A split special Lagrangian calibration associated with frame vorticity
Abstract
Let M be an oriented three-dimensional Riemannian manifold. We define a notion of vorticity of local sections of the bundle SO(M) --> M of all its positively oriented orthonormal tangent frames. When M is a space form, we relate the concept to a suitable invariant split pseudo-Riemannian metric on Iso_o (M) \cong SO(M): A local section has positive vorticity if and only if it determines a space-like submanifold. In the Euclidean case we find explicit homologically volume maximizing sections using a split special Lagrangian calibration. We introduce the concept of optimal frame vorticity and give an optimal screwed global section for the three-sphere. We prove that it is also homologically volume maximizing (now using a common one-point split calibration). Besides, we show that no optimal section can exist in the Euclidean and hyperbolic cases.
Cite
@article{arxiv.2205.03754,
title = {A split special Lagrangian calibration associated with frame vorticity},
author = {Marcos Salvai},
journal= {arXiv preprint arXiv:2205.03754},
year = {2023}
}
Comments
We changed the title (the nomenclature "vorticity" is more appropriate for the concept we define in the article than "helicity"). We corrected some typos and an assertion in the paragraph after display number 20. To be published in Advances in Calculus of Variations