What  is  an  example  of a $n$ dimensional   manifold $M$ which  is  not  a  lie  group or $S^{7}$ but  satisfies the  following  property?:


>There is  an $n$  dimensional   sub vector space $V\subset \chi^{\infty}(M)$ such that every $0 \neq X \in  V$ is  a  nonvanishing  vector  field  on $M$.



So  this is a  motivation to  define  an  invariant  of  manifolds  as  follows:
$$ \text{The  maximum number $k\leq n$ with a k dimensional subvector  space$$ $V\subset\chi^{\infty}(M)$  with the  above property} $$