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

Validity
Under the assumptions that Santaló's formula is valid for all $$f\in C^\infty(M)$$. In this case it is equivalent to the following identity of measures:
 * 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 $$ x \in \partial M $$),
 * $$ \Phi^*d \mu (x,v,t) = \langle \nu(x),x\rangle d \sigma(x,v) d t, $$

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 $$ I f(x,v) = \int_0^{\tau(x,v)} f(\varphi_t(x,v)) \, dt  $$  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 $$\partial_0SM=\{(x,v):\langle \nu(x), v\rangle =0 \}$$ 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 $$X u = - f$$:
 * $$ \exists R: C_c^\infty( SM\smallsetminus\partial_0 SM) \rightarrow C^\infty(SM): XRf = - f \text{ and } Rf\vert_{\partial_+ SM} = If \quad \text{for all } f\in C_c^\infty( SM\smallsetminus\partial_0 SM)  $$

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.