Universität zu Köln

Geiges, Hansjorg ORCID: 0000-0002-1360-1062 (2022). What does a vector field know about volume? J. Fixed Point Theory Appl., 24 (2). BASEL: SPRINGER BASEL AG. ISSN 1661-7746

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Abstract

This note provides an affirmative answer to a question of Viterbo concerning the existence of nondiffeomorphic contact forms that share the same Reeb vector field. Starting from an observation by Croke-Kleiner and Abbondandolo that such contact forms define the same total volume, we discuss various related issues for the wider class of geodesible vector fields. In particular, we define an Euler class of a geodesible vector field in the associated basic cohomology and give a topological characterisation of vector fields with vanishing Euler class. We prove the theorems of Gauss-Bonnet and Poincare-Hopf for closed, oriented 2-dimensional orbifolds using global surfaces of section and the volume determined by a geodesible vector field. This volume is computed for Seifert fibred 3-manifolds and for some transversely holomorphic flows.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Geiges, Hansjorg UNSPECIFIED orcid.org/0000-0002-1360-1062 UNSPECIFIED
URN:	urn:nbn:de:hbz:38-659049
DOI:	10.1007/s11784-022-00946-9
Journal or Publication Title:	J. Fixed Point Theory Appl.
Volume:	24
Number:	2
Date:	2022
Publisher:	SPRINGER BASEL AG
Place of Publication:	BASEL
ISSN:	1661-7746
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language REEB FLOWS; MANIFOLDS; EXAMPLES; METRICS Multiple languages Mathematics, Applied; Mathematics Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/65904