Vector Fields on Spheres and Clifford Algebras

Speaker: Balázs Csikós

Abstract: By the well-known "hedgehog theorem", every (tangential) vector field on the sphere S^2 must vanish somewhere. More generally, Poincaré proved that one can find a nowhere zero vector field on the n-dimensional sphere S^n if and only if n is odd. This result raised the question how many pointwise linearly independent vector fields can be found on S^n for a given n. The solution of this problem consists of two parts. First, we have to construct as many linearly independent vector fields on S^n as we can, then it has to be proved that the maximal number is reached. In this lecture we focus on the first part and show how representation theory of Clifford algebras tells us the maximal number of pointwise linearly independent linear vector fields on S^n. This number cannot be exceeded even if we drop the linearity condition on the vector fields, by a theorem of Adams.

[shalvagang]

I want to know that this like algebra of vector feild on a reimaan sphere is a type of course and at what level we study about it and what are the prewuesties for it

[tensorific]

He doesn't take a single look at any notes!