THEORY OF VALUATIONS ON MANIFOLDS, III. MULTIPLICATIVE STRUCTURE IN THE GENERAL CASE

By Semyon Alesker, Joseph H. G. Fu. 2005

83 citations recorded by the source corpus.

This is the third part of a series of articles where the theory of valuations on manifolds is constructed. In the second part of this series the notion of a smooth valuation on a manifold was introduced. The goal of this article is to put a canonical multiplicative structure on the space of smooth valuations on general manifolds, thus extending some of the affine constructions from the first author's 2004 paper and, from the first part of this series.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/66c5b2b5c77f53d792b5b0124a2f00718f388a33

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In Applied Mathematics

1. A. Bernig
2. Joseph H. G. Fu
3. Semyon Alesker
4. G. Solanes
5. Franz E. Schuster
6. Thomas Wannerer
7. Monika Ludwig
8. Judit Abardia

In Geometry and Topology

1. A. Bernig
2. Joseph H. G. Fu
3. Semyon Alesker
4. G. Solanes
5. Franz E. Schuster
6. Monika Ludwig
7. Thomas Wannerer
8. Judit Abardia

Citations in the corpus

• The product on smooth and generalized valuations (2009, methodology): “special and non-uniqueway of presentation of smooth valuations; the general case in [7] uses a choice of affine coordinate charts. In both steps analytical difficulties have arose, and in the second step [7] they were resolved using some geometric measure theory. In this paper we give a new construction”
• Theory of valuations on manifolds, I. Linear spaces (2005, background): “The proof of this result can be found in [ 6 ].”
• A fourier-type transform on translation-invariant valuations on convex sets (2007, background): “Moreover some parts of the classical theory of valuations on affine spaces have been generalized to more general context of arbitrary manifolds [8]-[10], [12] (see also a survey [11] of these results).”
• Algebraic Integral Geometry (2010, background): “The theory of valuations on manifolds was recently worked out, mainly by Alesker [11, 12, 18, 14, 13, 24, 23, 16].”
• Minkowski valuations intertwining the special linear group (2012, unknown): “, [5–7,9,10,12–14,31,32,63]).”
• Integral geometry of complex space forms (2012, background): “One of the main features of V(M) is the existence of a canonical product structure, see [12] for its construction and [11] for an alternative approach.”