 For an ordinary (2-dimensional) sphere in three-dimensional space, it can be shown that the index of any vector field on the sphere must be 2.This shows that every such vector field must have a zero.This implies the hairy ball theorem, which states that if a vector in R3 is assigned to each point of the unit sphere S2 in a continuous manner, then it is impossible to "comb the hairs flat", i.e., to choose the vectors in a continuous way such that they are all non-zero and tangent to S2.