Group Of Orientation Preserving Diffeomorphisms Research Articles

Constructing the Virasoro Groups Using Differential Cohomology

Abstract The Virasoro groups are a family of central extensions of $\textrm{Diff}^+(S^1)$, the group of orientation-preserving diffeomorphisms of $S^1$, by the circle group $\mathbb T$. We give a novel, geometric construction of these central extensions using “off-diagonal” differential lifts of the first Pontryagin class, thus affirmatively answering a question of Freed–Hopkins.

On Moduli Spaces of Positive Scalar Curvature Metrics on Highly Connected Manifolds

Abstract Let $M$ be a simply connected spin manifold of dimension at least six, which admits a metric of positive scalar curvature. We show that the observer moduli space of positive scalar curvature metrics on $M$ has non-trivial higher homotopy groups. Moreover, denote by $\mathcal{M}_0^+(M)$ the moduli space of positive scalar curvature metrics on $M$ associated to the group of orientation-preserving diffeomorphisms of $M$. We show that if $M$ belongs to a certain class of manifolds that includes $(2n-2)$-connected $(4n-2)$-dimensional manifolds, then the fundamental group of $\mathcal{M}_0^+(M)$ is non-trivial.

On a class of immersions of spheres into space forms of nonpositive curvature

Let \( M^{n+1} \) (\( n \ge 2 \)) be a simply-connected space form of sectional curvature \( -\kappa ^2 \) for some \( \kappa \ge 0 \), and I an interval not containing \( [-\kappa ,\kappa ] \) in its interior. It is known that the domain of a closed immersed hypersurface of M whose principal curvatures lie in I must be diffeomorphic to the n-sphere \( \mathbb {S}^n \). These hypersurfaces are thus topologically rigid. The purpose of this paper is to show that they are also homotopically rigid. More precisely, for fixed I, the space \( \mathcal {F} \) of all such hypersurfaces is either empty or weakly homotopy equivalent to the group of orientation-preserving diffeomorphisms of \( \mathbb {S}^n \). An equivalence assigns to each element of \( \mathcal {F} \) a suitable modification of its Gauss map. For M not simply-connected, \( \mathcal {F} \) is the quotient of the corresponding space of hypersurfaces of the universal cover of M by a natural free proper action of the fundamental group of M.

The universal n-pointed surface bundle only has n sections

The classifying space BDiff[Formula: see text] of the orientation-preserving diffeomorphism group of a surface [Formula: see text] of genus [Formula: see text] fixing [Formula: see text] points pointwise has a universal bundle [Formula: see text] The [Formula: see text] fixed points provide [Formula: see text] sections [Formula: see text] of [Formula: see text]. In this paper we prove a conjecture of R. Hain that any section of [Formula: see text] is homotopic to some [Formula: see text]. Let [Formula: see text] be the space of ordered [Formula: see text]-tuple of distinct points on [Formula: see text]. As part of the proof of Hain’s conjecture, we prove a result of independent interest: any surjective homomorphism [Formula: see text] is equal to one of the forgetful homomorphisms [Formula: see text], possibly post-composed with an automorphism of [Formula: see text]. We also classify sections of the universal hyperelliptic surface bundle.

Euler-Poincaré-Arnold equations on semi-direct products II

We study the well-posedness of the Euler-Poincare-Arnold equations on the semi-direct products of the group of orientation-preserving diffeomorphisms of the circle with itself. To achieve this goal, according to the previous results obtained in [5], we had to extend the results obtained in [10] for the general case of inertia operators of pseudo-differential type.

Топологическая классификация функций Морса рода 1 на S^3

We study the question about a topological classification of Morse functions on the 3-sphere, all critical points of which lie on a different level surfaces. The classification provides with respect to the group Diff0(S3) x Diff0(R) - the group of orientation-preserving diffeomorphisms of the source and the target. We give a description of a corresponding oriented graphs (Kronrod-Reeb graphs). It is shown that these graphs completely classify genus 1 functions. These functions has a property that the genus of all the components of their level surfaces is not greater then 1. Moreover, all these graphs can be realized by a genus 1 functions, thus they can not distinguish a topological type of a more complex functions.

Controllability on Infinite-Dimensional Manifolds: A Chow–Rashevsky Theorem

One of the fundamental problems in control theory is that of controllability, the question of whether one can drive the system from one point to another with a given class of controls. A classical result in geometric control theory of finite-dimensional (nonlinear) systems is Chow–Rashevsky theorem that gives a sufficient condition for controllability on any connected manifold of finite dimension. In other words, the classical Chow–Rashevsky theorem, which is in fact a primary theorem in subriemannian geometry, gives a global connectivity property of a subriemannian manifold. In this paper, following the unified approach of Kriegl and Michor (The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, vol. 53, Am. Math. Soc., Providence, 1997) for a treatment of global analysis on a class of locally convex spaces known as convenient, we give a generalization of Chow–Rashevsky theorem for control systems in regular connected manifolds modelled on convenient (infinite-dimensional) locally convex spaces which are not necessarily normable. To indicate an application of our approach to the infinite-dimensional geometric control problems, we conclude the paper with a novel controllability result on the group of orientation-preserving diffeomorphisms of the unit circle.

The orientation-preserving diffeomorphism group of ${\mathbb{S}^{2}}$ deforms to SO(3) smoothly

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

Groups of volume-preserving diffeomorphisms of noncompact manifolds and mass flow toward ends

Suppose M is a noncompact connected oriented C ∞ n-manifold and ω is a positive volume form on M. Let D + (M) denote the group of orientation-preserving diffeomorphisms of M endowed with the compact-open C ∞ topology and let D(M; w) denote the subgroup of ω-preserving diffeomorphisms of M. In this paper we propose a unified approach for realization of mass transfer toward ends by diffeomorphisms of M. This argument, together with Moser's theorem, enables us to deduce two selection theorems for the groups D + (M) and D(M;ω). The first one is the extension of Moser's theorem to noncompact manifolds, that is, the existence of sections for the orbit maps under the action of D + (M) on the space of volume forms. This implies that D(M; w) is a strong deformation retract of the group D + (M;E ω M ) consisting of h ∈ D + (M), which preserves the set E ω M of ω-finite ends of M. The second one is related to the mass flow toward ends under volume-preserving diffeomorphisms of M. Let D EM (M; w) denote the subgroup consisting of all h ∈ D(M; ω) which fix the ends E M of M. S. R. Alpern and V. S. Prasad introduced the topological vector space S(M; ω) of end charges of M and the end charge homomorphism c ω : D EM (M; ω) → S(M; ω), which measures the mass flow toward ends induced by each h ∈ D EM (M; ω). We show that the homomorphism c ω has a continuous section. This induces the factorization D EM (M; ω) ≅ ker c ω x S(M; ω), and it implies that ker c ω is a strong deformation retract of D EM (M; ω).

Realization of the mapping class group of handlebody by diffeomorphisms

For the oriented 3-dimensional handlebody constructed from a 3-ball by attaching 1-handles, it is shown that the natural surjection from the group of orientation-preserving diffeomorphisms of it to the mapping class group of it has no section when is at least 6.

Lagrangian time-discretization of the Hunter–Saxton equation

We study Lagrangian time-discretizations of the Hunter–Saxton equation. Using the Moser–Veselov approach, we obtain such discretizations defined on the Virasoro group and on the group of orientation-preserving diffeomorphisms of the circle. We conjecture that one of these discretizations is integrable.

Actions de groupes de Kazhdan sur le cercle

We prove that every homomorphism from a discrete Kazhdan group to the group of orientation-preserving diffeomorphisms of the circle of class C 1+ α ( α>1/2) has a finite image.

A Note on Diffeomorphism Groups of Closed Manifolds

In this short note, we construct a retracting deformation from group of orientation-preserving diffeomorphisms onto the group of volume-preserving diffeomorphismsof a compact smooth manifold. The retracting deformation isrealized by a moment map type of flow with the aid of the theory of quasi-linear parabolic equations.

Projective unitary positive-energy representations of diff ( S1)

Let D be the group of orientation-preserving diffeomorphisms of the circle S 1. Then D is Fréchet Lie group with Lie algebra ( δ ∞) R the smooth real vector fields on S 1. Let δ R be the subalgebra of real vector fields with finite Fourier series. It is proved that every infinitesimally unitary projective positive-energy representation of δ R integrates to a continuous projective unitary representation of D . This result was conjectured by V. Kac.