Uniqueness of curvature measures in pseudo-Riemannian geometry

Bernig, Andreas; Faifman, Dmitry; Solanes, Gil

doi:10.48550/arxiv.2009.02230

2020

DOI: 10.48550/arxiv.2009.02230

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Uniqueness of curvature measures in pseudo-Riemannian geometry

Andreas Bernig,

Dmitry Faifman,

Gil Solanes

Abstract: The recently introduced Lipschitz-Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz-Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.

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“…Recently there were constructed analogues of intrinsic volumes in pseudo-Riemannian geometry and obtained non-trivial results for them [10], [11], [12].…”

mentioning

confidence: 99%

Some Conjectures on Intrinsic Volumes of Riemannian Manifolds and Alexandrov Spaces

Alesker

2018

Arnold Math J.

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For any closed smooth Riemannian manifold H. Weyl [21] has defined a sequence of numbers called today intrinsic volumes. They include volume, Euler characteristic, and integral of the scalar curvature. We conjecture that absolute values of all intrinsic volumes are bounded by a constant depending only on the dimension of the manifold, upper bound on its diameter, and lower bound on the sectional curvature. Furthermore we conjecture that intrinsic volumes can be defined for some (so called weakly smoothable) Alexandrov spaces with curvature bounded below and state few of the expected properties of them, particularly the behavior under the Gromov-Hausdorff limits. We suggest conjectural compactifications of the space of smooth closed connected Riemannian manifolds with given upper bounds on dimension and diameter and a lower bound on sectional curvature to which the intrinsic volumes extend by continuity. We discuss also known cases of some of these conjectures. The work is a joint project with A. Petrunin.

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“…Recently there were constructed analogues of intrinsic volumes in pseudo-Riemannian geometry and obtained non-trivial results for them [10], [11], [12].…”

mentioning

confidence: 99%

Some Conjectures on Intrinsic Volumes of Riemannian Manifolds and Alexandrov Spaces

Alesker

2018

Arnold Math J.

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show abstract

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Uniqueness of curvature measures in pseudo-Riemannian geometry

Cited by 1 publication

References 26 publications

Some Conjectures on Intrinsic Volumes of Riemannian Manifolds and Alexandrov Spaces

Some Conjectures on Intrinsic Volumes of Riemannian Manifolds and Alexandrov Spaces

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