A remark on vector fields on lens spaces

Abstract

Let M be a C°°-manif old. The (continuous) vector field v on M is a crosssection of the tangent bundle of M, and &-field on ikί is a set of k vector fields vι, • ••, vk such that the k vectors v± (x), • •-, vk (x) are linearly independent for each point x e M. We denote by span(M) the maximal number of k where M admits a £>field. In this note, it is remarked that span(L(p)), of the (2n + l)-dimensional mod p lens space L (p), is given partially by the following PROPOSITION. Let n + l = m2 (m: odd), (i) If c = 0, then 2t + l<,span(L(p))<:2t-{-2 (=span(S)). (ii) // c = l, 2, then span(L(p)) = 2t + l (=span (S)). (in) If c = 3, then 2t + l^span(L(p))<,2t + 3 (=span(S)). Here the lens space L(p) (p>l) is the quotient space S/Γ of the unit sphere s by the topological transformation groupΓ= {1, γ, • , γ~} defined by