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I just came across a chapter in Needham's Visual complex analysis; in particular, these diagrams:

enter image description here

(p. 153 - these happen to be on the cover as well)

They represent families of Möbius transformations acting on the Riemann sphere. The pictures reminded me of the famous Hairy ball theorem: one could imaging the combing as being a combined effect of the above transformations. The more striking connection is that they completely agree with the theorem: any such transformation has at least one fixed point. Is there an underlying connection here, either with the theorem itself or perhaps a dumbed-down version of it?

PS: I know little about the actual proof of the theorem $-$ please forgive me if the connection (or absence thereof) is obvious.

user118224
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    The images are slightly misleading. Mobius transformations are maps from the sphere to itself, not a continuous flow which is what the images suggest. It is true though that the invariant geodesics of a Mobius transformation can be generated from a geodesic flow which gives a canonical vector field (and so puts us in the realm of the hairy ball theorem). The fixed points of the transformations correspond then to the zeros of the respective vector field. – Dan Rust Jan 04 '14 at 00:33

2 Answers2

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As Daniel mentioned in comments, there is an intimate relation between flows, integral curves and vector fields (a collection of arrows at every point of the sphere). Given a flow, one can follow individual points to create curves on the surface which fall into distinct orbits in this case. Given a family of such curves, one can differentiate them to obtain the vector field.

Now, the Hairy ball theorem only concerns the vector fields and says that the sum of degrees of zeroes of any field is $2$ but thanks to the above connection this also applies to flows. In particular, a vector field has a zero precisely when the associated flow has a fixed point.

We still need to connect flows with Mobius transformations though. For example, a rotation by $90$ degrees around certain axis is a transformation that replaces a sphere with its rotated double but a flow is an actual animation of this rotation. It's possible to pass from one to the other the same way movie-makers do. Rotate the sphere by $1$ degree, take a picture, rotate by another degree, take a picture again, etc. At the end you will have stop motion film recording the rotation. By making the steps infinitesimal you'll obtain the flow. Still, for the fixed points this distinction doesn't matter because they don't move at all, so we are at the end of the road: the fixed points of the Mobius transformations are precisely the zeroes of the corresponding vector fields.

Note that the Hairy ball theorem also tells us that there must be precisely two fixed points (at least when the degrees of the zeroes are positive, as they are here), counted with multiplicity.

Marek
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A Mobius transformation can be continued to an isometry of hyperbolic space, which (being a self-map of the ball) has a fixed point. This can be in the interior of the ball (in which case the transformation is elliptic, so a rotation, which, in three dimensions always has an axis), or on the sphere. Of course, this is the hard way to get the fixed points -- the easy way just uses the explicit matrix description.

Igor Rivin
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