THE SYMMETRY OF spinC DIRAC SPECTRUMS ON RIEMANNIAN PRODUCT MANIFOLDS
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It is well-known that the spectrum of a spinC Dirac operator on a closed Riemannian spinC manifold M2k of dimension 2k for k ∈ N is symmetric. In this article, we prove that over an odd-dimensional Riemannian product M2p/1 × M2q+1/2 with a product spinC structure for p ≥ 1, q ≥ 0, the spectrum of a spinC Dirac operator given by a product connection is symmetric if and only if either the spinC Dirac spectrum of M2q+1/2 is symmetric or (e 1/2 c1(L1)Aˆ(M1))[M1] = 0, where L1 is the associated line bundle for the given spinC structure of M1.
Abstract
It is well-known that the spectrum of a spinC Dirac operator on a closed Riemannian spinC manifold M2k of dimension 2k for k ∈ N is symmetric. In this article, we prove that over an odd-dimensional Riemannian product M2p/1 × M2q+1/2 with a product spinC structure for p ≥ 1, q ≥ 0, the spectrum of a spinC Dirac operator given by a product connection is symmetric if and only if either the spinC Dirac spectrum of M2q+1/2 is symmetric or (e 1/2 c1(L1)Aˆ(M1))[M1] = 0, where L1 is the associated line bundle for the given spinC structure of M1.
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