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. Suppose that every vector in the unit tangent bundle S M {\displaystyle SM} can be reached via the geodesic flow starting from points on ∂ M {\displaystyle \partial M} . 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[\int _{0}^{\tau (x,v)}f(\varphi _{t}(x,v))\,dt\right]\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 [0,t]:~\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
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
∂ 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,}
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