(half-of a) hairy ball theorem

Question

so I know that, by the hairy ball theorem, there does not exist a smooth global frame for the n-sphere.

What about for the northern hemisphere? Can anyone comb half-of a sphere (provide a smooth global frame)?

score 2 · Accepted Answer · answered May 12 '14 at 07:04

2

The answer is yes. The northern hemisphere of $S^n$ is diffeomorphic to the Euclidean space $\mathbb R^n$, so it does admit a non-vanishing vector field, namely take the pushforward of the vector field $X=\frac{\partial}{\partial x^1}$ on $\mathbb R^n$.

answered May 12 '14 at 07:04

Dimitris

6,984
1
30
47

Wait, so they form a basis at each point that smoothly varies? – Dave May 12 '14 at 07:19

(half-of a) hairy ball theorem

1 Answers1