scholarly journals Commutator length of annulus diffeomorphisms

2012 ◽  
Vol 34 (3) ◽  
pp. 919-937
Author(s):  
E. MILITON
Keyword(s):  
Linear Space ◽  
Diffeomorphism Group ◽  
One Dimensional ◽  
Group Diff ◽  
Commutator Length

AbstractWe study the group Diffr0(𝔸) of Cr-diffeomorphisms of the closed annulus that are isotopic to the identity. We show that, for r≠2,3, the linear space of homogeneous quasi-morphisms on the group Diffr0(𝔸) is one-dimensional. Therefore, the commutator length on this group is (stably) unbounded. In particular, this provides an example of a manifold whose diffeomorphism group is unbounded in the sense of Burago, Ivanov and Polterovich.

The Centralizer

2018 ◽  
Author(s):  
Ercüment H. Ortaçgil
Keyword(s):  
Lie Group ◽  
Group Structure ◽  
Diffeomorphism Group ◽  
Group Diff ◽  
Local Solutions

The pseudogroup of local solutions in Chapter 3 defines another pseudogroup by taking its centralizer inside the diffeomorphism group Diff(M) of a manifold M. These two pseudogroups define a Lie group structure on M.

scholarly journals Applications of Square Roots of Diffeomorphisms

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 43
Author(s):  
Yoshihiro Sugimoto
Keyword(s):  
Smooth Manifold ◽  
Contact Manifold ◽  
Diffeomorphism Group ◽  
Square Root ◽  
Square Roots ◽  
Group Diff

In this paper, we prove that on any contact manifold ( M , ξ ) there exists an arbitrary C ∞ -small contactomorphism which does not admit a square root. In particular, there exists an arbitrary C ∞ -small contactomorphism which is not “autonomous”. This paper is the first step to study the topology of C o n t 0 ( M , ξ ) ∖ Aut ( M , ξ ) . As an application, we also prove a similar result for the diffeomorphism group Diff ( M ) for any smooth manifold M.

scholarly journals On the Integrable Chaplygin Type Hydrodynamic Systems and Their Geometric Structure

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 697
Author(s):  
Yarema Prykarpatskyy
Keyword(s):  
Geometric Structure ◽  
Algebraic Approach ◽  
Strong Relationship ◽  
Diffeomorphism Group ◽  
Differential Systems ◽  
One Dimensional ◽  
Infinite Hierarchy ◽  
Completely Integrable ◽  
Hydrodynamic Systems

A class of spatially one-dimensional completely integrable Chaplygin hydrodynamic systems was studied within framework of Lie-algebraic approach. The Chaplygin hydrodynamic systems were considered as differential systems on the torus. It has been shown that the geometric structure of the systems under analysis has strong relationship with diffeomorphism group orbits on them. It has allowed to find a new infinite hierarchy of integrable Chaplygin like hydrodynamic systems.

scholarly journals Homomorphisms between diffeomorphism groups

2013 ◽  
Vol 35 (1) ◽  
pp. 192-214 ◽  
Author(s):  
KATHRYN MANN
Keyword(s):  
Closed Manifold ◽  
Independent Interest ◽  
Diffeomorphism Group ◽  
Diffeomorphism Groups ◽  
One Dimensional ◽  
Compactly Supported ◽  
Trivial Group ◽  
Elementary Form ◽  
General Conditions

AbstractFor $r\geq 3$, $p\geq 2$, we classify all actions of the groups ${ \mathrm{Diff} }_{c}^{r} ( \mathbb{R} )$ and ${ \mathrm{Diff} }_{+ }^{r} ({S}^{1} )$ by ${C}^{p} $-diffeomorphisms on the line and on the circle. This is the same as describing all non-trivial group homomorphisms between groups of compactly supported diffeomorphisms on 1-manifolds. We show that all such actions have an elementary form, which we call topologically diagonal. As an application, we answer a question of Ghys in the 1-manifold case: if $M$ is any closed manifold, and ${\mathrm{Diff} }^{\infty } \hspace{-2.0pt} \mathop{(M)}\nolimits_{0} $ injects into the diffeomorphism group of a 1-manifold, must $M$ be one-dimensional? We show that the answer is yes, even under more general conditions. Several lemmas on subgroups of diffeomorphism groups are of independent interest, including results on commuting subgroups and flows.

PROJECTIVE REPRESENTATIONS OF INFINITE-DIMENSIONAL ORTHOGONAL AND SYMPLECTIC GROUPS

1994 ◽  
Vol 06 (01) ◽  
pp. 1-17 ◽  
Author(s):  
LARS-ERIK LUNDBERG
Keyword(s):  
Loop Group ◽  
Diffeomorphism Group ◽  
Symplectic Groups ◽  
Second Quantization ◽  
Infinite Dimensional ◽  
Group Diff ◽  
Projective Representations

We consider some particular projective representations of the restricted orthogonal and symplectic groups. These representations are related to so-called "second quantization". In particular, we apply our results to the loop group LS1 and the diffeomorphism group Diff + (S1).

scholarly journals On Some Properties of Solutions of Second Order Linear Functional Differential Equations

1994 ◽  
Vol 1 (5) ◽  
pp. 477-484
Author(s):  
I. Kiguradze
Keyword(s):  
Differential Equations ◽  
Linear Space ◽  
Measurable Function ◽  
Linear Functional ◽  
Functional Differential Equations ◽  
Absolutely Continuous ◽  
Order Linear ◽  
One Dimensional ◽  
Functional Differential ◽  
Necessary And Sufficient

Abstract The properties of solutions of the equation u″(t) = p 1 (t)u(τ 1(t)) + p 2(t)u′( τ 2(t)) are investigated where pi : [a, +∞[→ R (i = 1, 2) are locally summable functions, τ 1 : [a, +∞[→ R is a measurable function and τ 2 : [a, +∞[→ R is a nondecreasing locally absolutely continuous one. Moreover, τ i (t) ≥ t (i = 1, 2), p 1 (t) ≥ 0, , ε = const > 0 and . In particular, it is proved that solutions whose derivatives are square integrable on [a, +∞ [ form a one-dimensional linear space and for any such solution to vanish at infinity it is necessary and sufficient that .

scholarly journals Gradual transitivity in orthogonality spaces of finite rank

2020 ◽  
Author(s):  
Thomas Vetterlein
Keyword(s):  
Automorphism Group ◽  
Linear Space ◽  
Binary Relation ◽  
Finite Rank ◽  
Positive Definite ◽  
Inner Product ◽  
Orthogonality Space ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Divisible Subgroup

Abstract An orthogonality space is a set together with a symmetric and irreflexive binary relation. Any linear space equipped with a reflexive and anisotropic inner product provides an example: the set of one-dimensional subspaces together with the usual orthogonality relation is an orthogonality space. We present simple conditions to characterise the orthogonality spaces that arise in this way from finite-dimensional Hermitian spaces. Moreover, we investigate the consequences of the hypothesis that an orthogonality space allows gradual transitions between any pair of its elements. More precisely, given elements e and f, we require a homomorphism from a divisible subgroup of the circle group to the automorphism group of the orthogonality space to exist such that one of the automorphisms maps e to f, and any of the automorphisms leaves the elements orthogonal to e and f fixed. We show that our hypothesis leads us to positive definite quadratic spaces. By adding a certain simplicity condition, we furthermore find that the field of scalars is Archimedean and hence a subfield of the reals.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

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