The Dirac operator on manifold admitting parallel one-form

In this note, we get estimates on the eigenvalues of the Dirac operator on the manifold admitting a nontrivial parallel one-form, in terms of the eigenvalues of the Laplace Beltrami operator and the scalar curvature. These estimates are sharp, in the sense that, for the first eigenvalue, they reduce to the result of Alexandrov et al. (1998). (C) 2017 Elsevier B.V. All rights reserved.