Lower bounds for the first eigenvalue of the Dirac operator on compact Riemannian manifolds

Some new, improved, curvature depending lower bounds for the first eigenvalue of the Dirac operator on compact Riemannian manifolds are proved. If certain curvature conditions are satisfied, then these lower bounds are also useful in cases where the scalar curvature has zeros or attains negative values. This implies stronger vanishing theorems for harmonic spinors.