The symmetry of Spin^c Dirac spectrums on Riemannian product manifolds

2013/08/24 by Hong, Kyusik, Sung, Chanyoung
#53C27 #58C40 #59J28 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1308.5279

Abstract

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 M₁^2p × M₂^2q+1 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 M₂^2q+1 is symmetric or (e^(1)/(2)c₁(L₁) A(M₁))[M₁]=0, where L₁ is the associated line bundle for the given spin^ℂ structure of M₁.

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