Valuations on manifolds and integral geometry
Abstract
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.
Cite
@article{arxiv.0905.4046,
title = {Valuations on manifolds and integral geometry},
author = {Semyon Alesker},
journal= {arXiv preprint arXiv:0905.4046},
year = {2014}
}
Comments
77 pages. In several places the claim of continuity of some operators on distributions and valuations is replaced by the claim of sequential continuity. Few related modifications in appropriate places are made