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. Suppose that every vector in the unit tangent bundle $SM$ can be reached via the geodesic flow starting from points on $\partial M$. Then for a function $f:SM\to ℂ$, Santaló's formula takes the form

${\int }_{SM}f(x,v)d\mu (x,v)={\int }_{{\partial }_{+}SM}[{\int }_0^{\tau (x,v)}f({\phi }_t(x,v))dt]⟨v,\nu (x)⟩d\sigma (x,v),$

where

• $({\phi }_t{)}_t$ is the geodesic flow and $\tau (x,v)=sup\{t\ge 0:\mathrm{\forall }s\in [0,t]:{\phi }_s(x,v)\in SM\}$ 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 ${\partial }_{+}SM:=\{(x,v)\in SM:x\in \partial M,⟨v,\nu (x)⟩\ge 0\}$ 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. $\tau (x,v)<\mathrm{\infty }$ for all $(x,v)\in SM$) and
2. $\partial M$ is strictly convex (i.e. the second fundamental form $I{I}_{\partial M}(x)$ is positive definite for every $x\in \partial M$),

Santaló's formula is valid for all $f\in C^{\mathrm{\infty }}(M)$. In this case it is equivalent to the following identity of measures:

${\mathrm{\Phi }}^{\ast }d\mu (x,v,t)=⟨\nu (x),x⟩d\sigma (x,v)dt,$

where $\mathrm{\Omega }=\{(x,v,t):(x,v)\in {\partial }_{+}SM,t\in (0,\tau (x,v))\}$ and $\mathrm{\Phi }:\mathrm{\Omega }\to SM$ is defined by $\mathrm{\Phi }(x,v,t)={\phi }_t(x,v)$. In particular this implies that the geodesic X-ray transform $If(x,v)={\int }_0^{\tau (x,v)}f({\phi }_t(x,v))dt$ extends to a bounded linear map $I:L^1(SM,\mu )\to L^1({\partial }_{+}SM,{\sigma }_{\nu })$, where $d{\sigma }_{\nu }(x,v)=⟨v,\nu (x)⟩d\sigma (x,v)$ and thus there is the following, $L^1$-version of Santaló's formula:

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

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

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

${\int }_{SM}Xud\mu =-{\int }_{{\partial }_{+}SM}ud{\sigma }_{\nu }\text{for all }u\in C^{\mathrm{\infty }}(SM)$

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

$\mathrm{\exists }R:C_c^{\mathrm{\infty }}(SM\setminus {\partial }_{0}SM)\to C^{\mathrm{\infty }}(SM):XRf=-f\text{ and }Rf{|}_{{\partial }_{+}SM}=If\text{for all }f\in C_c^{\mathrm{\infty }}(SM\setminus {\partial }_{0}SM)$

For the integration by parts formula, recall that $X$ leaves the Liouville-measure $\mu$ invariant and hence $Xu={\mathrm{div}}_G(uX)$, the divergence with respect to the Sasaki-metric $G$. The result thus follows from the divergence theorem and the observation that $⟨X(x,v),N(x,v){⟩}_G=⟨v,\nu (x){⟩}_g$, where $N$ is the inward-pointing unit-normal to $\partial SM$. The resolvent is explicitly given by $Rf(x,v)={\int }_0^{\tau (x,v)}f({\phi }_t(x,v))dt$ and the mapping property $C_c^{\mathrm{\infty }}(SM\setminus {\partial }_{0}SM)\to C^{\mathrm{\infty }}(SM)$ follows from the smoothness of $\tau :SM\setminus {\partial }_{0}SM\to [0,\mathrm{\infty })$, which is a consequence of the non-trapping and the convexity assumption.

References

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