Vector Fields on the n-Sphere.

Publisher Summary This chapter discusses vector fields on the n-sphere. Any set of 2 k continuous vector fields tangent to S n are somewhere dependent. The case k = 0 is the classical result that a vector field on a sphere of even dimension has at least one zero. The case k = 2 was based on the erroneous assertion of π 5 (S 3 ) =0. If n and k are with r > 0, then the fiber bundle h: V n+1, 2h+1 → S n does not have a cross-section. If n is not of the form 2 k — 1, then the fiber bundle R n+1 → S n does not have a cross-section.