New Structures on Valuations and Applications

Alesker, Semyon

doi:10.1007/978-3-0348-0874-3_1

Cited by 8 publications

(10 citation statements)

References 74 publications

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“…These are the principal examples of the concept of convex valuation. Alesker's recent work [1][2][3][4][5][6][7] introduces for any smooth manifold M the space V(M ) of smooth valuations on M , equipped with a natural commutative product. With respect to this product, the intrinsic volumes then appear (up to scale) as powers of µ 1 .…”

Section: Introductionmentioning

confidence: 99%

Riemannian curvature measures

Wannerer

2019

Geom. Funct. Anal.

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A famous theorem of Weyl states that if M is a compact submanifold of euclidean space, then the volumes of small tubes about M are given by a polynomial in the radius r, with coefficients that are expressible as integrals of certain scalar invariants of the curvature tensor of M with respect to the induced metric. It is natural to interpret this phenomenon in terms of curvature measures and smooth valuations, in the sense of Alesker, canonically associated to the Riemannian structure of M . This perspective yields a fundamental new structure in Riemannian geometry, in the form of a certain abstract module over the polynomial algebra R[t] that reflects the behavior of Alesker multiplication. This module encodes a key piece of the array of kinematic formulas of any Riemannian manifold on which a group of isometries acts transitively on the sphere bundle. We illustrate this principle in precise terms in the case where M is a complex space form.

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Section: Introductionmentioning

confidence: 99%

Riemannian curvature measures

Wannerer

2019

Geom. Funct. Anal.

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“…[6,11]). There exists a canonical filtration of V ∞ (M ) by closed subspacesV ∞ (M ) = W 0 ⊃ W 1 ⊃ · · · ⊃ W n , such that the associated graded space gr W V ∞ (M ) := n k=0 W k /W k+1 is canonically isomorphic to the space C ∞ (M, Val ∞ (T M )) of smooth sections of the infinite-dimensional vector bundle Val ∞ (T M ) −→ M .…”

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confidence: 99%

Spin-invariant valuations on the octonionic plane

Bernig

Voide

2016

Isr. J. Math.

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“…We will only consider the case of n − k ≡ 0 mod 4, as the other case can be treated similarly. We will use the identification given by equation (13). Use Q to identify V ∼ = V * .…”

Section: Global Extensionsmentioning

confidence: 99%

Valuation theory of indefinite orthogonal groups

Bernig

Faifman

2017

Journal of Functional Analysis

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Let $\mathrm{SO}^+(p,q)$ denote the identity connected component of the real orthogonal group with signature $(p,q)$. We give a complete description of the spaces of continuous and generalized translation- and $\mathrm{SO}^+(p,q)$-invariant valuations, generalizing Hadwiger's classification of Euclidean isometry-invariant valuations. As a result of independent interest, we identify within the space of translation-invariant valuations the class of Klain-Schneider continuous valuations, which strictly contains all continuous translation-invariant valuations. The operations of pull-back and push-forward by a linear map extend naturally to this class.Comment: 65 pages; Some details in proofs added and minor mistakes corrected, an appendix on wave front sets of G-invariant distributions and a section on the linear algebra of O(p,q) added. Accepted for publication in Journal of Functional Analysi

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New Structures on Valuations and Applications

Cited by 8 publications

References 74 publications

Riemannian curvature measures

Riemannian curvature measures

Spin-invariant valuations on the octonionic plane

Valuation theory of indefinite orthogonal groups

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