Energy-Momentum tensor on foliations

TL;DR

This paper establishes a new lower bound for Dirac operator eigenvalues on compact spin manifolds, linking geometric tensors to spectral properties, with applications to Riemannian flows and Sasakian manifolds.

Contribution

It introduces a novel eigenvalue estimate involving the O'Neill tensor and characterizes the 3D case via Dirac equation solutions.

Findings

Derived a new lower bound for Dirac eigenvalues.

Identified geometric tensors related to Riemannian flows.

Characterized 3D case through Dirac equation solutions.

Abstract

In this paper, we give a new lower bound for the eigenvalues of the Dirac operator on a compact spin manifold. This estimate is motivated by the fact that in its limiting case a skew-symmetric tensor (see Equation \eqref{eq:16}) appears that can be identified geometrically with the O'Neill tensor of a Riemannian flow, carrying a transversal parallel spinor. The Heisenberg group which is a fibration over the torus is an example of this case. Sasakian manifolds are also considered as particular examples of Riemannian flows. Finally, we characterize the 3-dimensional case by a solution of the Dirac equation