Santaló's formula explained

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^[1] and rigidity results.^[2] The formula is named after Luis Santaló, who first proved the result in 1952.^{[3] [4]}

Formulation

Let

(M,\partial M,g) be a compact, oriented Riemannian manifold with boundary. Then for a function f: SM → C , Santaló's formula takes the form\int _{SM} f(x,v) d\mu(x,v) = \int _{\partial _{+} SM} \left[ \int ^{\tau(x,v)} _{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\{t\ge 0 : \forall s\in [ 0 ,t]:~ \varphi _{s(x,v)\in} SM \} 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 : = \{(x,v) \in SM: x \in \partial M, \langle v,\nu(x) \rangle \ge 0 \} 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) < inf ty for all (x,v)\in SM ) and \partial M is strictly convex (i.e. the second fundamental form II _{\partial} (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 = \{(x,v,t): (x,v)\in \partial _{+SM,} t\in ( 0 ,\tau(x,v)) \} and \Phi:\Omega → SM is defined by \Phi(x,v,t) = \varphi _{t(x,v)} . In particularthis implies that the geodesic X-ray transform I f(x,v) = \int ^{\tau(x,v)} _{0} f(\varphi _{t(x,v))} dt extends to a bounded linear map I: L ^{1(SM,} \mu) → 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 _{SM} f d \mu = \int _{\partial _{+} SM} If ~ d \sigma _{\nu} forall 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 [<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. </ref> 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=\{(x,v):\langle} \nu(x), v\rangle =0 \} has measure zero.

• An integration by parts formula for the geodesic vector field

X :\int _{SM} Xu ~ d \mu = - \int _{\partial _{+} SM} u ~ d \sigma _{\nu} forall u \in C ^{infty(SM)}

• The construction of a resolvent for the transport equation

X u = - f :\exists R: C ^{inf ty(} _{c} SM\smallsetminus\partial _{0} SM) → C ^{infty(SM):} XRf = - f and Rf\vert _{\partial _{+} SM} = If forall f\in C ^{inf ty(} _{c} SM\smallsetminus\partial _{0} SM)

For the integration by parts formula, recall that

X leaves the Liouville-measure \mu invariant and hence Xu = \operatorname{div} _{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 ^{\tau(x,v)} _{0} f(\varphi _{t(x,v))} dt and the mapping property C ^{inf ty(} _{c} SM\smallsetminus\partial _{0} SM) → C ^{infty(SM)} follows from the smoothness of \tau: SM\smallsetminus\partial _{0} SM → [ 0 , inf ty) , which is a consequence of the non-trapping and the convexity assumption.

References

1. Croke, Christopher B. "A sharp four dimensional isoperimetric inequality." Commentarii Mathematici Helvetici 59.1 (1984): 187–192.
2. Ilmavirta, Joonas, and François Monard. "4 Integral geometry on manifolds with boundary and applications." The Radon Transform: The First 100 Years and Beyond 22 (2019): 43.
3. Santaló, Luis Antonio. Measure of sets of geodesics in a Riemannian space and applications to integral formulas in elliptic and hyperbolic spaces. 1952
4. Santaló, Luis A. Integral geometry and geometric probability. Cambridge university press, 2004

• Book: Isaac Chavel. Riemannian Geometry: A Modern Introduction. Cambridge University Press. 1995. Cambridge Tracts in Mathematics. 108. 5.2 Santalo's formula. 0-521-48578-9.