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I have seen several posts on the tangent bundles of $S^1$, and $S^3$. Basically, it seems that the idea is to find $n$ linearly independent smooth vector fields. In particular, for $S^1$, we pick $x\to ix$. For $S^3$, we pick $x\to ix,jx,kx$ via quaternions etc. However, I have 2 questions.

  1. Why are these vector fields? My definition is that a vector field maps an element of the smooth manifold say $x\in S^1$ to an element of the tangent bundle $(x,v)\in TS^1$.

  2. Why does finding $n$ linearly independent vector fields tell us that there is a diffeomorphism between $TS^n$ and $S^n\times \mathbb{R}$.

Ook
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    For Q1, and for Q2, in general if you have a rank $k$ vector bundle $E$ over a base manifold $M$ and you find $k$ linearly independent sections (which you can always do locally) then they provide you with the trivialization in the same way that specifying a basis for a $k$-dimensional real vector space $V$ is equivalent to specifying an isomorphism $V\to\Bbb{R}^k$. So essentially you’re repeating this linear algebra fact fiber-by-fiber. – peek-a-boo Oct 07 '23 at 17:10

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  1. If you embed $S^{n-1}$ into $\mathbb{R}^n$ as the unit sphere you can embed the tangent bundle as the collection of all hyperplanes tangent to the unit sphere. Said more explicitly the tangent space at a unit vector $v \in S^{n-1}$ can be identified with the space of all vectors in $\mathbb{R}^n$ orthogonal to $v$, and the tangent bundle can be identified with the space of pairs $(v, x) \in (\mathbb{R}^n)^2 \cong T(\mathbb{R}^n)$ such that $\| v \| = 1$ and $\langle v, x \rangle = 0$. So you can specify a vector field on $S^{n-1}$ by writing down a continuous function $S^{n-1} \to \mathbb{R}^n$ sending a vector $v \in S^{n-1} \subset \mathbb{R}^n$ to any vector orthogonal to it. For $S^1$ and $S^3$ the ambient $\mathbb{R}^2$ and $\mathbb{R}^4$ should be thought of as identified with $\mathbb{C}$ and $\mathbb{H}$ respectively, and then you can check that multiplication by $i, j, k$ always produces a vector orthogonal to a given vector.

  2. The diffeomorphism is given by taking linear combinations of the vector fields. At each point they evaluate to a basis of the tangent space. So if the vector fields are $X_1, \dots X_n$ we have a map which takes the vector $(c_1, \dots c_n) \in \mathbb{R}^n$, in the fiber of the trivial bundle with fiber $\mathbb{R}^n$ at any point $p$, to the tangent vector given by taking $\sum c_i X_i$ and then evaluating it at $p$.

Qiaochu Yuan
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