Curvature Measures of Pseudo-Riemannian Manifolds

Bernig, Andreas; Faifman, Dmitry; Solanes, Gil

doi:10.48550/arxiv.1910.09635

Cited by 3 publications

(13 citation statements)

References 47 publications

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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].…”

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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].…”

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confidence: 99%

Some Conjectures on Intrinsic Volumes of Riemannian Manifolds and Alexandrov Spaces

Alesker

2018

Arnold Math J.

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“…Intrinsic volumes on pseudo-Riemannian manifolds. In [14] we constructed a sequence of complex-valued generalized valuations µ M 0 , . .…”

Section: 4mentioning

confidence: 99%

“…We say that (X, g) is LC-regular if 0 is a regular value of g ∈ C ∞ (T X \ 0). It was shown in [14,Proposition 4.9] that the extrinsic notion of LC-transversality and the intrinsic notion of LC-regularity coincide: a submanifold of a pseudo-Riemannian manifold, equipped with the field of quadratic forms induced from the metric, is LC-regular if and only if it is LC-transversal.…”

Section: 4mentioning

confidence: 99%

“…Note that the spherical Crofton formula (2) is a special case of our theorem. We also note that we will prove a slightly more general statement in Corollary 5.11 using the notion of LC-regularity from [14].…”

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confidence: 94%

“…The right hand side of the Crofton formula will be expressed in terms of the recently introduced intrinsic volumes on pseudo-Riemannian manifolds [14], which are complex-valued generalized valuations on M . They satisfy a Hadwiger-type classification [15], which allows us to use the template method to compute the coefficients in this formula.…”

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confidence: 99%

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Crofton formulas in pseudo-Riemannian space forms

Bernig¹,

Faifman²,

Solanes³

2021

Preprint

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Crofton formulas on simply-connected Riemannian space forms allow to compute the volumes, or more generally the Lipschitz-Killing curvature integrals of a submanifold with corners, by integrating the Euler characteristic of its intersection with all geodesic submanifolds. We develop a framework of Crofton formulas with distributions replacing measures, which has in its core Alesker's Radon transform on valuations. We then apply this framework, and our recent Hadwiger-type classification, to compute explicit Crofton formulas for all isometry-invariant valuations on all pseudospheres, pseudo-Euclidean and pseudohyperbolic spaces. We find that, in essence, a single measure which depends analytically on the metric, gives rise to all those Crofton formulas through its distributional boundary values at parts of the boundary corresponding to the different indefinite signatures. In particular, the Crofton formulas we obtain are formally independent of signature.

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Uniqueness of curvature measures in pseudo-Riemannian geometry

Bernig,

Faifman,

Solanes

2020

Preprint

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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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Curvature Measures of Pseudo-Riemannian Manifolds

Cited by 3 publications

References 47 publications

Some Conjectures on Intrinsic Volumes of Riemannian Manifolds and Alexandrov Spaces

Some Conjectures on Intrinsic Volumes of Riemannian Manifolds and Alexandrov Spaces

Crofton formulas in pseudo-Riemannian space forms

Uniqueness of curvature measures in pseudo-Riemannian geometry

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