Santaló formula

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)