Hofer-Like Geometry and Flux Theory. Issue 2 (3rd July 2021)

• Record Type: Journal Article
• Authors: Tchuiaga, Stéphane
• Is Part Of: Journal of dynamical systems and geometric theories. Volume 19:Issue 2(2021)
• Page Start: 227
• Page End: 270
• Publication Date: 2021-07-03
• Subjects: 53C24 -- 53D05 -- 57R50; Rigidity -- Symplectic manifolds -- Differential topological aspects of volume-preserving diffeomorphisms -- Isotopies -- Homotopy -- Metrics; Differentiable dynamical systems -- Periodicals; Geometry -- Periodicals; Differentiable dynamical systems; Geometry; Periodicals; 515.39
• DOI: 10.1080/1726037X.2021.2011110
• Languages: English
• ISSNs: 1726-037X

Abstract

Abstract: This paper meticulously revisit and study the flux geometry of any compact connected oriented manifold ( M, Ω). We generalize several well- known factorization results, exhibit some orbital conditions under which flux geometry can be studied, give a proof of the discreteness of the flux group for volume-preserving diffeomorphisms, derive that any smooth isotopy in the group of all vanishing-flux volume-preserving diffeomorphisms is a vanishing- flux path, and show that the kernel of flux for volume-preserving diffeomorphisms is C 1 −closed inside the group of all volume-preserving diffeomorphisms isotopic to the identity map: We recover several well-known results from symplectic geometry. We use the above studies to construct a right-invariant metric on the group of all volume-preserving diffeomorphisms isotopic to the identity map and study the induced geometry. In the case of a symplectic volume form, the restriction of our metric to the group Ham ( N, ω ), of all Hamiltonian diffeomorphisms of a closed symplectic manifold ( N, ω ), is controlled from above by the usual Hofer metric in general, while the Hofer-like metric control our metric in the case where the Riemannian structure is compatible with the symplectic structure (in particular, our construction implies the non-degeneracy of the Hofer and Hofer-like norms).