TY - JOUR

T1 - Crofton formulas in pseudo-Riemannian space forms

AU - Bernig, Andreas

AU - Faifman, Dmitry

AU - Solanes, Gil

N1 - Publisher Copyright:
© 2022 The Author(s).

PY - 2022/10/28

Y1 - 2022/10/28

N2 - 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.

AB - 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.

KW - Crofton formula

KW - Lipschitz-Killing curvature measures

KW - pseudo-Riemannian space form

KW - valuation

UR - http://www.scopus.com/inward/record.url?scp=85142322840&partnerID=8YFLogxK

U2 - 10.1112/S0010437X22007722

DO - 10.1112/S0010437X22007722

M3 - מאמר

AN - SCOPUS:85142322840

VL - 158

SP - 1935

EP - 1979

JO - Compositio Mathematica

JF - Compositio Mathematica

SN - 0010-437X

IS - 10

ER -