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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)

C.F.G
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    Which theorem in that paper are you referring to? I ask because some hypotheses are missing: Assuming only that $M$ is a Riemannian manifold of non-positive curvature, there are simple counterexamples. – Lee Mosher Sep 27 '20 at 19:22
  • Proposition 4.2. I think the authors considered all manifolds to be of non-positive curvature. What example? – C.F.G Sep 27 '20 at 19:31
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    There are isometries of compact hyperbolic surfaces which have no fixed point, do not translate any geodesic, and $f_\varphi$ has a minimum. Take, for example, a nontrivial deck transformation of any connected, regular covering space of degree $\ge 2$ over any closed hyperbolic surface. – Lee Mosher Sep 27 '20 at 20:50
  • @Lee: I agree your isometries don't satisfy 1 and 3, but how do you see it doesn't translate a geodesic? – Jason DeVito - on hiatus Sep 28 '20 at 00:20
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    What I wrote regarding 2 is not quite right: it should have been "do not translate a unique geodesic", i.e. there are examples that translate more than one closed geodesic. – Lee Mosher Sep 28 '20 at 00:55
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    One such example is the 180 degree rotation of this genus $3$ surface around an axis that goes through the middle hole. – Lee Mosher Sep 28 '20 at 00:58
  • @LeeMosher: Sorry, I did not actually read the Bishop and O'Neill paper, and I saw this theorem in another paper referred to the above paper. here are the assumptions (in introduction section of second paper): $M$ will denote a simply-connected, complete, Riemannian manifold of sectional curvature $K \leq C < 0$ – C.F.G Sep 28 '20 at 07:24

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