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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[1] and rigidity results.[2] The formula is named after Luis Santaló, who first proved the result in 1952.[3][4]

Formulation

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Let be a compact, oriented Riemannian manifold with boundary. Then for a function , Santaló's formula takes the form

where

  • is the geodesic flow and is the exit time of the geodesic with initial conditions ,
  • and are the Riemannian volume forms with respect to the Sasaki metric on and respectively ( is also called Liouville measure),
  • is the inward-pointing unit normal to and the influx-boundary, which should be thought of as parametrization of the space of geodesics.

Validity

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Under the assumptions that

  1. is non-trapping (i.e. for all ) and
  2. is strictly convex (i.e. the second fundamental form is positive definite for every ),

Santaló's formula is valid for all . In this case it is equivalent to the following identity of measures:

where and is defined by . In particular this implies that the geodesic X-ray transform extends to a bounded linear map , where and thus there is the following, -version of Santaló's formula:

If the non-trapping or the convexity condition from above fail, then there is a set of positive measure, such that the geodesics emerging from either fail to hit the boundary of or hit it non-transversely. In this case Santaló's formula only remains true for functions with support disjoint from this exceptional set .

Proof

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The following proof is taken from [[5] 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 has measure zero.

  • An integration by parts formula for the geodesic vector field :
  • The construction of a resolvent for the transport equation :

For the integration by parts formula, recall that leaves the Liouville-measure invariant and hence , the divergence with respect to the Sasaki-metric . The result thus follows from the divergence theorem and the observation that , where is the inward-pointing unit-normal to . The resolvent is explicitly given by and the mapping property follows from the smoothness of , which is a consequence of the non-trapping and the convexity assumption.

References

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  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.
  • Isaac Chavel (1995). "5.2 Santalo's formula". Riemannian Geometry: A Modern Introduction. Cambridge Tracts in Mathematics. Vol. 108. Cambridge University Press. ISBN 0-521-48578-9.