A split special Lagrangian calibration associated with frame vorticity

TL;DR

This paper introduces a new notion of vorticity for orthonormal frames on 3D Riemannian manifolds, relating it to split calibrations and identifying volume-maximizing sections in Euclidean space, with implications for geometric analysis.

Contribution

It defines a novel concept of frame vorticity, connects it to split pseudo-Riemannian metrics, and constructs explicit volume-maximizing sections in Euclidean space using split special Lagrangian calibrations.

Findings

Positive vorticity sections correspond to space-like submanifolds.

Explicit homologically volume-maximizing sections are found in Euclidean space.

No optimal sections exist in Euclidean and hyperbolic cases.

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.