Products of constant curvature spaces with a Brownian independence property

The time and place Brownian motion on the product of constant curvature spaces first exits a normal ball of radius $\epsilon$ centered at the starting point of the Brownian motion are considered. The asymptotic expansions, as $\epsilon$ decreases to zero, for joint moments of the first exit time and place random variables are computed with error $O(\epsilon ^{10})$. It is shown that the first exit time and place are independent random variables only if each factor space is locally flat or of dimension three.