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

Abstract

It is well-known that the spectrum of a ${\text{spin}}^{\mathbb{C}}$ Dirac operator on a closed Riemannian ${\text{spin}}^{\mathbb{C}}$ manifold $M^{2k}$ of dimension $2k$ for $k\in \mathbb{N}$ is symmetric. In this article, we prove that over an odd-dimensional Riemannian product $M_1^{2p}\times M_2^{2q+1}$ with a product ${\text{spin}}^{\mathbb{C}}$ structure for $p\ge 1,q\ge 0$, the spectrum of a ${\text{spin}}^{\mathbb{C}}$ Dirac operator given by a product connection is symmetric if and only if either the ${\text{spin}}^{\mathbb{C}}$ Dirac spectrum of $M_2^{2q+1}$ is symmetric or $(e^{\frac{1}{2}{c}_1(L_1)}\hat{A}(M_1))[M_1]=0$, where $L_1$ is the associated line bundle for the given ${\text{spin}}^{\mathbb{C}}$ structure of $M_1$.