Spectrum of the Dirac operator on Compact Riemannian Manifolds

In this paper, we consider the eigenvalue problem of Dirac operator on a compact Riemannian manifold isometrically immersed into Euclidean space and derive some extrinsic estimates for the sum of arbitrary consecutive $n$ eigenvalues of the square of the Dirac operator acting on some Dirac invariant subbundles. As some applications, we deduce some eigenvalue inequalities on the compact submanifolds immersed into Euclidean space, unit sphere or projective spaces and further get some bounds of general Reilly type. In addition, we also establish some universal bounds under certain curvature condition and on the meanwhile provide an alternative proof for Anghel's result. In particular, utilizing Atiyah-Singer index theorem, we drive an upper bound estimate for the sum of the first $n$ nontrivial eigenvalues of Atiyah-Singer Laplacian acting on the spin manifolds without dimensional assumption.