Lower bounds on volumes of vector fields
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.
volumes of vector fields on hypersurfaces of euclidean space
scientific article; zbMATH DE number 4199077