J. Korean Math. Soc. 2015; 52(5): 1037-1049
Printed September 1, 2015
https://doi.org/10.4134/JKMS.2015.52.5.1037
Copyright © The Korean Mathematical Society.
Kyusik Hong and Chanyoung Sung
Korea Institute for Advanced Study, Korea National University of Education
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 $p \geq 1$, $q \geq 0$, 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 $M_1$.
Keywords: Dirac operator, $\text{spin}^{\mathbb{C}}$ manifold, spectrum, eta invariant
MSC numbers: 53C27, 58C40
2023; 60(5): 999-1021
2017; 54(6): 1801-1816
2016; 53(6): 1347-1370
1995; 32(2): 341-350
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd