Santaló's Formula

In differential geometry, Santaló's formula describes how to integrate a function on the unit sphere bundle of a Riemannian manifold by first integrating along every geodesic separately and then
over the space of all geodesics. It is a standard tool in integral geometry and has applications in isoperimetric and rigidity results. The formula is named after Luis Santaló, who first proved the result in 1952.

Formulation

Let $(M,\partial M,g)$ be a compact, oriented Riemannian manifold with boundary. Then for a function $f: SM \rightarrow \mathbb$, Santaló's formula takes the form
: \int_ f(x,v) \, d\mu(x,v) = \int_ \left \int_0^ f(\varphi_t(x,v)) \, dt \right\langle v, \nu(x) \rangle \, d \sigma(x,v),
where

* $(\varphi_t)_t$ is the geodesic flow and $\tau(x,v) = \sup\$ is the exit time of the geodesic with initial conditions $(x,v)\in SM$,
* $\mu$ and $\sigma$ are the Riemannian volume forms with respect to the Sasaki metric on $SM$ and $\partial S M$ respectively ($\mu$ is also called Liouville measure),
* $\nu$ is the inward-pointing unit normal to $\partial M$ and $\partial_+ SM := \$ the ''influx-boundary'', which should be thought of as parametrization of the space of geodesics.

Validity

Under the assumptions that
# $M$ is ''non-trapping'' (i.e. $\tau(x,v) <\infty$ for all $(x,v)\in SM$) and
# $\partial M$ is ''strictly convex'' (i.e. the second fundamental form $II_(x)$ is positive definite for every $x \in \partial M$),
Santaló's formula is valid for all $f\in C^\infty(M)$. In this case it is equivalent to the following identity of measures:
:$\Phi^*d \mu (x,v,t) = \langle \nu(x),x\rangle d \sigma(x,v) d t,$
where $\Omega=\$ and $\Phi:\Omega \rightarrow SM$ is defined by $\Phi(x,v,t)=\varphi_t(x,v)$. In particular
this implies that the ''geodesic X-ray transform'' $I f(x,v) = \int_0^ f(\varphi_t(x,v)) \, dt$ extends to a bounded linear map $I: L^1(SM, \mu) \rightarrow L^1(\partial_+ SM, \sigma_\nu)$, where $d\sigma_\nu(x,v) = \langle v, \nu(x) \rangle \, d \sigma(x,v)$ and thus there is the following, $L^1$-version of Santaló's formula:
:$\int_ f \, d \mu = \int_ If ~ d \sigma_\nu \quad \text f \in L^1(SM,\mu).$

If the non-trapping or the convexity condition from above fail, then there is a set $E\subset SM$ of positive measure, such that the geodesics emerging from $E$ either fail to hit the boundary of $M$ or hit it non-transversely. In this case Santaló's formula only remains true for functions with support disjoint from this exceptional set $E$.

Proof

The following proof is taken from Lemma 3.3">ref>Guillarmou, Colin, Marco Mazzucchelli, and Leo Tzou. "Boundary and lens rigidity for non-convex manifolds." American Journal of Mathematics 143 (2021), no. 2, 533-575.
Lemma 3.3 adapted to the (simpler) setting when conditions 1) and 2) from above are true. Santaló's formula follows from the following two ingredients, noting that $\partial_0SM=\$ has measure zero.
* An integration by parts formula for the geodesic vector field $X$:
:$\int_ Xu ~ d \mu = - \int_ u ~ d \sigma_\nu \quad \text u \in C^\infty(SM)$
* The construction of a resolvent for the transport equation $X u = - f$:
:$\exists R: C_c^\infty( SM\smallsetminus\partial_0 SM) \rightarrow C^\infty(SM): XRf = - f \text Rf\vert_ = If \quad \text f\in C_c^\infty( SM\smallsetminus\partial_0 SM)$

For the integration by parts formula, recall that $X$ leaves the Liouville-measure $\mu$ invariant and hence $Xu = \operatorname_G (uX)$, the divergence with respect to the Sasaki-metric $G$. The result thus follows from the divergence theorem and the observation that $\langle X(x,v), N(x,v)\rangle_G = \langle v, \nu(x)\rangle_g$, where $N$ is the inward-pointing unit-normal to $\partial SM$. The resolvent is explicitly given by $Rf(x,v) = \int_0^ f(\varphi_t(x,v)) \, dt$ and the mapping property $C_c^\infty( SM\smallsetminus\partial_0 SM) \rightarrow C^\infty(SM)$ follows from the smoothness of $\tau: SM\smallsetminus\partial_0 SM \rightarrow [0,\infty)$, which is a consequence of the non-trapping and the convexity assumption.

References

*{{cite book, author=Isaac Chavel, url=https://books.google.com/books?id=Wg-gQcvS25sC&q=Santalo%27s+formula, title=Riemannian Geometry: A Modern Introduction, publisher=Cambridge University Press, year=1995, series=Cambridge Tracts in Mathematics, volume=108, chapter=5.2 Santalo's formula, isbn=0-521-48578-9

Differential geometry
Integrals
Riemannian geometry