THE SYMMETRY OF spin^ℂDIRAC SPECTRUMS ON RIEMANNIAN PRODUCT MANIFOLDS

Abstract: It is well-known that the spectrum of a

e x t {s p i n}^{m a t h b b C}

Dirac operator on a closed Riemannian

e x t {s p i n}^{m a t h b b C}

manifold

M^{2 k}

of dimension

2 k

for

k i n m a t h b b N

is symmetric. In this article, we prove that over an odd-dimensional Riemannian product

M_{1}^{2 p} i m e s M_{2}^{2 q + 1}

with a product

e x t {s p i n}^{m a t h b b C}

structure for

p g e q 1, q g e q 0

, the spectrum of a

e x t {s p i n}^{m a t h b b C}

Dirac operator given by a product connection is symmetric if and only if either the

e x t {s p i n}^{m a t h b b C}

Dirac spectrum of

M_{2}^{2 q + 1}

is symmetric or

(e^{f r a c 12 c_{1} (L_{1})} h a t A (M_{1})) [M_{1}] = 0

, where

L_{1}

is the associated line bundle for the given

e x t {s p i n}^{m a t h b b C}

structure of

M_{1}

.

THE SYMMETRY OF spinℂDIRAC SPECTRUMS ON RIEMANNIAN PRODUCT MANIFOLDS