View source for Santaló formula

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).

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.

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:

<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>

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.

<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>

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:

<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>

One immediate application, by simply putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130020/s13002048.png" />, is:

<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>

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" />.

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====
<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>