Theorem (Weinstein - Synge) Let $(M,g)$ be a compact oriented positively curved $n$-dimensional Riemannian manifold. Supose $f$ is an isometry that preserves orientation if $n$ is even or reverses it if $n$ is odd. Then $f$ has a fixed point.
This result is also true for conformal diffeomorphisms and implies classical Synge's theorem for compact manifolds. Using the celebrated Soul Theorem we see that the compactness hypotesis in Synge's theorem can be replaced by completness.
Question Can we do the same on Weinstein's theorem?
PS: the even-dimensional version of Synge's theorem is that a compact positively curved oriented even-dimensional Riemannian manifold be simply connected.