manifold diffeomorphic (?) to SO(3) Consider the set of all pairs $(\boldsymbol{n},\boldsymbol{v})$ of vectors in $\mathbb{R}^3$ such that $\boldsymbol{n}$ is a vector on the unit sphere centered at the origin and $\boldsymbol{v}$ is a unit vector tangent to the sphere at the point $\boldsymbol{n}.$
i. Introduce a structure of smooth manifold on this set.
ii. Prove that this manifold is diffeomorphic to the group $SO(3).$
To my understanding, this manifold is $S^2 \times S^1,$ which gives a parametrization
of $SO(3),$
but it is far from being a diffeomorphism, i.e. the exercise is false: do you agree?
 A: The exercise is fine. The manifold described in the exercise is called the unit tangent bundle of $S^2$ and it is not diffeomorphic to $S^2 \times S^1$ [one way to see this is to observe that you could produce a nowhere vanishing vector field on $S^2$ if they were diffeomorphic. This is impossible by the hairy ball theorem.]
Here's a slightly more detailed outline:
By definition the set $M$ is given as a subset of $\mathbb{R}^3 \times \mathbb{R}^3 \ni (\boldsymbol{n}, \boldsymbol{v})$ subject to the equations
$$
\begin{align*}
 1 & = \boldsymbol{n} \cdot \boldsymbol{n} && \boldsymbol{n} \text{ is a unit vector}\\
1 & = \boldsymbol{v} \cdot \boldsymbol{v} && \boldsymbol{v} \text{ is a unit vector}\\
 0 & = \boldsymbol{n} \cdot \boldsymbol{v} && \boldsymbol{n} \text{ is perpendicular to }\boldsymbol{v}.
\end{align*}
$$
Can you use this information to turn $M$ into a manifold? (implicit functions, regular values, etc)
The map $M \to SO(3)$ given by $(\boldsymbol{n},\boldsymbol{v}) \mapsto [\boldsymbol{n},\boldsymbol{v},\boldsymbol{n} \times \boldsymbol{v}]$ is well-defined smooth and bijective. You can exhibit an explicit smooth inverse, so $M$ and $SO(3)$ are diffeomorphic.
A: Yes, you're looking at the manifold of unit tangent vectors of $S^2$. You can go the other way: $SO(3)$ acts smoothly and transitively on $S^2$; moreover, it also acts smoothly and transitively on your manifold $M$. But the stabilizer subgroup of a point is trivial, so the map $SO(3)\to M$ is a diffeomorphism.
