2024/03/19 by Tilman Becker, Becker, Tilman
Engineering · #53C22 #53D35 #57K33 #Advanced Numerical Analysis Techniques #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Differential Geometry (math.DG) #Dynamics and Control of Mechanical Systems #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2403.12903
openalex publication_date 2024/03/19 · openalex created_date 2024/03/21 · openalex updated_date 2026/07/28
In this paper, we provide new and simpler proofs of two theorems of Gluck and Harrison on contact structures induced by great circle or line fibrations. Furthermore, we prove that a geodesic vector field whose Jacobi tensor is parallel along flow lines (e.g. if the underlying manifold is locally symmetric) induces a contact structure if the 'mixed' sectional curvatures are nonnegative, and if a certain nondegeneracy condition holds. Additionally, we prove that in dimension three, contact structures admitting a Reeb flow which is either periodic, isometric, or free and proper, must be universally tight. In particular, we generalise an earlier result of Geiges and the author, by showing that every contact form on ℝ3 whose Reeb vector field spans a line fibration is necessarily tight. Furthermore, we provide a characterisation of isometric Reeb vector fields. As an application, we recover a result of Kegel and Lange on Seifert fibrations spanned by Reeb vector fields, and we classify closed contact 3-manifolds with isometric Reeb flows (also known as R-contact manifolds) up to diffeomorphism.