Santaló formula

A formula describing the Liouville measure on the unit tangent bundle of a Riemannian manifold in terms of the geodesic flow and the measure of a codimension-one submanifold (see [a5] and [a6], Chap. 19).

Let be an -dimensional Riemannian manifold, let be the unit tangent bundle of , let be the Liouville measure on , and let be the geodesic flow. One way to define is to start with the standard contact form (cf. Contact structure) and define . Liouville's theorem says that is invariant under the geodesic flow (since is). Locally, is just the product measure where is the Riemannian volume form and is the standard volume form on the unit -sphere.

For any (locally defined) codimension-one submanifold , let be the Riemannian volume element of the submanifold. Let , and, for each , let be a unit normal to at . Then there is a smooth mapping , given by . Santaló's formula says:

The formula is used to convert integrals over subsets of the unit tangent bundle to iterated integrals, first over a fixed unit-speed geodesic (say parametrized on ) and then over the space of geodesics which are parametrized by their intersections with a fixed codimension-one submanifold and endowed with the measure , i.e.

One of the most important applications is to the study of Riemannian manifolds with smooth boundary. In this case , is the inwardly pointing unit normal vector and . For any , set . Note that means that is defined for all . Let , i.e. means for some some and some . In this setting, Santaló's formula takes the form:

One immediate application, by simply putting , is:

Since the Liouville measure is locally a product measure, in the special case this says .

The formula is often used to prove isoperimetric and rigidity results. A sample of such applications can be found in the references. See [a1] for Santaló's formula for time-like geodesic flow on Lorentzian surfaces.

References

[a1]	L. Andersson, M. Dahl, R. Howard, "Boundary and lens rigidity of Lorentzian surfaces" Trans. Amer. Math. Soc. , 348 : 6 (1996) pp. 2307–2329
[a2]	C. Croke, "A sharp four-dimensional isoperimetric inequality" Comment. Math. Helv. , 59 : 2 (1984) pp. 187–192
[a3]	C. Croke, "Some isoperimetric inequalities and eigenvalue estimates" Ann. Sci. École Norm. Sup. , 13 : 4 (1980) pp. 419–435
[a4]	C. Croke, N. Dairbekov, V. Sharafutdinov, "Local boundary rigidity of a compact Riemannian manifold with curvature bounded above" Trans. Amer. Math. Soc. (to appear)
[a5]	L.A. Santaló, "Measure of sets of geodesics in a Riemannian space and applications to integral formulas in elliptic and hyperbolic spaces" Summa Brasil. Math. , 3 (1952) pp. 1–11
[a6]	L.A. Santaló, "Integral geometry and geometric probability (With a foreword by Mark Kac)" , Encyclopedia Math. Appl. , 1 , Addison-Wesley (1976)