Clifford structures, bilegendrian surfaces, and extrinsic curvature

TL;DR

This paper introduces a Clifford algebra-based framework for analyzing surfaces with constant extrinsic curvature in 3-manifolds, providing classifications of bilegendrian surfaces in the unit tangent bundle of the 3-sphere.

Contribution

It develops a unified formalism using Clifford algebras for studying such surfaces and classifies bilegendrian surfaces in specific 3-manifold bundles.

Findings

Complete classifications of bilegendrian surfaces in U(S^3)

Unified Clifford algebra formalism for extrinsic curvature surfaces

Applications to riemannian and semi-riemannian 3-manifolds

Abstract

We use Clifford algebras to construct a unified formalism for studying constant extrinsic curvature immersed surfaces in riemannian and semi-riemannian $3$-manifolds in terms of immersed bilegendrian surfaces in their unitary bundles. As an application, we provide full classifications of both complete and compact immersed bilegendrian surfaces in the unit tangent bundle $\text{U}\Bbb{S}^3$ of the $3$-sphere.