In Lemma 6.13, is $\tau (p,w)=[w]$ defined as follows?
$$T_pM\to T_p\Bbb R^N\cong \Bbb R^N\setminus\{0\}\to \Bbb {RP}^{N-1}$$
$$w\mapsto w^i\frac{\partial}{\partial x^i}|_p\mapsto (w^1,\cdots,w^N)\mapsto [ (w^1,\cdots,w^N)]$$
Then $\tau (p,w)=[w]= [(w^1,\cdots,w^N)]$.