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  be a compact, oriented Riemannian manifold with boundary. 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 [, 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 

Differential geometry
Mathematical concepts
Integrals
Riemannian geometry