The Vector Field Problem for Homogeneous Spaces

Abstract

Let M be a smooth connected manifold of dimension \(n\ge 1\). A vector field on M is an association \(p\rightarrow v(p)\) of a tangent vector \(v(p)\in T_pM\) for each \(p\in M\) which varies continuously with p. In more technical language, it is a (continuous) cross section of the tangent bundle \(\tau (M)\). The vector field problem asks: Given M, what is the largest possible number r such that there exist vector fields \(v_1,\ldots , v_r\) which are everywhere linearly independent, that is, \(v_1(x),\ldots ,v_r(x)\in T_xM\) are linearly independent for every \(x\in M\). The number r is called the span of M, written \({\text {span}}(M)\). It is clear that \(0\le {\text {span}}(M)\le \dim (M)\). The vector field problem is an important and classical problem in differential topology. In this survey, we shall consider the vector field problem focussing mainly on the class of compact homogeneous spaces.