Santaló's formula

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 $(M,\partial M,g)$ be a compact, oriented Riemannian manifold with boundary. Then for a function $f:SM\rightarrow \mathbb {C}$, Santaló's formula takes the form

$\int _{SM}f(x,v)\,d\mu (x,v)=\int _{\partial _{+}SM}\left[\int _{0}^{\tau (x,v)}f(\varphi _{t}(x,v))\,dt\right]\langle v,\nu (x)\rangle \,d\sigma (x,v),$

where

• $(\varphi _{t})_{t}$ is the geodesic flow and $\tau (x,v)=\sup\{t\geq 0:\forall s\in [0,t]:~\varphi _{s}(x,v)\in SM\}$ 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 $\partial SM$ respectively ($\mu$ is also called Liouville measure),
• $\nu$ is the inward-pointing unit normal to $\partial M$ and $\partial _{+}SM:=\{(x,v)\in SM:x\in \partial M,\langle v,\nu (x)\rangle \geq 0\}$ the influx-boundary, which should be thought of as parametrization of the space of geodesics.

Validity

Under the assumptions that

1. $M$ is non-trapping (i.e. $\tau (x,v)<\infty$ for all $(x,v)\in SM$) and
2. $\partial M$ is strictly convex (i.e. the second fundamental form $II_{\partial M}(x)$ 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:

$\Phi ^{*}d\mu (x,v,t)=\langle \nu (x),x\rangle d\sigma (x,v)dt,$

where $\Omega =\{(x,v,t):(x,v)\in \partial _{+}SM,t\in (0,\tau (x,v))\}$ and $\Phi :\Omega \rightarrow SM$ is defined by $\Phi (x,v,t)=\varphi _{t}(x,v)$. In particular this implies that the geodesic X-ray transform $If(x,v)=\int _{0}^{\tau (x,v)}f(\varphi _{t}(x,v))\,dt$ extends to a bounded linear map $I:L^{1}(SM,\mu )\rightarrow L^{1}(\partial _{+}SM,\sigma _{\nu })$, 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:

$\int _{SM}f\,d\mu =\int _{\partial _{+}SM}If~d\sigma _{\nu }\quad {\text{for all }}f\in L^{1}(SM,\mu ).$

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 [ 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 $\partial _{0}SM=\{(x,v):\langle \nu (x),v\rangle =0\}$ has measure zero.

• An integration by parts formula for the geodesic vector field $X$:

$\int _{SM}Xu~d\mu =-\int _{\partial _{+}SM}u~d\sigma _{\nu }\quad {\text{for all }}u\in C^{\infty }(SM)$

• The construction of a resolvent for the transport equation $Xu=-f$:

$\exists R:C_{c}^{\infty }(SM\smallsetminus \partial _{0}SM)\rightarrow C^{\infty }(SM):XRf=-f{\text{ and }}Rf\vert _{\partial _{+}SM}=If\quad {\text{for all }}f\in C_{c}^{\infty }(SM\smallsetminus \partial _{0}SM)$

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 $Rf(x,v)=\int _{0}^{\tau (x,v)}f(\varphi _{t}(x,v))\,dt$ and the mapping property $C_{c}^{\infty }(SM\smallsetminus \partial _{0}SM)\rightarrow C^{\infty }(SM)$ follows from the smoothness of $\tau :SM\smallsetminus \partial _{0}SM\rightarrow [0,\infty )$, which is a consequence of the non-trapping and the convexity assumption.

References

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