Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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THE SYMMETRY OF spinℂ DIRAC SPECTRUMS ON RIEMANNIAN PRODUCT MANIFOLDS

  • HONG, KYUSIK (School of Mathematics Korea Institute for Advanced Study) ;
  • SUNG, CHANYOUNG (Department of Mathematics Education Korea National University of Education)
  • Received : 2014.12.07
  • Published : 2015.09.01
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Abstract

It is well-known that the spectrum of a $spin^{\mathbb{C}}$ Dirac operator on a closed Riemannian $spin^{\mathbb{C}}$ 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 $M^{2p}_1{\times}M^{2q+1}_2$ with a product $spin^{\mathbb{C}}$ structure for $p{\geq}1$, $q{\geq}0$, the spectrum of a $spin^{\mathbb{C}}$ Dirac operator given by a product connection is symmetric if and only if either the $spin^{\mathbb{C}}$ Dirac spectrum of $M^{2q+1}_2$ 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 $spin^{\mathbb{C}}$ structure of $M_1$.

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References

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