  The effective one-loop potential on $R^{m+1}\times S^N$ spaces for massless
tensor fields is evaluated. The Casimir energy is given as a value of $\zeta-$
function by means of which regularization is made. In even- dimensional spaces
the vacuum energy contains divergent terms coming from poles of $\zeta(s,q)$ at
$s=1$, whereas in odd-dimensional spaces it becomes finite.
