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Curvature measures of pseudo-Riemannian manifolds

  • Andreas Bernig , Dmitry Faifman EMAIL logo und Gil Solanes
Veröffentlicht/Copyright: 25. Mai 2022

Abstract

The Weyl principle is extended from the Riemannian to the pseudo-Riemannian setting, and subsequently to manifolds equipped with generic symmetric (0,2)-tensors. More precisely, we construct a family of generalized curvature measures attached to such manifolds, extending the Riemannian Lipschitz–Killing curvature measures introduced by Federer. We then show that they behave naturally under isometric immersions, in particular they do not depend on the ambient signature. Consequently, we extend Theorema Egregium to surfaces equipped with a generic metric of changing signature, and more generally, establish the existence as distributions of intrinsically defined Lipschitz–Killing curvatures for such manifolds of arbitrary dimension. This includes in particular the scalar curvature and the Chern–Gauss–Bonnet integrand. Finally, we deduce a Chern–Gauss–Bonnet theorem for pseudo-Riemannian manifolds with generic boundary.

Award Identifier / Grant number: RGPIN-2016-06764

Award Identifier / Grant number: BE 2484/5-2

Award Identifier / Grant number: 1750/20

Award Identifier / Grant number: MTM2015-66165-P

Award Identifier / Grant number: IEDI-2015-00634

Award Identifier / Grant number: PGC2018-095998-B-I00

Funding statement: Andreas Bernig was supported by DFG grant BE 2484/5-2. Dmitry Faifman was partially supported by an NSERC Discovery Grant and ISF Grant 1750/20. Gil Solanes was supported by the Serra Húnter Programme and FEDER/MICINN grants MTM2015-66165-P, IEDI-2015-00634 and PGC2018-095998-B-I00.

Acknowledgements

We would like to thank Bo’az Klartag for his insightful input on distributional integrals, and the referee for numerous valuable comments. This work was partially done during Dmitry Faifman’s term at the University of Toronto as Coxeter Assistant Professor, as well as a CRM-ISM postdoctoral fellow in Montreal, which we gratefully acknowledge.

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Received: 2021-03-30
Revised: 2022-03-10
Published Online: 2022-05-25
Published in Print: 2022-07-01

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