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 $M_1$.
Categories: math.DG