Crofton formulas in isotropic pseudo-Riemannian spaces

Abstract: The length of a curve in the plane can be computed by counting the intersection points with a line, and integrating over all lines. More generally, the intrinsic volumes (quermassintegrals) of a subset of Euclidean space can be computed by Crofton integrals, bringing forth their fundamental role in integral geometry. In spherical and hyperbolic geometry, such formulas are also known and classical. In pseudo-Riemannian isotropic spaces, such as de Sitter or anti-de Sitter space, one can similarly ask for an integral-geometric formula for the volume of a submanifold, or more generally for the intrinsic volumes of a subset, which were introduced only recently. I will explain how to obtain and apply such formulas, and how in fact there is a universal Crofton formula depending on a complex parameter extending the Riemannian Crofton formulas, for which all indefinite signatures are distributional boundary values. This is a joint work in progress with Andreas Bernig and Gil Solanes.

analysis of PDEsmetric geometryprobability

Audience: researchers in the topic

Online asymptotic geometric analysis seminar

Series comments: The link: technion.zoom.us/j/99202255210

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