Eigenvalue estimates for Dirac operators with parallel characteristic torsion

Abstract

Assume that the compact Riemannian spin manifold ( M n , g ) admits a G-structure with characteristic connection ∇ and parallel characteristic torsion ( ∇ T = 0 ), and consider the Dirac operator D 1 / 3 corresponding to the torsion T / 3 . This operator plays an eminent role in the investigation of such manifolds and includes as special cases Kostant's “cubic Dirac operator” and the Dolbeault operator. In this article, we describe a general method of computation for lower bounds of the eigenvalues of D 1 / 3 by a clever deformation of the spinorial connection. In order to get explicit bounds, each geometric structure needs to be investigated separately; we do this in full generality in dimension 4 and for Sasaki manifolds in dimension 5.