Projective span of a manifold

Question

Recall that the span of a smooth manifold $M$, denoted $\operatorname{span}(M)$, is the largest $k$ such that $M$ admits $k$ linearly independent vector fields. Equivalently, $\operatorname{span}(M)$ is the largest $k$ such that the tangent bundle $TM$ splits off a trivial bundle of dimension $k$.

I am interested in a related notion, which I've been calling projective span and denoting $\operatorname{pspan}(M)$. It is defined to be the largest $k$ such that $TM$ splits off a direct sum of $k$ (not necessarily trivial) line bundles.

Has this concept been studied before (perhaps under a different name)?

A few observations: Clearly $\operatorname{pspan}(M)\geq \operatorname{span}(M)$. The Klein bottle $K$ has $\operatorname{pspan}(K)=2>1=\operatorname{span}(K)$. If a manifold has projective span $k$, then some cover of it has span at least $k$.

Answer 1

With apologies for the self-promotion, my student Baylee Schutte and I now have a paper on this topic called Projective span of Wall manifolds. We calculate the projective span of a family of manifolds $Q(m,n)$ defined by C.T.C. Wall in the course of his determination of the oriented cobordism ring; they are mapping tori of the Dold manifolds $P(m,n)$. A key idea is that $G$-quasi-invariant vector fields on a free $G$-manifold $M$, meaning vector fields that are $G$-invariant up to a sign, define line fields on on the quotient $M/G$.

It turns out that the question of line element parallelizability, or whether the tangent bundle splits into a sum of line bundles, had previously been studied by Massey and Szczarba, Auslander and Szczarba, and others (see the references in our paper).