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Let $M$ be a smooth manifold of dimension $\geq 2$. Let $g$ be a complete Riemannian metric on $M$. Let $C \subseteq M \times M$ be the set of pairs of $g$-conjugate points.

The set $C$ doesn't generally determine $g$, even up to scaling. For example hyperbolic space and Euclidean space both have no conjugate points.

Question: When can the metric $g$ be recovered (up to scaling) from knowledge of the pair $(M,C)$? Does this happen when $M$ is

  1. A compact manifold?

  2. A compact homogeneous space?

  3. A compact symmetric space?

Do curvature conditions help? And in the noncompact case, can $g$ sometimes be recovered (up to scaling) from $C$?

Note: I've been a bit vague with the special cases I'm suggesting -- e.g. it's not clear if I'm asking whether for $M$ a compact homogeneous space, the conjugate points uniquely determine the metric among all metrics or just among homogeneous metrics. Ultimately I'd be interested in both questions, but I think the latter, more restricted question, is already interesting.

EDIT: As an example, I think I've almost convinced myself that for $n \geq 2$, the round metric on $\mathbb R \mathbb P^n$ can be reconstructed (up to scaling) from knowledge of the conjugate points.

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I guess the most famous cases are: (1) Riemannian tori without conjugate points are flat (Hopf, Burago and Ivanov) and (2) Auf-Wiedersehen metrics on the sphere (Green, Berger) are round (i.e. your projective space case).

You can then do other things. For example Victor Bangert recently showed that a plane without conjugate point and with the area-growth of a Euclidean metric must be flat.

I think you may find something of interest in Besse's book "Manifold all of whose geodesics are closed". Possibly the same conjugate points as the Fubini study metric in CP^n + Kahler implies that the metric is a multiple of the Fubiny-Study metric, etc.

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