SCHWEITZER 2018 Problem 11

We call an $m$ -dimensional smooth manifold parallelizable if it admits $m$ smooth tangent vector fields that are linearly independent at all points. Show that if $M$ is a closed orientable $2n$ -dimensional smooth manifold of Euler characteristic $0$ that has an immersion into a parallelizable smooth $(2n+1)$ -dimensional manifold $N$ , then $M$ is itself parallelizable.