In this paper, we investigate the question of whether a typical vector field
on a compact connected Riemannian manifold Md has a `small' centralizer. In
the C1 case, we give two criteria, one of which is C1-generic, which
guarantees that the centralizer of a C1-generic vector field is indeed
small, namely \textit{collinear}. The other criterion states that a C1
\textit{separating} flow has a collinear C1-centralizer. When all the
singularities are hyperbolic, we prove that the collinearity property can
actually be promoted to a stronger one, refered as \textit{quasi-triviality}.
In particular, the C1-centralizer of a C1-generic vector field is
quasi-trivial. In certain cases, we obtain the triviality of the centralizer of
a C1-generic vector field, which includes C1-generic Axiom A (or
sectional Axiom A) vector fields and C1-generic vector fields with countably
many chain recurrent classes. For sufficiently regular vector fields, we also
obtain various criteria which ensure that the centralizer is \textit{trivial}
(as small as it can be), and we show that in higher regularity, collinearity
and triviality of the Cd-centralizer are equivalent properties for a generic
vector field in the Cd topology. We also obtain that in the non-uniformly
hyperbolic scenario, with regularity C2, the C1-centralizer is trivial.Comment: This is the final version, accepted in Mathematische Zeitschrift. New
introduction and some proofs where rewritten and/or expanded, according to
referee's suggestion. Also, a new appendix was adde