Open Access
Research Article
Hofer-Like Geometry and Flux Theory
*Stéphane TchuiagaCorresponding authortchuiagas@gmail.comUniversity of BueaSouth West Region, CameroonView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 26 Aug 2021
- Accepted:
- 26 Oct 2021
- Published Online:
- 03 Jul 2021
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- 1726037X.2021.2011110
- Pages:
- 227–270
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 C1−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).
Keywords
Subject Classifications
53C2453D0557R50
References
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