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).