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 isoperimetricCroke, Christopher B. "A sharp four dimensional isoperimetric inequality." Commentarii Mathematici Helvetici 59.1 (1984): 187–192. and rigidity results.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. The formula is named after Luis Santaló, who first proved the result in 1952.Santaló, Luis Antonio. Measure of sets of geodesics in a Riemannian space and applications to integral formulas in elliptic and hyperbolic spaces. 1952Santaló, Luis A. Integral geometry and geometric probability. Cambridge university press, 2004
Formulation
Let be a compact, oriented Riemannian manifold with boundary. Suppose that every vector in the unit tangent bundle can be reached via the geodesic flow starting from points on . Then for a function , Santaló's formula takes the form
where
is the geodesic flow and is the exit time of the geodesic with initial conditions , and are the Riemannian volume forms with respect to the Sasaki metric on and respectively ( is also called Liouville measure), is the inward-pointing unit normal to and the influx-boundary, which should be thought of as parametrization of the space of geodesics.
Validity Under the assumptions that is non-trapping (i.e. for all ) and is strictly convex (i.e. the second fundamental form is positive definite for every ), Santaló's formula is valid for all . In this case it is equivalent to the following identity of measures:
where and is defined by . In particular this implies that the geodesic X-ray transform extends to a bounded linear map , where and thus there is the following, -version of Santaló's formula:
If the non-trapping or the convexity condition from above fail, then there is a set of positive measure, such that the geodesics emerging from either fail to hit the boundary of or hit it non-transversely. In this case Santaló's formula only remains true for functions with support disjoint from this exceptional set .
Proof The following proof is taken from [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 has measure zero. An integration by parts formula for the geodesic vector field :
The construction of a resolvent for the transport equation :
For the integration by parts formula, recall that leaves the Liouville-measure invariant and hence , the divergence with respect to the Sasaki-metric . The result thus follows from the divergence theorem and the observation that , where is the inward-pointing unit-normal to . The resolvent is explicitly given by and the mapping property follows from the smoothness of , which is a consequence of the non-trapping and the convexity assumption.
References