Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-02-08T23:19:36.750Z Has data issue: false hasContentIssue false
  • English
  • Français

Crofton formulas in pseudo-Riemannian space forms

Part of: General convexity Local differential geometry Global 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 [email protected]
Dmitry Faifman
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel [email protected]
Gil Solanes
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain [email protected] Centre de Recerca Matemàtica, Campus de Bellaterra, 08193 Bellaterra, Spain
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Akhiezer, D. and Kazarnovskii, B., Average number of zeros and mixed symplectic volume of Finsler sets, Geom. Funct. Anal. 28 (2018), 1517–1547.CrossRefGoogle Scholar
Alesker, S., Description of translation invariant valuations on convex sets with solution of P. McMullen's conjecture, Geom. Funct. Anal. 11 (2001), 244–272.CrossRefGoogle Scholar
Alesker, S., Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations, J. Differential Geom. 63 (2003), 63–95.CrossRefGoogle Scholar
Alesker, S., Theory of valuations on manifolds. I. Linear spaces, Israel J. Math. 156 (2006), 311–339.CrossRefGoogle Scholar
Alesker, S., Theory of valuations on manifolds. II, Adv. Math. 207 (2006), 420–454.CrossRefGoogle Scholar
Alesker, S., Theory of valuations on manifolds. IV. New properties of the multiplicative structure, in Geometric aspects of functional analysis, Lecture Notes in Mathematics, vol. 1910 (Springer, Berlin, 2007), 1–44.CrossRefGoogle Scholar
Alesker, S., Valuations on manifolds and integral geometry, Geom. Funct. Anal. 20 (2010), 1073–1143.CrossRefGoogle Scholar
Alesker, S. and Bernig, A., The product on smooth and generalized valuations, Amer. J. Math. 134 (2012), 507–560.CrossRefGoogle Scholar
Alesker, S. and Bernstein, J., Range characterization of the cosine transform on higher Grassmannians, Adv. Math. 184 (2004), 367–379.CrossRefGoogle Scholar
Álvarez Paiva, J. C. and Fernandes, E., Gelfand transforms and Crofton formulas, Selecta Math. (N.S.) 13 (2007), 369–390.CrossRefGoogle Scholar
Bernig, A., Valuations with Crofton formula and Finsler geometry, Adv. Math. 210 (2007), 733–753.CrossRefGoogle Scholar
Bernig, A. and Bröcker, L., Valuations on manifolds and Rumin cohomology, J. Differ. Geom. 75 (2007), 433–457.CrossRefGoogle Scholar
Bernig, A. and Faifman, D., Valuation theory of indefinite orthogonal groups, J. Funct. Anal. 273 (2017), 2167–2247.CrossRefGoogle Scholar
Bernig, A., Faifman, D. and Solanes, G., Uniqueness of curvature measures in pseudo-Riemannian geometry, J. Geom. Anal. 31 (2021), 11819–11848.CrossRefGoogle Scholar
Bernig, A., Faifman, D. and Solanes, G., Curvature measures of pseudo-Riemannian manifolds, J. Reine Angew. Math. 788 (2022), 77–127.CrossRefGoogle Scholar
Bernig, A., Fu, J. H. G. and Solanes, G., Integral geometry of complex space forms, Geom. Funct. Anal. 24 (2014), 403–492.CrossRefGoogle Scholar
Birman, G. S., Crofton's and Poincaré's formulas in the Lorentzian plane, Geom. Dedicata 15 (1984), 399–411.CrossRefGoogle Scholar
Boman, J., Differentiability of a function and of its compositions with functions of one variable, Math. Scand. 20 (1967), 249–268.CrossRefGoogle Scholar
Brouder, C., Dang, N. V. and Hélein, F., Continuity of the fundamental operations on distributions having a specified wave front set (with a counterexample by Semyon Alesker), Studia Math. 232 (2016), 201–226.Google Scholar
Croke, C. B., A sharp four-dimensional isoperimetric inequality, Comment. Math. Helv. 59 (1984), 187–192.CrossRefGoogle Scholar
Dabrowski, Y. and Brouder, C., Functional properties of Hörmander's space of distributions having a specified wavefront set, Comm. Math. Phys. 332 (2014), 1345–1380.CrossRefGoogle Scholar
Duistermaat, J. J., Fourier integral operators, Modern Birkhäuser Classics (Birkhäuser/Springer, New York, 2011). Reprint of the 1996 edition [MR 1362544], based on the original lecture notes published in 1973 [MR 0451313].CrossRefGoogle Scholar
Faifman, D., Crofton formulas and indefinite signature, Geom. Funct. Anal. 27 (2017), 489–540.CrossRefGoogle Scholar
Fu, J. H. G., Algebraic integral geometry, in Integral geometry and valuations, Advanced Courses in Mathematics – CRM Barcelona, eds. E. Gallego and G. Solanes (Springer, Basel, 2014), 47–112.Google Scholar
Fu, J. H. G., Intersection theory and the Alesker product, Indiana Univ. Math. J. 65 (2016), 1347–1371.CrossRefGoogle Scholar
Fu, J. H. G. and Wannerer, T., Riemannian curvature measures, Geom. Funct. Anal. 29 (2019), 343–381.CrossRefGoogle Scholar
Gårding, L., Extension of a formula by Cayley to symmetric determinants, Proc. Edinb. Math. Soc. (2) 8 (1948), 73–75.CrossRefGoogle Scholar
Guillemin, V. and Sternberg, S., Geometric asymptotics, Mathematical Surveys, vol. 14 (American Mathematical Society, Providence, RI, 1977).CrossRefGoogle Scholar
Hörmander, L., The analysis of linear partial differential operators. I, Classics in Mathematics (Springer, Berlin, 2003). Distribution theory and Fourier analysis, reprint of the second (1990) edition [Springer, Berlin; MR 1065993 (91m:35001a)].CrossRefGoogle Scholar
Kiderlen, M. and Jensen, E. B. V., (eds.), Tensor valuations and their applications in stochastic geometry and imaging, Lecture Notes in Mathematics, vol. 2177 (Springer, 2017).Google Scholar
Klain, D. A. and Rota, G.-C., Introduction to geometric probability, Lezioni Lincee. [Lincei Lectures] (Cambridge University Press, Cambridge, 1997).Google Scholar
Langevin, R., Chaves, R. M. B. and Bianconi, R., Formulas of Cauchy and Crofton in Lorentz-Minkowski and de Sitter spaces. Homage to Luis Santaló. Vol. 1 (Spanish), Math. Notae 41(2001/02) (2003), 99–113.Google Scholar
Muro, M., Singular invariant hyperfunctions on the space of real symmetric matrices, Tohoku Math. J. (2) 51 (1999), 329–364.CrossRefGoogle Scholar
Oh, Y.-G., Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds, Invent. Math. 101 (1990), 501–519.CrossRefGoogle Scholar
O'Neill, B., Semi-Riemannian geometry with applications to relativity, Pure and Applied Mathematics, vol. 103 (Academic Press, New York, 1983).Google Scholar
Schneider, R., Convex bodies: the Brunn-Minkowski theory, expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151 (Cambridge University Press, Cambridge, 2014).Google Scholar
Solanes, G. and Teufel, E., Integral geometry in constant curvature Lorentz spaces, Manuscripta Math. 118 (2005), 411–423.CrossRefGoogle Scholar
Steiner, J., Über parallele Flächen, Monatsber. Preuß. Akad. Wiss., 114–118 (1840). Ges. Werke, vol. 2 (Reimer, Berlin, 1882), 171–176.Google Scholar
Teufel, E., Kinematische Berührung im Äquiaffinen, Geom. Dedicata 33 (1990), 317–323.CrossRefGoogle Scholar
Treibergs, A., Estimates of volume by the length of shortest closed geodesics on a convex hypersurface, Invent. Math. 80 (1985), 481–488.CrossRefGoogle Scholar
Vladimirov, V. S., Methods of the theory of functions of many complex variables (MIT Press, Cambridge, MA, 1966).Google Scholar
Weyl, H., On the volume of tubes, Amer. J. Math. 61 (1939), 461–472.CrossRefGoogle Scholar
Wolf, J., Homogeneous manifolds of constant curvature, Comment. Math. Helv. 36 (1961), 112–147.CrossRefGoogle Scholar
Ye, N., Ma, X. and Wang, D., The Fenchel-type inequality in the 3-dimensional Lorentz space and a Crofton formula, Ann. Global Anal. Geom. 50 (2016), 249–259.CrossRefGoogle Scholar