THE SYMMETRY OF spinℂDIRAC SPECTRUMS ON RIEMANNIAN PRODUCT MANIFOLDS (original) (raw)

It is well-known that the spectrum of a spin C Dirac operator on a closed Riemannian spin C manifold M 2k of dimension 2k for k ∈ N is symmetric. In this article, we prove that over an odd-dimensional Riemannian product M 2p 1 × M 2q+1 2 with a product spin C structure for p ≥ 1, q ≥ 0, the spectrum of a spin C Dirac operator given by a product connection is symmetric if and only if either the spin C Dirac spectrum of M 2q+1 2 is symmetric or (e 1 2 c 1 (L 1)Â (M 1))[M 1 ] = 0, where L 1 is the associated line bundle for the given spin C structure of M 1 .