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^[1] and rigidity results.^[2] The formula is named after Luis Santaló, who first proved the result in 1952.^[3][4]

Formulation

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

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

where

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

Validity

Under the assumptions that

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

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

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

where ${\displaystyle \Omega =\{(x,v,t):(x,v)\in \partial _{+}SM,t\in (0,\tau (x,v))\}}$ and ${\displaystyle \Phi$ :\Omega \rightarrow SM} is defined by ${\displaystyle \Phi (x,v,t)=\varphi _{t}(x,v)}$. 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 ${\displaystyle I:L^{1}(SM,\mu )\rightarrow L^{1}(\partial _{+}SM,\sigma _{\nu })}$, where ${\displaystyle d\sigma _{\nu }(x,v)=\langle v,\nu (x)\rangle \,d\sigma (x,v)}$ and thus there is the following, ${\displaystyle L^{1}}$-version of Santaló's formula:

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

If the non-trapping or the convexity condition from above fail, then there is a set ${\displaystyle E\subset SM}$ of positive measure, such that the geodesics emerging from ${\displaystyle E}$ either fail to hit the boundary of ${\displaystyle M}$ or hit it non-transversely. In this case Santaló's formula only remains true for functions with support disjoint from this exceptional set ${\displaystyle E}$.

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

• An integration by parts formula for the geodesic vector field ${\displaystyle X}$:

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

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

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

For the integration by parts formula, recall that ${\displaystyle X}$ leaves the Liouville-measure ${\displaystyle \mu }$ invariant and hence ${\displaystyle Xu=\operatorname {div} _{G}(uX)}$, the divergence with respect to the Sasaki-metric ${\displaystyle G}$. The result thus follows from the divergence theorem and the observation that ${\displaystyle \langle X(x,v),N(x,v)\rangle _{G}=\langle v,\nu (x)\rangle _{g}}$, where ${\displaystyle N}$ is the inward-pointing unit-normal to ${\displaystyle \partial SM}$. 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 ${\displaystyle C_{c}^{\infty }(SM\smallsetminus \partial _{0}SM)\rightarrow C^{\infty }(SM)}$ follows from the smoothness of ${\displaystyle \tau :SM\smallsetminus \partial _{0}SM\rightarrow [0,\infty )}$, which is a consequence of the non-trapping and the convexity assumption.