New lower bound for eigenvalues of the Dirac operator

In this talk, we give a new estimate for the eigenvalues of the Dirac operator on a compact spin manifold in terms of an appropriate endomorphism $E^{\psi }$ of the tangent bundle associated with an eigenspinor $\psi$. We then show that, for isometric immersions and Riemannian flows (local Riemannian submersions), the limiting case could be achieved. In this case the tensor $E^{\psi }$ is identified with the second fundamental form of the immersion while it is identified with the O'Neill tensor of the flow.