Lower bounds on volumes of vector fields
type: Item
instance of: scholarly article
title: Lower bounds on volumes of vector fields
zbMATH Open document ID: 0726.58018
DOI: 10.1007/BF01190122
author: Alexander G. Reznikov
published in: Archiv der Mathematik
publication date: 1992-01-01T00:00:00Z
review text: The notion of volume of a unit vector field on a smooth Riemannian manifold was introduced recently H. Gluck and W. Ziller. They defined it as a volume of the corresponding submanifold in the tangent bundle endowed by Sasaki metric. Generators of Hopf fibrations in the standard three-dimensional sphere minimize the volume functional, as was shown by Gluck and Ziller. In the spheres of higher dimensions, these generators do not minimize volume even locally, which was proved by D. Johnson. Here we give lower bounds on volumes of vector fields on hypersurfaces of euclidean space. We also give lower bounds on volumes of generators of geodesic flows on unit tangent bundles to compact manifolds and generators of geodesic fibrations with closed fibers. This generalizes inequalities, obtained by Gluck and Ziller.
Mathematics Subject Classification ID: 58E35
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
zbMATH DE Number: 4199077
zbMATH Keywords: lower bounds
zbMATH Keywords: volumes of vector fields on hypersurfaces of euclidean space
zbMATH Keywords: geodesic flows
MaRDI profile type: Publication
cites work: Q4169382
cites work: On the volume of a unit vector field on the three-sphere
cites work: Q5538378
cites work: Volumes of Flows
cites work: Volumes of Vector Fields on Spheres
cites work: Blaschke manifolds of the projective plane type
full work available at URL: https://doi.org/10.1007/bf01190122
OpenAlex ID: W2033303259
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description: scientific article; zbMATH DE number 4199077