Berger inequality

For a compact Riemannian manifold $M = M ^ { n }$, let

\begin{equation*} \operatorname {inj} M = \operatorname { inf } _ { p \in M } \operatorname { sup } \{ r : \operatorname { exp } _ { p } \text { injective on } B _ { r } ( 0 ) \subset T _ { p } M \}, \end{equation*}

where $B _ { r } ( 0 )$ is the ball around $0$ with radius $r$, be the injectivity radius, and set $\alpha ( n ) = \text { Vol } ( S ^ { n } )$. Then the inequality

\begin{equation*} \operatorname {Vol} ( M ) \geq \alpha ( n ) \left( \frac { \operatorname { inj } M } { \pi } \right) ^ { n } \end{equation*}

holds, with equality if and only if $M$ is isometric to the standard sphere with diameter $\operatorname { inj} M$.

This inequality relies on the Kazdan inequality applied to the Jacobi equation $X ^ { \prime \prime } ( t ) + {\cal {R}} ( t ) \circ X ( t ) = 0$ for operators $X ( t )$ on $v ^ { \perp }$ for a unit vector $v \in T _ { p } M$. Here, $R ( t ) = R ( \gamma ^ { \prime } ( t ) , . ) \gamma ^ { \prime } ( t )$ is the curvature operator, $\tau _ { t , v } : T _ { p } M \rightarrow T _ { \gamma ( t ) } M$ is the parallel transport along the geodesic ray $\gamma ( t ) = \operatorname { exp } _ { p } ( t v )$, and $\mathcal{R} ( t ) = \tau ^ { - 1 _ { t , v } } \circ R ( t ) \circ \tau _ { t , v }$ is the parallel translated curvature operator on $v ^ { \perp } \subset T _ { p } M$.

References

[a1]	M. Berger, "Une borne inférieure pour le volume d'une variété riemannienes en fonction du rayon d'injectivité" Ann. Inst. Fourier (Grenoble) , 30 (1980) pp. 259–265
[a2]	I. Chavel, "Riemannian geometry: A modern introduction" , Cambridge Univ. Press (1995)