I am studying for a differential geometry exam and our professor gave us the following practice question but the whole class is having trouble with it. Any help would be greatly appreciated!
Suppose that C $\subset$ $\mathbb{R}^3$ is a smooth embedded curve. Let M be the set of all pair (p,v) where p is a point of C and v $\in$ $\mathbb{R}^3$ is a vector perpendicular to the tangent line C at p.
(a) Describe a smooth structure on M in which M is a manifold. Hint: you might need more than one chart. Hint: Don't write out the transition maps.
(b) Explain why M is an embedded submanifold of $\mathbb{R}^3$ x $\mathbb{R}^3$.
(c) Suppose that C is connected. Prove that the map (p,v) $\in$ M $\rightarrow$ p+v $\in$ $\mathbb{R}^3$ is an embedding just when C is a subset of a line in $\mathbb{R}^3$.