More on $r$-cross $t$-intersecting families for vector spaces

Let $V$ be a finite dimensional vector space over a finite field. Suppose that ${\mathcal{F}}_1$, ${\mathcal{F}}_2$, $\dots$, ${\mathcal{F}}_r$ are $r$-cross $t$-intersecting families of $k$-subspaces of $V$. In this paper, we determine the extremal structure when ${\prod }_{i=1}^r\mathrm{\mid }{\mathcal{F}}_i\mathrm{\mid }$ is maximum under the condition that $\dim ({\bigcap }_{F\in {\mathcal{F}}_i}F)<t$ for each $i$.