Is there a simple proof that there is no Anosov flow on $S^2$? Where can I find it?



Anosov flow on the 2-sphere
The usual definition of Anosov flow requires three invariant sub-bundles, so I guess you are actually asking about the 3-sphere?


Plante and Thurston have proved in


*Plante, J. F.; Thurston, W. P.*, [**Anosov flows and the fundamental group**](https://doi.org/10.1016/0040-9383(72)90002-X), Topology 11, 147-150 (1972). [ZBL0246.58014](https://zbmath.org/?q=an:0246.58014).


that if a manifold admits a codimension 1 Anosov flow, then its fundamental group has exponential growth.


