Santaló's formula

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← Previous revision Revision as of 18:14, 12 April 2026 Line 5: Line 5: == Formulation == == Formulation ==

Let <math>(M,\partial M,g)</math> be a compact, oriented Riemannian manifold with boundary. Suppose that every vector in the unit tangent bundle <math>SM</math> can be reached via the geodesic flow starting from points on <math>\partial M</math>. Then for a function <math> f: SM \rightarrow \mathbb{C} </math>, Santaló's formula takes the form Let <math>(M,\partial M,g)</math> be a compact, oriented Riemannian manifold with boundary. Suppose that every vector in the unit [[tangent bundle]] <math>SM</math> can be reached via the geodesic flow starting from points on <math>\partial M</math>. Then for a function <math> f: SM \rightarrow \mathbb{C} </math>, Santaló's formula takes the form :<math> \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> :<math> \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 where Line 20: Line 20: :<math> \Phi^*d \mu (x,v,t) = \langle \nu(x),x\rangle d \sigma(x,v) d t, </math> :<math> \Phi^*d \mu (x,v,t) = \langle \nu(x),x\rangle d \sigma(x,v) d t, </math> where <math> \Omega=\{(x,v,t): (x,v)\in \partial_+SM, t\in (0,\tau(x,v)) \}</math> and <math>\Phi:\Omega \rightarrow SM</math> is defined by <math>\Phi(x,v,t)=\varphi_t(x,v)</math>. In particular where <math> \Omega=\{(x,v,t): (x,v)\in \partial_+SM, t\in (0,\tau(x,v)) \}</math> and <math>\Phi:\Omega \rightarrow SM</math> is defined by <math>\Phi(x,v,t)=\varphi_t(x,v)</math>. In particular this implies that the ''geodesic X-ray transform'' <math> I f(x,v) = \int_0^{\tau(x,v)} f(\varphi_t(x,v)) \, dt </math> extends to a bounded linear map <math> I: L^1(SM, \mu) \rightarrow L^1(\partial_+ SM, \sigma_\nu)</math>, where <math> d\sigma_\nu(x,v) = \langle v, \nu(x) \rangle \, d \sigma(x,v) </math> and thus there is the following, <math>L^1</math>-version of Santaló's formula: this implies that the ''geodesic X-ray transform'' <math> I f(x,v) = \int_0^{\tau(x,v)} f(\varphi_t(x,v)) \, dt </math> extends to a bounded [[linear map]] <math> I: L^1(SM, \mu) \rightarrow L^1(\partial_+ SM, \sigma_\nu)</math>, where <math> d\sigma_\nu(x,v) = \langle v, \nu(x) \rangle \, d \sigma(x,v) </math> and thus there is the following, <math>L^1</math>-version of Santaló's formula: :<math> \int_{SM} f \, d \mu = \int_{\partial_+ SM} If ~ d \sigma_\nu \quad \text{for all } f \in L^1(SM,\mu). </math> :<math> \int_{SM} f \, d \mu = \int_{\partial_+ SM} If ~ d \sigma_\nu \quad \text{for all } f \in L^1(SM,\mu). </math>

Line 28: Line 28: The following proof is taken from [<ref>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. The following proof is taken from [<ref>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. </ref> 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>\partial_0SM=\{(x,v):\langle \nu(x), v\rangle =0 \}</math> has measure zero. </ref> 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>\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> X </math>: * An integration by parts formula for the geodesic [[vector field]] <math> X </math>: :<math> \int_{SM} Xu ~ d \mu = - \int_{\partial_+ SM} u ~ d \sigma_\nu \quad \text{for all } u \in C^\infty(SM) </math> :<math> \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>X u = - f</math>: * The construction of a resolvent for the transport equation <math>X u = - f</math>: