If $\alpha$ is a section of $T^*M\odot T^*M$, where $M$ is a smooth manifold, the rank of $\alpha$ at $m\in M$ is the codimension of the kernel of $\alpha_m$, i.e. the subspace of vectors $v_m\in T_mM$ that satisfy
$$\alpha(v_m,w_m)=0$$
for all $w_m\in T_mM$.
Given an integer $0\leq r\leq\dim(M)$, is there ever an obstruction to the existence of global constant-rank sections of $T^*M\odot T^*M$? The answer is ``no" when $r=0$ (obviously) or $r=\dim(M)$ (because we can put a Riemannian metric on $M$).
My initial reaction is that the existence of such a tensor field is equivalent (by the usual partition of unity game) to having a distinguished subbundle $E\subset TM$ of rank $n-r$. (And, with a metric, of course, we can take $E^\perp$ and get a splitting of $TM$.) Equivalently, we get an embedding of $M$ into $G(n-r,TM)$, for what it's worth. Perhaps more is true?