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Hello respected members of Mathoverflow. I was reading the paper "Flots d’Anosov dont les feuilletages stables sont différentiables" by Etienne Ghys and there was a statement which he remarked was trivial, but which seem not to me.

In the 13th page ( 262 journal page number) of the PDF of the paper he says:

" Mais ceci est impossible car il est bien connu (et élémentaire) qu'un champ de vecteurs qui commute avec un champ d'Anosov est nécessairement un multiple constant de ce champ."

This translates to,

"But this is impossible because it is well known (and elementary) that a vector field that commutes with an Anosov flow is necessarily a constant multiple of the later."

Even though it seems to be a very standard fact in the literature, I seemed to have gotten stuck on it. Is it possible to slightly outline the method or suggest some reference.

Thanks.

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    $\begingroup$ Intuitively, if the vector field had some component which wasn't parallel to the Anosov vector field, then one may decompose the vector field into a part which is tangent to the stable, unstable, and flow directions. Flowing along the Anosov flow, the unstable part would expand unboundedly. But this is impossible if the manifold is compact. Similar for unstable component by the negative flow. $\endgroup$
    – Ian Agol
    Commented Feb 27, 2018 at 1:24
  • $\begingroup$ Yes I have finally understood. I had to do a bit more work after showing with the help of your suggestion that the flows are parallel. Thanksa lot for the hint. $\endgroup$
    – hakunamatata
    Commented Mar 3, 2018 at 17:04

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