Extrinsic upper bounds for eigenvalues of Dirac-type operators

Abstract

Extrinsic estimates from above for eigenvalues of generalized Dirac operators on compact manifolds are given. They depend on the second fundamental form of any isometric immersion of the manifold in some Euclidean space and the curvature term in the Bochner-Weitzenböck formula for the square of the Dirac operator. Most of the known extrinsic upper bounds for the first eigenvalue of the Laplacian are in this way easily recovered and extended.