Flag structures on real 3-manifolds

TL;DR

This paper introduces flag structures on real 3-manifolds, constructs adapted connections, and defines a global invariant analogous to known secondary classes, unifying various geometric structures.

Contribution

It formalizes flag structures on 3-manifolds, constructs adapted connections, and introduces a new global invariant similar to Chern-Simons and Burns-Epstein invariants.

Findings

Null curvature models correspond to totally real submanifolds in flag space.

A global invariant is defined and shown to be constant on homotopy classes.

Includes special cases like path geometries and CR structures.

Abstract

We define flag structures on a real three manifold M as the choice of two complex lines on the complexified tangent space at each point of M. We suppose that the plane field defined by the complex lines is a contact plane and construct an adapted connection on an appropriate principal bundle. This includes path geometries and CR structures as special cases. We prove that the null curvature models are given by totally real submanifolds in the flag space SL(3, C)/B, where B is the subgroup of upper triangular matrices. We also define a global invariant which is analogous to the Chern-Simons secondary class invariant for three manifolds with a Riemannian structure and to the Burns-Epstein invariant in the case of CR structures. It turns out to be constant on homotopy classes of totally real immersions in flag space.