Hairy ball theorem for $S^2$

Question

It is well-known that there is "no" nowhere vanishing continuous tangent vector field on $S^2$, by the so-called Hairy-ball theorem. But then, is there a continuous tangent vector field on $S^2$ which vanishes at only one point?

Answer 1

Consider the non-zero continuous tangent field to $D^1$ (the open unit disk in the plane)

$$T(r, \theta) = (1-r) \cdot \hat{x},$$

where $r$ is the distance from the origin, and $\hat{x}$ the rightward pointing unit vector.

There exists a diffeomorphism from $D^1$ to $S^2 \setminus \{ NP \}$, where NP is the north pole. Use this map to map the tangent field to a (non-zero!) tangent field $\hat{T}$ on $S^2 \setminus \{NP \}$, and complete it to a tangent field on $S^2$ by setting $\hat{T}(NP) = \vec{0}$.