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Conic sub-Hilbert-Finsler structure on a Banach manifold. (English) Zbl 1428.58006

Kielanowski, Piotr (ed.) et al., Geometric methods in physics XXXVI. Workshop and summer school, Białowieża, Poland, July 2–8, 2017. Selected papers of the 36th workshop (WGMPXXXVI) and extended abstracts of lectures given at the 6th “School of geometry and physics”. Cham: Birkhäuser. Trends Math., 237-244 (2019).
Summary: A Hilbert-Finsler metric \(\mathcal{F}\) on a Banach bundle \(\pi : E \rightarrow M\) is a classical Finsler metric on \(E\) whose fundamental tensor is positive definite. Following [M. A. Javaloyes and M. Sánchez, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 13, No. 3, 813–858 (2014; Zbl 1317.53102)], a conic Hilbert-Finsler metric \(\mathcal{F}\) on \(E\) is a Hilbert-Finsler metric which is defined on an open conic submanifold of \(E\). In the particular case where we have a (strong) Riemannian metric \(g\) on \(E\), then \(\sqrt{g}\) is a natural example of Hilbert-Finsler metric on \(E\). According to [S. Arguillere, “Sub-Riemannian geometry and geodesics in Banach manifolds”, Prepirnt, arXiv:1601.00827], if, moreover, we have an anchor \(\rho : E \to T M\) we get a sub-Riemannian structure on \(M\), that is, \(g\) induces a “singular” Riemannian metric on the distribution \(\mathcal{D} = \rho(E)\) on \(M\). By analogy, a sub-Hilbert-Finsler structure on \(M\) is the data of a conic Hilbert-Finsler metric \(\mathcal{F}\) on a Banach bundle \(\pi : E \to M\) and an anchor \(\rho : E \to T M\). Of course, we get a “singular” conic Hilbert-Finsler metric on \(\mathcal{D} = \rho(E)\). In the finite-dimensional sub-Riemannian framework, it is well known that “normal extremals” are projections of Hamiltonian trajectories, and any such extremal is locally minimizing (relatively to the associated distance). Analogous results in the context of sub-Riemannian Banach manifold were obtained in [Arguillère, loc. cit.]. By an adaptation of his arguments, we generalize these properties to the sub-Hilbert-Finsler framework.
For the entire collection see [Zbl 1417.53002].

MSC:

58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
58B25 Group structures and generalizations on infinite-dimensional manifolds
37K06 General theory of infinite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, conservation laws
53C17 Sub-Riemannian geometry
53C22 Geodesics in global differential geometry

Citations:

Zbl 1317.53102
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