Extrinsic eigenvalue estimates of Dirac operators on Riemannian manifolds

Abstract

AbstractFor eigenvalues of generalized Dirac operators on compact Riemannian manifolds, we obtain a general inequality. By using this inequality, we study eigenvalues of generalized Dirac operators on compact submanifolds of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces. We obtain explicit bounds for the (k + 1)‐th eigenvalue of generalized Dirac operators on such objects in terms of its first k eigenvalues, which depend on the mean curvature of the embedding and the curvature term in the Bochner‐Weitzenböck formula for the square of the Dirac operator. These inequalities of eigenvalues extend the recent results in 15. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim