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Crofton formulas in pseudo-Riemannian space forms

Part of: General convexity Global differential geometry Local differential geometry

Published online by Cambridge University Press:  28 October 2022

Andreas Bernig ,
Dmitry Faifman  and
Gil Solanes
Andreas Bernig
Affiliation:
Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, 60629 Frankfurt, Germany bernig@math.uni-frankfurt.de
Dmitry Faifman
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel faifmand@tauex.tau.ac.il
Gil Solanes
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain solanes@mat.uab.cat Centre de Recerca Matemàtica, Campus de Bellaterra, 08193 Bellaterra, Spain
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Abstract

Crofton formulas on simply connected Riemannian space forms allow the volumes, or more generally the Lipschitz–Killing curvature integrals of a submanifold with corners, to be computed 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.

Keywords

Crofton formulapseudo-Riemannian space formvaluationLipschitz–Killing curvature measures

MSC classification

Primary: 53C65: Integral geometry 53B30: Lorentz metrics, indefinite metrics 53C50: Lorentz manifolds, manifolds with indefinite metrics 52A22: Random convex sets and integral geometry 52A39: Mixed volumes and related topics
Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 10 , October 2022 , pp. 1935 - 1979
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Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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