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
be a compact, oriented Riemannian manifold with boundary. Then for a function
, Santaló's formula takes the form
![{\displaystyle \int _{SM}f(x,v)\,d\mu (x,v)=\int _{\partial _{+}SM}\left[\int _{0}^{\tau (x,v)}f(\varphi _{t}(x,v))\,dt\right]\langle v,\nu (x)\rangle \,d\sigma (x,v),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb2f146cd234f37408cb5d7f32777ec36ef760dc)
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:
![{\displaystyle \Phi ^{*}d\mu (x,v,t)=\langle \nu (x),x\rangle d\sigma (x,v)dt,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25b359a8be5c270ee454d6349e584ef313fc68fe)
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:
![{\displaystyle \int _{SM}f\,d\mu =\int _{\partial _{+}SM}If~d\sigma _{\nu }\quad {\text{for all }}f\in L^{1}(SM,\mu ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5815c98859d46f7307069c4fefa0550cf793ddbe)
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 [[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
has measure zero.
- An integration by parts formula for the geodesic vector field
:
![{\displaystyle \int _{SM}Xu~d\mu =-\int _{\partial _{+}SM}u~d\sigma _{\nu }\quad {\text{for all }}u\in C^{\infty }(SM)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f02e88d0b6f8ef070b5ee1f0136d48b0891e687e)
- The construction of a resolvent for the transport equation
:
![{\displaystyle \exists R:C_{c}^{\infty }(SM\smallsetminus \partial _{0}SM)\rightarrow C^{\infty }(SM):XRf=-f{\text{ and }}Rf\vert _{\partial _{+}SM}=If\quad {\text{for all }}f\in C_{c}^{\infty }(SM\smallsetminus \partial _{0}SM)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44ffe1f49936900d9fb01534ab5cdfd3932f72d2)
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
- ^ Croke, Christopher B. "A sharp four dimensional isoperimetric inequality." Commentarii Mathematici Helvetici 59.1 (1984): 187–192.
- ^ 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.
- ^ Santaló, Luis Antonio. Measure of sets of geodesics in a Riemannian space and applications to integral formulas in elliptic and hyperbolic spaces. 1952
- ^ Santaló, Luis A. Integral geometry and geometric probability. Cambridge university press, 2004
- ^ 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.