Theory of valuations on manifolds, II

By Semyon Alesker. 2005

76 citations recorded by the source corpus.

Abstract This article is the second part in the series of articles where we are developing theory of valuations on manifolds. Roughly speaking valuations could be thought as finitely additive measures on a class of nice subsets of a manifold which satisfy some additional assumptions. The goal of this article is to introduce a notion of a smooth valuation on an arbitrary smooth manifold and establish some of the basic properties of it.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/63f14f94019901bf0fc0ee26a507e3a6e6c0c5fa

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Valuations on Sobolev spaces (2012, background): “Valuations were also investigated on star shaped sets [27, 28], on manifolds [6, 7, 8, 10, 11] and on Lebesgue spaces [37, 56, 57].”
Harmonic Analysis of Translation Invariant Valuations (2010, methodology): “representation theory, diﬀerential geometry and geometric measure theory. To be more speciﬁc, we use a representation of smooth translation invariant valuations via integral currents ﬁrst obtained in [8] and later reﬁned in [14] and [13] as well as an analysis of the action of SO(n) on the space of ”
Algebraic Integral Geometry (2010, background): “o valuations and not to curvature measures. (2) We have described in detail the theory of valuations on an aﬃne space. The theory of valuations on manifolds was recently worked out, mainly by Alesker [11, 12, 18, 14, 13, 24, 23, 16]. This gives the appropriate framework to study integral geometry of”
THEORY OF VALUATIONS ON MANIFOLDS, III. MULTIPLICATIVE STRUCTURE IN THE GENERAL CASE (2005, background): “5 of [5] any smooth valuation is defined uniquely by its values on sets from K(V )∩P(V ).”
Crofton Measures and Minkowski Valuations (2010, background): “However, we remark that Alesker has recently introduced a broader notion of valuation in the more general setting of smooth manifolds, see [6–8, 10, 13].”
The product on smooth and generalized valuations (2009, methodology): “This definition was extended in [3], Lemma 4.”
Hermitian integral geometry (2008, background): “The following implies that δξμ extends by continuity to a smooth (but not translation-invariant) valuation in the sense of [5, 6, 7, 8, 9] (although we will not make use of this fact).”
Kinematic formulas for tensor valuations (2014, background): “ce Val0(V) is spanned by the Euler characteristic χ (which satisﬁes χ(K) = 1 for all K∈ K(V)\{∅}), while Valn(V) is spanned by a Lebesgue measure voln on V. The Euler-Verdier involution σis deﬁned in [9] by (σφ)(K) := (−1)kφ(−K), φ∈ Valk(V),K∈ K(V). Let V ∼= Rn be an n-dimensional Euclidean vector s”
Integral geometry of complex space forms (2012, background): “7 of [7], dimVnλ = dimVn0 = dimAngU(n).”
Theory of valuations on manifolds, I. Linear spaces (2005, background): “The proof of this result can be found in [ 6 ].”
Theory of Valuations on Manifolds: A Survey (2006, background): “In [ A11 ] the notion of smooth valuation on a smooth manifold was introduced.”
Theory of Valuations on Manifolds, IV. New Properties of the Multiplicative Structure (2005, background): “A1,...,Ak with smooth boundaries, and any K∈ K(V) one has T (µ⊠hA1 ⊠···⊠hA k )(K) = ∂k ∂λ1 ...∂λk k 0 µ(K+ X i=1 λiAi) where λi ≥ 0 in the right hand side. Moreover the map T is an epimorphism. 10 In [5] one has introduced for any smooth manifold Xa class of ﬁnitely additive measures on the family o”