The Hairy Ball Theorem


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The Hairy Ball Theorem

  1. 1. The hairy ball theorem Vladimir Cuesta † Instituto de Ciencias Nucleares, Universidad Nacional Aut´ noma de M´ xico, 70-543, Ciudad de o e M´ xico, M´ xico e e Abstract. I show that for the sphere S 2 , if I choose a finite number of points then I can define a tangent vector for each point. If I consider all the points in the sphere S 2 , I can not define a tangent vector for all together. 1. The hairy ball theorem If I take one point p1 of S 2 I can define a vector on it independent of everything. Now, if I take two different points p1 and p2 on S 2 , I can put p1 on a first sphere S 2 and p2 on a second sphere S 2 , in this case a tangent vector can exist on the first sphere and the point p1 and a tangent vector can exist (locally) for the second sphere and the point p2 in an independent way with a given direction, finally I can put it together without problems because p1 and p2 are different. Like third case, I take the different points p1 , p2 and p3 and I can put it on three different spheres S 2 and I can construct tangent vector (locally) for the three points with a given direction and finally put all together on the same sphere. For the case on n different points I can follow the previous procedure without problems. However, for the case of all the points on S 2 , the previous algorithm will fail because I can not separate tangent vectors at the final step. I mean, It does not exist space for separating different tangent vectors. Sometimes, you can describe the theorem like: You can not comb the hairs without the presence of cowlicks for a ball with hairs all over it. In fact, all the hairs will be vertical. 2. Conclusions and remarks The previous argument can be applied on different 2-surfaces, 3-surfaces, 4-surfaces and so on, making the difference for the case of finite points or the case of complete surfaces. †
  2. 2. The hairy ball theorem 2 The reader can note the importance for computer studies on wind, storms, fluids, fields and so on. I present the following example, you can think currents of wind on a portion of the plane, probably all the wind on the same direction (in this case I have a finite number of points and I can define a vector for all the points on the portion of the plane). For an unknown reason, the wind can be accumulated on a little 2-surface and according to the hairy ball theorem, there is not enough space for doing that and the wind will leave the plane and the air will transform into a twister. I mean, the hair will be vertical for complete 2-surfaces. For higher dimensions, the reasoning is similar. 3. Acknowledgments I appreciate very fruitful discussions with my children and their mothers, my brothers, friends and some members of the Department of Mathematics of the Centro de Investigaci´ n y de Estudios o Avanzados del Instituto Polit´ cnico Nacional (in particular some people of the football team). e