Abstract
General theorems on pin structures on products of manifolds and on homogeneous (pseudo-) Riemannian spaces are given and used to find explicitly all such structures on odd-dimensional real projective quadrics, which are known to be non-orientable (Cahen et al. 1993). It is shown that the product of two manifolds has a pin structure if and only if both are pin and at least one of them is orientable. This general result is illustrated by the example of the product of two real projective planes. It is shown how the Dirac operator should be modified to make it equivariant with respect to the twisted adjoint action of the Pin group. A simple formula is derived for the spectrum of the Dirac operator on the product of two pin manifolds, one of which is orientable, in terms of the eigenvalues of the Dirac operators on the factor spaces.
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