Rigidity results for spin manifolds with foliated boundary

TL;DR

This paper establishes rigidity results for spin manifolds with foliated boundaries, showing under certain curvature conditions that solutions to the basic Dirac equation imply the manifold's local product structure.

Contribution

It proves that solutions to the basic Dirac equation correspond to restrictions of parallel spinors, leading to a characterization of the manifold's geometry and flow structure.

Findings

Solutions to the basic Dirac equation are restrictions of parallel spinors.

The flow on the boundary is a local product under curvature assumptions.

Characterization of manifold geometry when solutions are basic Killing spinors.

Abstract

In this paper, we consider a compact Riemannian manifold whose boundary is endowed with a Riemannian flow. Under a suitable curvature assumption depending on the O'Neill tensor of the flow, we prove that any solution of the basic Dirac equation is the restriction of a parallel spinor field defined on the whole manifold. As a consequence, we show that the flow is a local product. In particular, in the case where solutions of the basic Dirac equation are given by basic Killing spinors, we characterize the geometry of the manifold and the flow.