On the minimal dimension of the orbits of a $\mathbb R^n$-action

Consider a smooth action of $\mathbb R^n$ on a connected manifold $M$, not necessarily compact, of dimension $m$ and rank $k$. Assume that $M$ is not a cylinder. Then there exists an orbit of the action of dimension $<(m+k)/2$. As a consequence, one shows that if there is a non-zero element of the ring of Pontrjagin classes of $M$ of degree $4\ell\geq 4$, then there exists an orbit of the action of dimension $\leq m-\ell-1$.