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.
be a compact, oriented Riemannian manifold with boundary.
Then for a function
, Santaló's formula takes the form where Under the assumptions that Santaló's formula is valid for all
In this case it is equivalent to the following identity of measures: where
is defined by
In particular this implies that the geodesic X-ray transform
{\displaystyle If(x,v)=\int _{0}^{\tau (x,v)}f(\varphi _{t}(x,v))\,dt}
extends to a bounded linear map
σ
-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
The following proof is taken from [[5] 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
For the integration by parts formula, recall that
leaves the Liouville-measure
div
{\displaystyle Xu=\operatorname {div} _{G}(uX)}
, the divergence with respect to the Sasaki-metric
The result thus follows from the divergence theorem and the observation that
is the inward-pointing unit-normal to
The resolvent is explicitly given by
{\displaystyle Rf(x,v)=\int _{0}^{\tau (x,v)}f(\varphi _{t}(x,v))\,dt}
and the mapping property
, which is a consequence of the non-trapping and the convexity assumption.