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A formula describing the Liouville measure on the unit tangent bundle of a [[Riemannian manifold|Riemannian manifold]] in terms of the [[Geodesic flow|geodesic flow]] and the measure of a codimension-one submanifold (see [[#References|[a5]]] and [[#References|[a6]]], Chap. 19).
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s1300201.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s1300202.png" />-dimensional Riemannian manifold, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s1300203.png" /> be the unit tangent bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s1300204.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s1300205.png" /> be the Liouville measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s1300206.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s1300207.png" /> be the geodesic flow. One way to define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s1300208.png" /> is to start with the standard contact form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s1300209.png" /> (cf. [[Contact structure|Contact structure]]) and define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002010.png" />. Liouville's theorem says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002011.png" /> is invariant under the geodesic flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002012.png" /> (since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002013.png" /> is). Locally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002014.png" /> is just the product measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002015.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002016.png" /> is the Riemannian volume form and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002017.png" /> is the standard volume form on the unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002018.png" />-sphere.
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For any (locally defined) codimension-one submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002019.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002020.png" /> be the Riemannian volume element of the submanifold. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002021.png" />, and, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002022.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002023.png" /> be a unit normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002024.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002025.png" />. Then there is a smooth mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002026.png" />, given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002027.png" />. Santaló's formula says:
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A formula describing the Liouville measure on the unit tangent bundle of a [[Riemannian manifold]] in terms of the [[geodesic flow]] and the measure of a codimension-one submanifold (see [[#References|[a5]]] and [[#References|[a6]]], Chap. 19).
  
<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/s/s130/s130020/s13002028.png" /></td> </tr></table>
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Let $M$ be an $n$-dimensional Riemannian manifold, let $\pi : U M \rightarrow M$ be the unit tangent bundle of $M$, let $d u$ be the Liouville measure on $UM$, and let $g _ { t } : U M \rightarrow U M$ be the geodesic flow. One way to define $d u$ is to start with the standard contact form $\alpha$ (cf. [[Contact structure]]) and define $d u = \alpha \wedge d \alpha ^ { n - 1 }$. Liouville's theorem says that $d u$ is invariant under the geodesic flow $g_{ t }$ (since $\alpha$ is). Locally, $d u$ is just the product measure $dm \times dv$ where $dm$ is the Riemannian volume form and $dv$ is the standard volume form on the unit $( n - 1 )$-sphere.
  
The formula is used to convert integrals over subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002029.png" /> of the unit tangent bundle to iterated integrals, first over a fixed unit-speed geodesic (say parametrized on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002030.png" />) and then over the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002031.png" /> of geodesics which are parametrized by their intersections with a fixed codimension-one submanifold and endowed with the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002032.png" />, i.e.
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For any (locally defined) codimension-one submanifold $N \subset M$, let $d x$ be the Riemannian volume element of the submanifold. Let $S N = \pi ^ { - 1 } ( N ) \subset U M$, and, for each $x \in N$, let $N_x$ be a unit normal to $N$ at $x$. Then there is a smooth mapping $G : S N \times R \rightarrow U M$, given by $G ( v , t ) = g _ { t } ( v )$. Santaló's formula says:
  
<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/s/s130/s130020/s13002033.png" /></td> </tr></table>
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\begin{equation*} G ^ { * } ( d u ) = | \langle v , N _ { x } \rangle | d t d v d x. \end{equation*}
  
One of the most important applications is to the study of Riemannian manifolds with smooth boundary. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002035.png" /> is the inwardly pointing unit normal vector and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002036.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002037.png" />, set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002038.png" />. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002039.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002040.png" /> is defined for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002041.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002042.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002043.png" /> means <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002044.png" /> for some some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002045.png" /> and some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002046.png" />. In this setting, Santaló's formula takes the form:
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The formula is used to convert integrals over subsets $Q \subset U M$ of the unit tangent bundle to iterated integrals, first over a fixed unit-speed geodesic (say parametrized on $I ( \gamma ) \subset R$) and then over the space $\Gamma$ of geodesics which are parametrized by their intersections with a fixed codimension-one submanifold and endowed with the measure $d \gamma = | \langle v , N _ { x } \rangle | d v d x$, i.e.
  
<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/s/s130/s130020/s13002047.png" /></td> </tr></table>
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\begin{equation*} \int _ { Q } f ( u ) d u = \int _ { \gamma \in \Gamma} \int_{ I ( \gamma ) } f ( \gamma ^ { \prime } ( t ) ) d t d \gamma. \end{equation*}
  
One immediate application, by simply putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002048.png" />, is:
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One of the most important applications is to the study of Riemannian manifolds with smooth boundary. In this case $N = \partial M$, $N_x$ is the inwardly pointing unit normal vector and $U ^ { + } \partial M = \{ v \in S N : \langle v , N _ { x } \rangle &gt; 0 \}$. For any $u \in U M$, set $l ( u ) = \operatorname { sup } \{ t \geq 0 : g_t ( u ) \text { is defined} \}$. Note that $l ( u ) = \infty$ means that $g _ { t } ( u )$ is defined for all $t &gt; 0$. Let $\overline { U M } = \{ u \in U M : l ( - u ) &lt; \infty \} \cup U ^ { + } \partial M$, i.e. $u \in \overline { UM }$ means $u = g _ { t } ( v )$ for some some $v \in U ^ { + } \partial M$ and some $t \geq 0$. In this setting, Santaló's formula takes the form:
  
<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/s/s130/s130020/s13002049.png" /></td> </tr></table>
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\begin{equation*} \int _ { \overline{U M} } f ( u ) d u = \int _ { U ^ { + } \partial M  } \int _ { 0 } ^ { l ( v ) } f ( g _ { t } ( v ) ) d t \langle v , N _ { x } \rangle d v d x. \end{equation*}
  
Since the Liouville measure is locally a product measure, in the special case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002050.png" /> this says <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002051.png" />.
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One immediate application, by simply putting $f ( u ) = 1$, is:
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\begin{equation*} \operatorname { Vol } ( \overline { U M } ) = C _ { 1 } ( n ) \int _ { U ^ { + } \partial M } l ( v ) \langle v , N _ { x } \rangle d v d x. \end{equation*}
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Since the Liouville measure is locally a product measure, in the special case $\overline { UM } = UM$ this says <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002051.png"/>.
  
 
The formula is often used to prove isoperimetric and rigidity results. A sample of such applications can be found in the references. See [[#References|[a1]]] for Santaló's formula for time-like geodesic flow on Lorentzian surfaces.
 
The formula is often used to prove isoperimetric and rigidity results. A sample of such applications can be found in the references. See [[#References|[a1]]] for Santaló's formula for time-like geodesic flow on Lorentzian surfaces.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Andersson,  M. Dahl,  R. Howard,  "Boundary and lens rigidity of Lorentzian surfaces"  ''Trans. Amer. Math. Soc.'' , '''348''' :  6  (1996)  pp. 2307–2329</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Croke,  "A sharp four-dimensional isoperimetric inequality"  ''Comment. Math. Helv.'' , '''59''' :  2  (1984)  pp. 187–192</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C. Croke,  "Some isoperimetric inequalities and eigenvalue estimates"  ''Ann. Sci. École Norm. Sup.'' , '''13''' :  4  (1980)  pp. 419–435</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Croke,  N. Dairbekov,  V. Sharafutdinov,  "Local boundary rigidity of a compact Riemannian manifold with curvature bounded above"  ''Trans. Amer. Math. Soc.''  (to appear)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L.A. Santaló,  "Measure of sets of geodesics in a Riemannian space and applications to integral formulas in elliptic and hyperbolic spaces"  ''Summa Brasil. Math.'' , '''3'''  (1952)  pp. 1–11</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  L.A. Santaló,  "Integral geometry and geometric probability (With a foreword by Mark Kac)" , ''Encyclopedia Math. Appl.'' , '''1''' , Addison-Wesley  (1976)</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  L. Andersson,  M. Dahl,  R. Howard,  "Boundary and lens rigidity of Lorentzian surfaces"  ''Trans. Amer. Math. Soc.'' , '''348''' :  6  (1996)  pp. 2307–2329</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  C. Croke,  "A sharp four-dimensional isoperimetric inequality"  ''Comment. Math. Helv.'' , '''59''' :  2  (1984)  pp. 187–192</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  C. Croke,  "Some isoperimetric inequalities and eigenvalue estimates"  ''Ann. Sci. École Norm. Sup.'' , '''13''' :  4  (1980)  pp. 419–435</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  C. Croke,  N. Dairbekov,  V. Sharafutdinov,  "Local boundary rigidity of a compact Riemannian manifold with curvature bounded above"  ''Trans. Amer. Math. Soc.''  (to appear)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  L.A. Santaló,  "Measure of sets of geodesics in a Riemannian space and applications to integral formulas in elliptic and hyperbolic spaces"  ''Summa Brasil. Math.'' , '''3'''  (1952)  pp. 1–11</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  L.A. Santaló,  "Integral geometry and geometric probability (With a foreword by Mark Kac)" , ''Encyclopedia Math. Appl.'' , '''1''' , Addison-Wesley  (1976)</td></tr></table>

Latest revision as of 06:04, 5 April 2023

A formula describing the Liouville measure on the unit tangent bundle of a Riemannian manifold in terms of the geodesic flow and the measure of a codimension-one submanifold (see [a5] and [a6], Chap. 19).

Let $M$ be an $n$-dimensional Riemannian manifold, let $\pi : U M \rightarrow M$ be the unit tangent bundle of $M$, let $d u$ be the Liouville measure on $UM$, and let $g _ { t } : U M \rightarrow U M$ be the geodesic flow. One way to define $d u$ is to start with the standard contact form $\alpha$ (cf. Contact structure) and define $d u = \alpha \wedge d \alpha ^ { n - 1 }$. Liouville's theorem says that $d u$ is invariant under the geodesic flow $g_{ t }$ (since $\alpha$ is). Locally, $d u$ is just the product measure $dm \times dv$ where $dm$ is the Riemannian volume form and $dv$ is the standard volume form on the unit $( n - 1 )$-sphere.

For any (locally defined) codimension-one submanifold $N \subset M$, let $d x$ be the Riemannian volume element of the submanifold. Let $S N = \pi ^ { - 1 } ( N ) \subset U M$, and, for each $x \in N$, let $N_x$ be a unit normal to $N$ at $x$. Then there is a smooth mapping $G : S N \times R \rightarrow U M$, given by $G ( v , t ) = g _ { t } ( v )$. Santaló's formula says:

\begin{equation*} G ^ { * } ( d u ) = | \langle v , N _ { x } \rangle | d t d v d x. \end{equation*}

The formula is used to convert integrals over subsets $Q \subset U M$ of the unit tangent bundle to iterated integrals, first over a fixed unit-speed geodesic (say parametrized on $I ( \gamma ) \subset R$) and then over the space $\Gamma$ of geodesics which are parametrized by their intersections with a fixed codimension-one submanifold and endowed with the measure $d \gamma = | \langle v , N _ { x } \rangle | d v d x$, i.e.

\begin{equation*} \int _ { Q } f ( u ) d u = \int _ { \gamma \in \Gamma} \int_{ I ( \gamma ) } f ( \gamma ^ { \prime } ( t ) ) d t d \gamma. \end{equation*}

One of the most important applications is to the study of Riemannian manifolds with smooth boundary. In this case $N = \partial M$, $N_x$ is the inwardly pointing unit normal vector and $U ^ { + } \partial M = \{ v \in S N : \langle v , N _ { x } \rangle > 0 \}$. For any $u \in U M$, set $l ( u ) = \operatorname { sup } \{ t \geq 0 : g_t ( u ) \text { is defined} \}$. Note that $l ( u ) = \infty$ means that $g _ { t } ( u )$ is defined for all $t > 0$. Let $\overline { U M } = \{ u \in U M : l ( - u ) < \infty \} \cup U ^ { + } \partial M$, i.e. $u \in \overline { UM }$ means $u = g _ { t } ( v )$ for some some $v \in U ^ { + } \partial M$ and some $t \geq 0$. In this setting, Santaló's formula takes the form:

\begin{equation*} \int _ { \overline{U M} } f ( u ) d u = \int _ { U ^ { + } \partial M } \int _ { 0 } ^ { l ( v ) } f ( g _ { t } ( v ) ) d t \langle v , N _ { x } \rangle d v d x. \end{equation*}

One immediate application, by simply putting $f ( u ) = 1$, is:

\begin{equation*} \operatorname { Vol } ( \overline { U M } ) = C _ { 1 } ( n ) \int _ { U ^ { + } \partial M } l ( v ) \langle v , N _ { x } \rangle d v d x. \end{equation*}

Since the Liouville measure is locally a product measure, in the special case $\overline { UM } = UM$ this says .

The formula is often used to prove isoperimetric and rigidity results. A sample of such applications can be found in the references. See [a1] for Santaló's formula for time-like geodesic flow on Lorentzian surfaces.

References

[a1] L. Andersson, M. Dahl, R. Howard, "Boundary and lens rigidity of Lorentzian surfaces" Trans. Amer. Math. Soc. , 348 : 6 (1996) pp. 2307–2329
[a2] C. Croke, "A sharp four-dimensional isoperimetric inequality" Comment. Math. Helv. , 59 : 2 (1984) pp. 187–192
[a3] C. Croke, "Some isoperimetric inequalities and eigenvalue estimates" Ann. Sci. École Norm. Sup. , 13 : 4 (1980) pp. 419–435
[a4] C. Croke, N. Dairbekov, V. Sharafutdinov, "Local boundary rigidity of a compact Riemannian manifold with curvature bounded above" Trans. Amer. Math. Soc. (to appear)
[a5] L.A. Santaló, "Measure of sets of geodesics in a Riemannian space and applications to integral formulas in elliptic and hyperbolic spaces" Summa Brasil. Math. , 3 (1952) pp. 1–11
[a6] L.A. Santaló, "Integral geometry and geometric probability (With a foreword by Mark Kac)" , Encyclopedia Math. Appl. , 1 , Addison-Wesley (1976)
How to Cite This Entry:
Santaló formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Santal%C3%B3_formula&oldid=23516
This article was adapted from an original article by C. Croke (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article