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Santalo Formula

This post has been migrated from my old blog, the math-physics learning seminar.

Let MM be a simple Riemannian manifold with boundary M\partial M. For (x,v)SM(x,v) \in SM, let τ(x,v)\tau(x,v) denote the exit time of the geodesic starting at xx with tangent vector vv, i.e. τ(x,v)\tau(x,v) is the (necessarily unique, and finite) time at which expx(tv)M\exp_x(tv) \in \partial M.

We let +(SM)\partial_+(SM) denote the set

+(SM)=(x,v)SM  xM,v,ν>0 \partial_+(SM) = { (x,v) \in SM \ | \ x \in \partial M, \langle v, \nu\rangle > 0 }

where ν\nu denotes the inward unit normal to M\partial M in MM. The exponential map identifies SMSM with the set

Ω=(x,v,t)+(SM)×R  0tτ(x,v), \Omega = { (x,v,t) \in \partial_+(SM) \times \mathbf{R} \ | \ 0 \leq t \leq \tau(x,v) },

via (x,v,t)expx(tv)(x,v,t) \mapsto \exp_x(tv). Let Φ:ΩM\Phi: \Omega \to M denote this diffeomorphism. Then we have, for all fC(SM)f \in C^\infty(SM)

SMfdvol(SM)=Ω(Φf)(Φdvol(SM))=+(SM)0τ(x,v)f(ϕt(x,v))Φdvol(SM). \begin{aligned} \int_{SM} f dvol(SM) &= \int_\Omega (\Phi^\ast f) (\Phi^\ast dvol(SM)) \cr &= \int_{\partial_+(SM)} \int_0^{\tau(x,v)} f(\phi_t(x,v)) \Phi^\ast dvol(SM). \end{aligned}

Therefore, we can compute integrals of functions over SMSM by integrating along geodesics, provided that we can cmopute Φdvol(SM)\Phi^\ast dvol(SM). This is the content of the Santalo formula.

Theorem (Santalo formula). For all fC(SM)f \in C^\infty(SM), we have

SMfdvol(SM)=+(SM)0τ(x,v)f(ϕt(x,v))v,νdtdvol((SM)) \int_{SM} f dvol(SM) = \int_{\partial_+(SM)} \int_0^{\tau(x,v)} f(\phi_t(x,v)) \langle v, \nu\rangle dt dvol(\partial(SM))

Proof. Necessarily, we must have

Φ(dvol(SM))=a(x,v)dtdvol((SM))), \Phi^\ast(dvol(SM)) = a(x,v) dt \wedge dvol(\partial(SM))),

for some function a(x,v)a(x,v). The reason we can assume that aa is independent of tt is that Φ\Phi is defined via geodesic flow, and geodesic flow preserves the volume form on SMSM. To compute the factor a(x,v)a(x,v), we just need to compute

i/tΦ(dvol(SM))=Φ(iΦ(/t)dvol(SM)) i_{\partial / \partial t} \Phi^\ast(dvol(SM)) = \Phi^\ast(i_{\Phi_\ast(\partial / \partial t)} dvol(SM))

From the definition of Φ\Phi, we have that Φ(/t)\Phi_\ast(\partial / \partial t) is the Reeb vector field on SMSM, i.e. the vector field generating geodesic flow. Therefore, Φ(/t)\Phi_\ast(\partial / \partial t) is equal, at a point (x,v)(x,v) to the horizontal lift of the vector vv. Therefore, using the definition of the induced volume form on a hypersurface of a Riemannian manifold, we find

i/tΦ(dvol(SM))=v,νdvol((SM)) i_{\partial / \partial t} \Phi^\ast(dvol(SM)) = \langle v, \nu \rangle dvol(\partial(SM))

where ν\nu is the inward pointing unit normal to (SM)\partial(SM) in SMSM. This shows that a(x,v)=v,νa(x,v) = \langle v, \nu \rangle and completes the proof.