Some Estimates Over Spacelike Spin Hypersurfaces of Lorentzian Manifold

We generalized the lower bound estimates for eigenvalues of the Dirac operator on spacelike hypersurfaces of Lorentzian manifolds obtained by Yongfa Chen in (Sci China Ser A Math 52(11):2459-2468, 2009) based on the constraint between the scalar curvature of the manifold, energy-momentum tensor and the mean curvature of the manifold. Afterwards, we examined the geometric data in the case of estimation satisfies equality condition.