  On Euclidean spaces, the Gaussian kernel is one of the most widely used
kernels in applications. It has also been used on non-Euclidean spaces, where
it is known that there may be (and often are) scale parameters for which it is
not positive definite. Hope remains that this kernel is positive definite for
many choices of parameter. However, we show that the Gaussian kernel is not
positive definite on the circle for any choice of parameter. This implies that
on metric spaces in which the circle can be isometrically embedded, such as
spheres, projective spaces and Grassmannians, the Gaussian kernel is not
positive definite for any parameter.
