Let $M^n$ a differentiable manifold and $T_1M=\{(p,v) : p \in M \hbox{ and } v \in T_pM \hbox{ with } \|v\|=1 \}$. Consider $\{(U_\alpha,{\tt x}_{\alpha})\}$ a differentiable structure on $M$. I want to prove that $T_1M$ is a differentiable manifold with dimension $2n-1$. I defined the functions ${\tt y}_{\alpha}:U_\alpha \times S^{n-1} \rightarrow T_1M$ given by
${\tt y}_{\alpha}(q,v)=\left({\tt x}_\alpha(q),\frac{d({\tt x}_{\alpha})_q(v)}{\|d({\tt x}_{\alpha})_q(v)\|}\right)$.
It seems that the axioms of differentiable manifold are fulfilled. The problem is that $S^{n-1}$ is not a open subset of $\mathbb{R}^{n}$.