scholarly journals Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics

2021 ◽  
Vol 60 (1) ◽  
Author(s):  
Martin Bauer ◽  
Cy Maor
Keyword(s):  
Diffeomorphism Group

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

2012 ◽  
Vol 44 (1) ◽  
pp. 5-21 ◽  
Author(s):  
Martin Bauer ◽  
Martins Bruveris ◽  
Philipp Harms ◽  
Peter W. Michor
Keyword(s):  
Fractional Order ◽  
Geodesic Distance ◽  
Diffeomorphism Group

THE MINIMAL GENUS PROBLEM IN RULED MANIFOLDS

2008 ◽  
Vol 17 (04) ◽  
pp. 471-482
Author(s):  
XU-AN ZHAO ◽  
HONGZHU GAO
Keyword(s):  
Cohomology Class ◽  
Standard Form ◽  
Geometric Construction ◽  
Diffeomorphism Group ◽  
Minimal Genus ◽  
Adjunction Formula ◽  
Embedded Surfaces

In this paper, we consider the minimal genus problem in a ruled 4-manifold M. There are three key ingredients in the studying, the action of diffeomorphism group of M on H2(M,Z), the geometric construction of surfaces representing a cohomology class and the generalized adjunction formula. At first, we discuss the standard form (see Definition 1.1) of a class under the action of diffeomorphism group on H2(M,Z), we prove the uniqueness of the standard form. Then we construct some embedded surfaces representing the standard forms of some positive classes, the generalized adjunction formula is used to show that these surfaces realize the minimal genera.

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.

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