Let
$$TS^2 = \{ (a,b) \in \mathbb{R}^3 \times \mathbb{R}^3: ||a||^2=1 , a\cdot b=0 \}$$
be the tangent bundle of $S^2$.
How do I see that $TS^2$ with the subspace topology of $\mathbb{R}^3 \times \mathbb{R}^3$ is a topological 4-manifold? Secondly, why there is a $C^\infty$ atlas such that the projection $\pi: TS^2 \to S^2$, $(a,b) \mapsto a$ is $C^\infty$?
Thanks for your help!