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For a compact [[Riemannian manifold|Riemannian manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b1201201.png" />, let
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b1201202.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b1201203.png" /> is the ball around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b1201204.png" /> with radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b1201205.png" />, be the injectivity radius, and set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b1201206.png" />. Then the inequality
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For a compact [[Riemannian manifold|Riemannian manifold]] $M = M ^ { n }$, let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b1201207.png" /></td> </tr></table>
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\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*}
  
holds, with equality if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b1201208.png" /> is isometric to the standard sphere with diameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b1201209.png" />.
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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
  
This inequality relies on the [[Kazdan inequality|Kazdan inequality]] applied to the Jacobi equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b12012010.png" /> for operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b12012011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b12012012.png" /> for a unit vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b12012013.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b12012014.png" /> is the [[Curvature|curvature]] operator, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b12012015.png" /> is the parallel transport along the geodesic ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b12012016.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b12012017.png" /> is the parallel translated curvature operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120120/b12012018.png" />.
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\begin{equation*} \operatorname {Vol} ( M ) \geq \alpha ( n ) \left( \frac { \operatorname { inj } M } { \pi } \right) ^ { n } \end{equation*}
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holds, with equality if and only if $M$ is isometric to the standard sphere with diameter $\operatorname { inj} M$.
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This inequality relies on the [[Kazdan inequality|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|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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I. Chavel,  "Riemannian geometry: A modern introduction" , Cambridge Univ. Press  (1995)</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  I. Chavel,  "Riemannian geometry: A modern introduction" , Cambridge Univ. Press  (1995)</td></tr></table>

Latest revision as of 16:57, 1 July 2020

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)
How to Cite This Entry:
Berger inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Berger_inequality&oldid=50215
This article was adapted from an original article by H. Kaul (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article