Characterization of isometries on Riemannian manifolds of positive (non-negative) curvature analogous to Bishop-O'Neill theorem

Question

Bishop and O'Neill in

Bishop, R. L.; O’Neill, B., Manifolds of negative curvature, Trans. Am. Math. Soc. 145, 1-49 (1969). ZBL0191.52002.

characterized isometry of Riemannian manifolds of non-positive curvature which states that:

Theorem (Bishop-O'Neill): Let $\varphi$ be an isometry of $M$. Then exactly one of the following is true:

1. $\varphi$ has a fixed point;
2. $\varphi$ translates a (unique) geodesic;
3. $f_\varphi :M\to \Bbb R$ defined by $f_\varphi(x) = d^2(x,\varphi(x))$ has no minimum.

I wonder if there is any analogous for positive (non-negative) curvature? If the answer is no, What efforts have been made so far for similar theorems?

Edit (after @LeeMosher comment): $M$ is simply-connected, complete, Riemannian manifold of sectional curvature $K \leq C < 0$. Othewise there is a counterexample. (E.g. see below Lee Mosher's comment)