Assume that $M_1, M_2, \ldots , M_k$  are  $k$ disjoint  compact  submanifolds  of  $\mathbb{R}^n$ of  the  same  dimension $m$. Assume that $\lambda_{ij}, \; 1\leq i,j\leq k$  are  $k^2$ arbitrary  real numbers.

>Are there  $k$ differential $m$- forms $\alpha_1, \alpha_2,\ldots, \alpha_k$ with $\int_{M_{i}} \alpha_{j}= \lambda{ij}$?


The  motivation for this post is that a  positive  answer to the  above  question implies that
the  number  of  closed orbits  of  a planar vector  field $X$ is  less than the  codimension of the  range of  the  following  linear  operator:

$$L_X:\Omega^1(\mathbb{R}^2)\to  \Omega^1(\mathbb{R}^2) $$