Lower bounds of the Dirac eigenvalues on compact Riemannian spin   manifolds with locally product structure
  We study some similarities between almost product Riemannian structures and almost Hermitian structures. Inspired by the similarities, we prove lower eigenvalue estimates for the Dirac operator on compact Riemannian spin manifolds with locally product structures. We also provide some examples (limiting manifolds) for the limiting case of the estimates. 
Eui Chul Kim