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Jul 9, 2018 at 16:26 comment added coudy The curvature is zero at a point, the flow is Anosov, you make a perturbation such that the curvature becomes positive around the point and the flow stays Anosov.
Jul 9, 2018 at 7:54 comment added Ali Taghavi But a slight perturbation of a compact surface with strictly negative curvature is stile a surface with strictly negative curvature. Am I right? But i think that there is modification of your argument. I try to find the valuable reference you mentioned in your answer and learn them. thanks again for your attention to my question and these very helpful references
Jul 8, 2018 at 19:49 comment added Ali Taghavi Thank you very much for this very great answer. I am sorry that I can not accept the two answers simultaneously. meta.mathoverflow.net/questions/1491/…
Jul 8, 2018 at 19:37 history answered coudy CC BY-SA 4.0