if and only if the wrapped Floer cohomology vanishes. Consequently, we prove that the Lagrangian Hofer diameter of the orbit space for any fiber in the disk cotangent bundle of a closed manifold is infinite.

Geometric analysis of the generalized surface quasi-geostrophic equations

We investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation. This family can be realised as geodesic equations on groups of diffeomorphisms. We show precisely when the corresponding Riemannian exponential map is non-linear Fredholm of index 0. We further illustrate this by examining the distribution of conjugate points in these settings via a Morse theoretic approach

Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 2

Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 2

Smooth actions of connected compact Lie groups with a free point are determined by two vector fields

Consider a smooth action G×M→M of a compact connected Lie group G on a connected manifold M. Assume the existence of a point of M whose isotropy group has a single element (a free point). Then we prove that there exist two complete vector field X,X1 such that their group of automorphisms equals G regarded as a group of diffeomorphisms of M (the existence of a free point implies that the action of G is effective). Moreover, some examples of effective actions with no free point where this result fails are exhibited.

Long‐time existence of Brownian motion on configurations of two landmarks

AbstractWe study Brownian motion on the space of distinct landmarks in , considered as a homogeneous space with a Riemannian metric inherited from a right‐invariant metric on the diffeomorphism group. As of yet, there is no proof of long‐time existence of this process, despite its fundamental importance in statistical shape analysis, where it is used to model stochastic shape evolutions. We make some first progress in this direction by providing a full classification of long‐time existence for configurations of exactly two landmarks, governed by a radial kernel. For low‐order Sobolev kernels, we show that the landmarks collide with positive probability in finite time, whilst for higher‐order Sobolev and Gaussian kernels, the landmark Brownian motion exists for all times. We illustrate our theoretical results by numerical simulations.

A Metric Fixed Point Theorem and Some of Its Applications

A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new for isometries of convex sets of Banach spaces as well as for non-locally compact CAT(0)-spaces and injective spaces. Examples of actions on non-proper CAT(0)-spaces come from the study of diffeomorphism groups, birational transformations, and compact Kähler manifolds. A special case of the fixed point theorem provides a novel mean ergodic theorem that in the Hilbert space case implies von Neumann’s theorem. The theorem accommodates classically fixed-point-free isometric maps such as those of Kakutani, Edelstein, Alspach and Prus. Moreover, from the main theorem together with some geometric arguments of independent interest, one can deduce that every bounded invertible operator of a Hilbert space admits a nontrivial invariant metric functional on the space of positive operators. This is a result in the direction of the invariant subspace problem although its full meaning is dependent on a future determination of such metric functionals.

On Some Aspects of the Courant-Type Algebroids, the Related Coadjoint Orbits and Integrable Systems

Poisson structures related to affine Courant-type algebroids are analyzed, including those related with cotangent bundles on Lie-group manifolds. Special attention is paid to Courant-type algebroids and their related R structures generated by suitably defined tensor mappings. Lie–Poisson brackets that are invariant with respect to the coadjoint action of the loop diffeomorphism group are created, and the related Courant-type algebroids are described. The corresponding integrable Hamiltonian flows generated by Casimir functionals and generalizing so-called heavenly-type differential systems describing diverse geometric structures of conformal type in finite dimensional Riemannian manifolds are described.

On the Cauchy problem for a Kadomtsev-Petviashvili hierarchy on non-formal operators and its relation with a group of diffeomorphisms

On the Cauchy problem for a Kadomtsev-Petviashvili hierarchy on non-formal operators and its relation with a group of diffeomorphisms

Decorated Nonlinear Flags, Pointed Vortex Loops and the Dihedral Group

Abstract We identify pointed vortex loops in the plane with low dimensional nonlinear flags decorated with volume forms. We show how submanifolds of such decorated nonlinear flags can be identified with coadjoint orbits of the area pre- serving diffeomorphism group and relate them to coadjoint orbits of pointed vortex loops. The subgroup of the dihedral group preserving the vorticity data plays a role in the description of these coadjoint orbits.

On the Quantum Deformations of Associative Sato Grassmannian Algebras and the Related Matrix Problems

We analyze the Lie algebraic structures related to the quantum deformation of the Sato Grassmannian, reducing the problem to studying co-adjoint orbits of the affine Lie subalgebra of the specially constructed loop diffeomorphism group of tori. The constructed countable hierarchy of linear matrix problems made it possible, in part, to describe some kinds of Frobenius manifolds within the Dubrovin-type reformulation of the well-known WDVV associativity equations, previously derived in topological field theory. In particular, we state that these equations are equivalent to some bi-Hamiltonian flows on a smooth functional submanifold with respect to two compatible Poisson structures, generating a countable hierarchy of commuting to each other’s hydrodynamic flows. We also studied the inverse problem aspects of the quantum Grassmannian deformation Lie algebraic structures, related with the well-known countable hierarchy of the higher nonlinear Schrödinger-type completely integrable evolution flows.

Intrinsic Geometric Structure of Subcartesian Spaces

Every subset S of a Cartesian space Rd, endowed with differential structure C∞(S) generated by restrictions to S of functions in C∞(Rd), has a canonical partition M(S) by manifolds, which are orbits of the family X(S) of all derivations of C∞(S) that generate local one-parameter groups of local diffeomorphisms of S. This partition satisfies the frontier condition, Whitney’s conditions A and B. If M(S) is locally finite, then it satisfies all definitions of stratification of S. This result extends to Hausdorff locally Euclidean differential spaces. The partition M(S) of a subcartesian space S by smooth manifolds provides a measure for the applicability of differential geometric methods to the study of the geometry of S. If all manifolds in M(S) are single points, we cannot expect differential geometry to be an effective tool in the study of S. On the other extreme, if M(S) contains only one manifold M, then the subcartesian space S is a manifold, S=M, and it is a natural domain for differential geometric techniques.

Hofer geometry via toric degeneration

The main theme of this paper is to use toric degeneration to study Hofer geometry by producing distinct homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms. We focus on the (complex n-dimensional) quadric hypersurface and the del Pezzo surfaces, and study two classes of distinguished Lagrangian submanifolds that appear naturally in a toric degeneration, namely the Lagrangian torus which is the monotone fiber of a Lagrangian torus fibration, and the Lagrangian spheres that appear as vanishing cycles. For the quadrics, we prove that the group of Hamiltonian diffeomorphisms admits two distinct homogeneous quasimorphisms and derive some superheaviness results. Along the way, we show that the toric degeneration is compatible with the Biran decomposition. This implies that for n=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=2$$\\end{document}, the Lagrangian fiber torus (Gelfand–Zeitlin torus) is Hamiltonian isotopic to the Chekanov torus, which answers a question of Y. Kim. We prove analogous results for the del Pezzo surfaces. We also discuss applications to C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^0$$\\end{document} symplectic topology.