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

[math]\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), }[/math]

where

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

Validity

Under the assumptions that

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

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

[math]\displaystyle{ \Phi^*d \mu (x,v,t) = \langle \nu(x),x\rangle d \sigma(x,v) d t, }[/math]

where [math]\displaystyle{ \Omega=\{(x,v,t): (x,v)\in \partial_+SM, t\in (0,\tau(x,v)) \} }[/math] and [math]\displaystyle{ \Phi:\Omega \rightarrow SM }[/math] is defined by [math]\displaystyle{ \Phi(x,v,t)=\varphi_t(x,v) }[/math]. In particular this implies that the geodesic X-ray transform [math]\displaystyle{ I f(x,v) = \int_0^{\tau(x,v)} f(\varphi_t(x,v)) \, dt }[/math] extends to a bounded linear map [math]\displaystyle{ I: L^1(SM, \mu) \rightarrow L^1(\partial_+ SM, \sigma_\nu) }[/math], where [math]\displaystyle{ d\sigma_\nu(x,v) = \langle v, \nu(x) \rangle \, d \sigma(x,v) }[/math] and thus there is the following, [math]\displaystyle{ L^1 }[/math]-version of Santaló's formula:

[math]\displaystyle{ \int_{SM} f \, d \mu = \int_{\partial_+ SM} If ~ d \sigma_\nu \quad \text{for all } f \in L^1(SM,\mu). }[/math]

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

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 [math]\displaystyle{ \partial_0SM=\{(x,v):\langle \nu(x), v\rangle =0 \} }[/math] has measure zero.

• An integration by parts formula for the geodesic vector field [math]\displaystyle{ X }[/math]:

[math]\displaystyle{ \int_{SM} Xu ~ d \mu = - \int_{\partial_+ SM} u ~ d \sigma_\nu \quad \text{for all } u \in C^\infty(SM) }[/math]

• The construction of a resolvent for the transport equation [math]\displaystyle{ X u = - f }[/math]:

[math]\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) }[/math]

For the integration by parts formula, recall that [math]\displaystyle{ X }[/math] leaves the Liouville-measure [math]\displaystyle{ \mu }[/math] invariant and hence [math]\displaystyle{ Xu = \operatorname{div}_G (uX) }[/math], the divergence with respect to the Sasaki-metric [math]\displaystyle{ G }[/math]. The result thus follows from the divergence theorem and the observation that [math]\displaystyle{ \langle X(x,v), N(x,v)\rangle_G = \langle v, \nu(x)\rangle_g }[/math], where [math]\displaystyle{ N }[/math] is the inward-pointing unit-normal to [math]\displaystyle{ \partial SM }[/math]. The resolvent is explicitly given by [math]\displaystyle{ Rf(x,v) = \int_0^{\tau(x,v)} f(\varphi_t(x,v)) \, dt }[/math] and the mapping property [math]\displaystyle{ C_c^\infty( SM\smallsetminus\partial_0 SM) \rightarrow C^\infty(SM) }[/math] follows from the smoothness of [math]\displaystyle{ \tau: SM\smallsetminus\partial_0 SM \rightarrow [0,\infty) }[/math], 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. 108. Cambridge University Press. ISBN 0-521-48578-9. https://books.google.com/books?id=Wg-gQcvS25sC&q=Santalo%27s+formula.