Let $M$ be  a  differentiable  manifold. 

Is there a  terminology  for the maximum number  of  globally defined independent vector  fields on $M\to M$? Is there a terminology for the  maximum number of  globally defined independent vector  fields on $TM$ whose mutual  flow  commute, that is they mutually have  zero  lie  bracket? What kind of  characteristic  classes can be used to compute such quantitties? what are these numbers for  $M=S^n$?