TITLE: What is $T\mathbb{S}^2$?
QUESTION [11 upvotes]: I recently learned that the only parallelizable spheres are $\mathbb{S}^1$, $\mathbb{S}^3$, and $\mathbb{S}^7$.  This led me to wonder:
What is $T\mathbb{S}^2$?  Is it diffeomorphic to a more familiar space?  What about $T\mathbb{S}^n$ for $n \neq 1, 3, 7$?
EDIT (for precision): Is $T\mathbb{S}^2$ diffeomorphic to some finite product, connected sum, and/or quotient of spheres, projective spaces, euclidean spaces, and linear groups?

REPLY [7 votes]: A slightly random answer, but if we concretely identify $TS^{n-1}$ as the embedded real manifold $\{ (x, v) \in \mathbb{R}^n \times \mathbb{R}^n : \| x \| = 1, \langle x, v \rangle = 0 \}$, then it is diffeomorphic to the complex affine quadric $\{ (z_1, \ldots, z_n) \in \mathbb{C}^n : z_1^2 + \cdots + z_n^2 = 1 \}$. (This was an amusing homework exercise I did yesterday.)