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Let $(M, g)$ be a compact Riemannian manifold. Let $i(-)$ be the index of a geodesic and let $l(-)$ be the length. Is there an inequality of the form $i(\gamma) \leq C l(\gamma)$ for some $C>0$ depending only on $M$. (I am happy to assume that $g$ is "generic" if needed.)

My intuition for why this might be true is roughly as follows: the index of a geodesic is equal to the total number of conjugate points counted with multiplicity. Now the distance between two adjacent conjugate points is bounded from below by the injectivity radius of $(M,g)$. So if all conjugate points counted with multiplicity one, the desired inequality would follow. One might naively hope that some sort of genericity might suffice to ensure all conjugate points count with multiplicity one. Of course, I could imagine that the inequality is true/false for trivial reasons -- I am really not expert in this area.

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  • $\begingroup$ For the generic set of so-called bumpy metrics the closed geodesics satisfy your multiplicity condition, no? In general maybe you could try to argue by contradiction, and extract a subsequence converging to some limit geodesic $\gamma$. In naive terms one might hope to prove some continuity of the second variation, although I'm a bit skeptical... (This fails in the 'higher-dimensional case', when one works with minimal surfaces.) $\endgroup$
    – Leo Moos
    Commented Mar 16, 2021 at 3:31
  • $\begingroup$ @LeoMoos I was not aware of this fact concerning bumpy metrics. Do you know a reference where this property is mentioned? $\endgroup$
    – user142700
    Commented Mar 16, 2021 at 12:40

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