Valuations on manifolds and integral geometry

Details Valuations on manifolds and integral geometry Valuations on manifolds and integral geometry

2009-05-25

arxiv.org/abs/0905.4046v3

Geom. Funct. Anal. 20 (2010), no. 5, 1073--1143

Mathematics

Metric Geometry

76 pages; typos

Scientific paper

One constructs new operations of pull-back and push-forward on valuations on manifolds with respect to submersions and immersions. A general Radon type transform on valuations is introduced using these operations and the product on valuations. It is shown that the classical Radon transform on smooth functions, and the well known Radon transform on constructible functions with respect to the Euler characteristic are special cases of this new Radon transform. An inversion formula for the Radon transform on valuations has been proven in a specific case of real projective spaces. Relations of these operations to yet another classical type of integral geometry, Crofton and kinematic formulas, are indicated.

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