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 centered at the starting point of the Brownian motion are considered. The asymptotic expansions, as decreases to zero, for joint moments of the first exit time and place random variables are computed with error . 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.