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I need a hint to this problem, or anything that gives a pointer in the right direction would be helpful. The question I need to prove is:

Let $M$ be a compact $n$-manifold such that Ric$(U,U) \geq (n-1)k$ for all $U \in TM$ such that $||U||=1$. Prove that $\gamma$ is a geodesic in $M$ of length greater than $\frac{m\pi}{\sqrt{k}}$, then $\gamma$ has index at least $m$.

Now I have seen a proof of a similar result where if $\gamma$ is of length greater than $\frac{\pi}{\sqrt{k}}$, then $\gamma$ has a conjugate point. This is lemma 13.142 of Manifold and Differential Geometry by Jeffery M.Lee. However I can't see a way of using this to prove what I need. Any help is appreciated.

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