It is known since Bochner that for M compact with negative Ricci curvature, the group of isometries is discreet and hence finite. Are there any generalizations to compact non-positive sectional curvature manifolds? Of course, $T^n$ is a counterexample for strictly 0 curvature case. But what about sectional curvature <0 on a non-empty open set, and vanishing elsewhere?


As I understand Bochner's argument does break down in this case.


Known counterexamples?



Isometries of manifolds with non-positive sectional curvature
Bochner's theorem extends to nonpositive Ricci to give:



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> If $(M,g)$ is compact and has $\textrm{Ric}\leq 0$ then any Killing vector $X$ is parallel and $\textrm{Ric}(X,X) = 0$.
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> 
> 


See Petersen (3rd ed) Theorem 8.2.2.


Thus if $(M,g)$ is compact and has $\textrm{Ric}\leq 0$ with $<0$ then $X = 0$.


