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Santaló's formula

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

∫ S M f ( x , v ) d μ ( x , v ) = ∫ ∂ + S M [ ∫ 0 τ ( x , v ) f ( φ t ( x , v ) ) d t ] ⟨ v , ν ( x ) ⟩ d σ ( x , v ) , {\displaystyle \int _{SM}f(x,v)\,d\mu (x,v)=\int _{\partial _{+}SM}\left\langle v,\nu (x)\rangle \,d\sigma (x,v),}

where

  • ( φ t ) t {\displaystyle (\varphi _{t})_{t}} is the geodesic flow and τ ( x , v ) = sup { t ≥ 0 : ∀ s ∈ [ 0 , t ] :   φ s ( x , v ) ∈ S M } {\displaystyle \tau (x,v)=\sup\{t\geq 0:\forall s\in :~\varphi _{s}(x,v)\in SM\}} is the exit time of the geodesic with initial conditions ( x , v ) ∈ S M {\displaystyle (x,v)\in SM} ,
  • μ {\displaystyle \mu } and σ {\displaystyle \sigma } are the Riemannian volume forms with respect to the Sasaki metric on S M {\displaystyle SM} and ∂ S M {\displaystyle \partial SM} respectively ( μ {\displaystyle \mu } is also called Liouville measure),
  • ν {\displaystyle \nu } is the inward-pointing unit normal to ∂ M {\displaystyle \partial M} and ∂ + S M := { ( x , v ) ∈ S M : x ∈ ∂ M , ⟨ v , ν ( x ) ⟩ ≥ 0 } {\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. M {\displaystyle M} is non-trapping (i.e. τ ( x , v ) < ∞ {\displaystyle \tau (x,v)<\infty } for all ( x , v ) ∈ S M {\displaystyle (x,v)\in SM} ) and
  2. ∂ M {\displaystyle \partial M} is strictly convex (i.e. the second fundamental form I I ∂ M ( x ) {\displaystyle II_{\partial M}(x)} is positive definite for every x ∈ ∂ M {\displaystyle x\in \partial M} ),

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

Φ ∗ d μ ( x , v , t ) = ⟨ ν ( x ) , x ⟩ d σ ( x , v ) d t , {\displaystyle \Phi ^{*}d\mu (x,v,t)=\langle \nu (x),x\rangle d\sigma (x,v)dt,}

where Ω = { ( x , v , t ) : ( x , v ) ∈ ∂ + S M , t ∈ ( 0 , τ ( x , v ) ) } {\displaystyle \Omega =\{(x,v,t):(x,v)\in \partial _{+}SM,t\in (0,\tau (x,v))\}} and Φ : Ω → S M {\displaystyle \Phi :\Omega \rightarrow SM} is defined by Φ ( x , v , t ) = φ t ( x , v ) {\displaystyle \Phi (x,v,t)=\varphi _{t}(x,v)} . In particular this implies that the geodesic X-ray transform I f ( x , v ) = ∫ 0 τ ( x , v ) f ( φ t ( x , v ) ) d t {\displaystyle If(x,v)=\int _{0}^{\tau (x,v)}f(\varphi _{t}(x,v))\,dt} extends to a bounded linear map I : L 1 ( S M , μ ) → L 1 ( ∂ + S M , σ ν ) {\displaystyle I:L^{1}(SM,\mu )\rightarrow L^{1}(\partial _{+}SM,\sigma _{\nu })} , where d σ ν ( x , v ) = ⟨ v , ν ( x ) ⟩ d σ ( x , v ) {\displaystyle d\sigma _{\nu }(x,v)=\langle v,\nu (x)\rangle \,d\sigma (x,v)} and thus there is the following, L 1 {\displaystyle L^{1}} -version of Santaló's formula:

∫ S M f d μ = ∫ ∂ + S M I f   d σ ν for all  f ∈ L 1 ( S M , μ ) . {\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 E ⊂ S M {\displaystyle E\subset SM} of positive measure, such that the geodesics emerging from E {\displaystyle E} either fail to hit the boundary of M {\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 E {\displaystyle E} .

Proof

The following proof is taken from , 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 ∂ 0 S M = { ( x , v ) : ⟨ ν ( x ) , v ⟩ = 0 } {\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 X {\displaystyle X} :
∫ S M X u   d μ = − ∫ ∂ + S M u   d σ ν for all  u ∈ C ∞ ( S M ) {\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 X u = − f {\displaystyle Xu=-f} :
∃ R : C c ∞ ( S M ∖ ∂ 0 S M ) → C ∞ ( S M ) : X R f = − f  and  R f | ∂ + S M = I f for all  f ∈ C c ∞ ( S M ∖ ∂ 0 S M ) {\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 X {\displaystyle X} leaves the Liouville-measure μ {\displaystyle \mu } invariant and hence X u = div G ⁡ ( u X ) {\displaystyle Xu=\operatorname {div} _{G}(uX)} , the divergence with respect to the Sasaki-metric G {\displaystyle G} . The result thus follows from the divergence theorem and the observation that ⟨ X ( x , v ) , N ( x , v ) ⟩ G = ⟨ v , ν ( x ) ⟩ g {\displaystyle \langle X(x,v),N(x,v)\rangle _{G}=\langle v,\nu (x)\rangle _{g}} , where N {\displaystyle N} is the inward-pointing unit-normal to ∂ S M {\displaystyle \partial SM} . The resolvent is explicitly given by R f ( x , v ) = ∫ 0 τ ( x , v ) f ( φ t ( x , v ) ) d t {\displaystyle Rf(x,v)=\int _{0}^{\tau (x,v)}f(\varphi _{t}(x,v))\,dt} and the mapping property C c ∞ ( S M ∖ ∂ 0 S M ) → C ∞ ( S M ) {\displaystyle C_{c}^{\infty }(SM\smallsetminus \partial _{0}SM)\rightarrow C^{\infty }(SM)} follows from the smoothness of τ : S M ∖ ∂ 0 S M → [ 0 , ∞ ) {\displaystyle \tau :SM\smallsetminus \partial _{0}SM\rightarrow [0,\infty )} , which is a consequence of the non-trapping and the convexity assumption.

References

  1. Croke, Christopher B. "A sharp four dimensional isoperimetric inequality." Commentarii Mathematici Helvetici 59.1 (1984): 187–192.
  2. Ilmavirta, Joonas, and François Monard. "4 Integral geometry on manifolds with boundary and applications." The Radon Transform: The First 100 Years and Beyond 22 (2019): 43.
  3. Santaló, Luis Antonio. Measure of sets of geodesics in a Riemannian space and applications to integral formulas in elliptic and hyperbolic spaces. 1952
  4. Santaló, Luis A. Integral geometry and geometric probability. Cambridge university press, 2004
  5. Guillarmou, Colin, Marco Mazzucchelli, and Leo Tzou. "Boundary and lens rigidity for non-convex manifolds." American Journal of Mathematics 143 (2021), no. 2, 533-575.
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