Linking curves, sutured manifolds and the Ambrose conjecture for generic 3-manifolds

TL;DR

This paper introduces a novel approach to prove the Ambrose conjecture for generic 3-manifolds by utilizing linking curves and sutured manifolds, establishing that a residual set of metrics satisfies the conjecture.

Contribution

It develops a new strategy involving linking curves and sutured manifolds to prove the Ambrose conjecture for generic 3-manifolds, including explicit constructions.

Findings

Any sutured manifold satisfies the Ambrose conjecture.

Residual set of metrics on 3-manifolds satisfy the conjecture.

Construction of linking curves following the conjugate descending flow.

Abstract

We present a new strategy for proving the Ambrose conjecture, a global version of the Cartan local lemma. A linking curve is defined as a curve in the tangent space whose composition with the exponential map is tree formed. This key idea is used to define sutured manifolds. We prove first that any sutured manifold satisfies the Ambrose conjecture. We then prove that the set of sutured Riemmanian manifolds contains a residual set of the metrics on a given smooth manifold of dimension 3 by explicitely constructing a special type of linking curves that follow the "conjugate descending flow".