“Once the storm is over, you won’t remember how you made it through, how you managed to survive. You won’t even be sure in fact, whether the storm is really over. But one thing is certain, when you come out of the storm you won’t be the same person who walked in. That’s what the storm is all about.” – Haruki Murakami.

(It ain’t no matter if the stuff introduced in sections 1, 2, and 3 are unfamiliar to you, reader. You can enjoy the Hairy consequences just as much if you start right from section 4. Happy read!)

Vector fields, sections, tangent and vector bundles

Tangent bundle: Let $M$ be a smooth manifold, and for any point $x \in M$ let $T_xM$ typify the tangent space anchored at $x$. Then, the tangent bundle $TM$ of $M$ is defined as follows.

Vector bundle: A vector bundle is a triple $(\pi, E, B)$, where $E$ and $B$ are smooth manifolds, and $\pi\colon E \to B$ is a smooth map satisfying the following.

• $\pi$ is surjective.
• There is an open cover $(U_i)_{i \in I}$ of $B$ with a collection of diffeomorphisms $h_i\colon \pi^{-1}(U_i) \to U_i\times \mathbb{R}^n$ where $h_i(\pi^{-1}(x)) = \{x\}\times \mathbb{R}^n$.
• For all $i, j \in I$, the map $h_i\circ h_j^{-1}$ restricted to $(U_i \cap U_j) \times \mathbb{R}^n$ is smooth.

Section: A (smooth) section of a vector bundle $(\pi, E, B)$ is a (smooth) map $m\colon B \to E$ such that $\pi\circ m = \mathop{id}_B$.

Vector field: If $M$ is a smooth manifold, a (smooth) vector field $F$ on $M$ is a section $F\colon M \to TM$. For our purpose, a vector field on $S^{n - 1}$ is a section of the tangent bundle. In a more prosaic but visually vibrant term, it is a map $F\colon S^{n - 1} \to \mathbb{R}^n$ such that for all $p \in S^{n - 1}$, we have $\langle F(p), p \rangle = 0$ and $F$ is continuous, where $\langle\cdot,\cdot\rangle$ can be taken as the standard dot product.

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About the author

Soumya Das Gupta is a student of the Chennai Mathematical Institute. This particular article placed fifth in our Article Writing Contest, 2022.