I have an $n$ dimensional parallelizable manifold $M$. I know that it is the cartesian product of two parallelizable manifolds $M_1$ and $M_2$ but I do not know these two manifolds, not even their dimensions $n_1$ and $n_2$.
Note 1: the factorization into $M_1\times M_2$ could be non-unique. Please, assume that it is, for now, then read the note at the end. And assume that $M_1$ and $M_2$ have at least dimension 1.
I am wondering if I can get some information about $M_1$ and $M_2$, e.g. the dimensions $n_1$ and $n_2$, by knowing the following fact. I have $n$ vector fields $v_j$; I know that they generate the tangent space of $M$, and that each of them belongs either:
- to the tangent space of $M_1\times \{m_2\}$, where $m_2 \in M_2$, or
- to the tangent space of $M_2\times \{m_1\}$, where $m_1 \in M_1$.
However, I do not know which $v_j$ belongs to one or the other group.
Is there any way to evaluate at least the dimensions of $M_1$ or $M_2$? Any suggestion is welcome!
Note 2: in case the factorization is not unique, I would like to find one of the possible factorizations, using the $v_j$, or at least the dimensions $n_1$ and $n_2$ of a couple of possible factors.