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January 2021 Integral geometry of exceptional spheres
Gil Solanes, Thomas Wannerer
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J. Differential Geom. 117(1): 137-191 (January 2021). DOI: 10.4310/jdg/1609902019

Abstract

The algebras of valuations on $S^6$ and $S^7$ invariant under the actions of $G_2$ and $\operatorname{Spin}(7)$ are shown to be isomorphic to the algebra of translation-invariant valuations on the tangent space at a point invariant under the action of the isotropy group. This is in analogy with the cases of real and complex space forms, suggesting the possibility that the same phenomenon holds in all Riemannian isotropic spaces. Based on the description of the algebras the full array of kinematic formulas for invariant valuations and curvature measures in $S^6$ and $S^7$ is computed. A key technical point is an extension of the classical theorems of Klain and Schneider on simple valuations.

Funding Statement

Gil Solanes is a Serra Húnter fellow and was supported by FEDER/MINECO grants IEDI-2015-00634 and MTM2015-66165-P.
Thomas Wannerer was supported by DFG grant WA 3510/1-1.

Citation

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Gil Solanes. Thomas Wannerer. "Integral geometry of exceptional spheres." J. Differential Geom. 117 (1) 137 - 191, January 2021. https://doi.org/10.4310/jdg/1609902019

Information

Received: 18 August 2017; Published: January 2021
First available in Project Euclid: 6 January 2021

MathSciNet: MR4195754
Digital Object Identifier: 10.4310/jdg/1609902019

Rights: Copyright © 2021 Lehigh University

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Vol.117 • No. 1 • January 2021
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