  Denote the collection of all $k$-flats in $AG(n,\mathbb{F}_q)$ by
$\mathscr{M}(k,n)$. Let $\mathscr{F}_1\subset\mathscr{M}(k_1,n)$ and
$\mathscr{F}_2\subset\mathscr{M}(k_2,n)$ satisfy $\dim(F_1\cap F_2)\ge t$ for
any $F_1\in\mathscr{F}_1$ and $F_2\in\mathscr{F}_2$. We say they are cross
$t$-intersecting families. Moreover, we say they are trivial if each member of
them contains a fixed $t$-flats in $AG(n,\mathbb{F}_q)$. In this paper, we show
that cross $t$-intersecting families with maximum product of sizes are trivial.
We also describe the structure of non-trivial $t$-intersecting families with
maximum product of sizes.
