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Here $M$ is a smooth manifold and $V(M)$ is the space of vector fields on $M$, $V^*(M)$ is the space of covector fields on $M$ and $T_{(q,r)}(M)$ is the space of $(q,r)$ tensor fields on $M$. What does it mean the example in picture? Namely, what does it mean by impossible to comb a sphere? Could anyone please explain?

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    $\begingroup$ As a side note, this is completely wrong that a vector field on $S^2$ must vanish at least at two points, it is only one point. For the construction of a vector field on $S^2$ with only one zero, see here. $\endgroup$ – C. Falcon Apr 24 '18 at 14:43

The visualization of the vector field being invoked here is that of attaching a strand of hair to each point of the manifold, and laying the hair flat in the direction of the vector field.

Any point where the vector field is zero is a point where you can't do that, since there isn't a direction for the hair to "point"; and furthermore all of the nearby hairs around it are all pointing in different directions.

Since "combing" a region of hair would cause all of the hairs to lay flat1, the fact any vector field must have a zero means it's impossible to comb an entire sphere covered in of hair.

1: And if we choose any metric on the region, such a combing would even define a vector field by choosing unit vectors in the direction of the hair


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