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Let $(M^n,g)$ be a complete, n dimensional Riemannian manifold. For $p \in M$, consider the map $\exp_p: T_p M \to M$. For $u \in T_pM$, we know $$ \exp^*_p vol|_{tu} = Det(D_{tu}\exp_p)dx^1...dx^n=J(u,t)t^{n-1}dtdu. $$ Here, du denotes the canonical measure of the unit sphere of $T_pM$.

For any unit speed geodesic $\gamma(t)$ with $\gamma(0)=p$ and $\gamma(l)=q$, let $u=\gamma'(0)$, $a(u,t)=J(u,t)t^{n-1}$ and $b(t)=a^{\frac{1}{n-1}}$. We have $b(0)=0, \quad b'(0)=1$ and $$ b''(t)+Ric(\gamma'(t),\gamma'(t))b \leqslant 0. $$ From now on, we assume that q is a conjugate point of p, then $D_{lu} \exp_p$ degenerate, so $b(l)=0$. What other information can we get? Do we have the left derivative $b'(l)=-1$? If $Ric\geqslant 0$, we know b is concave, do we have other information for $b'(l)$ except that $b'(l)\leqslant 1$?

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  • $\begingroup$ Have you tried to compute easy examples like products of two spheres of (intrinsic) diameter $l$ and $L>l$? $\endgroup$ Sep 8, 2017 at 8:38

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