Abstract
We relate the Lipschitz–Killing measures of a definable set X ⊂ ℝn in an o-minimal structure to the volumes of generic polar images. For smooth submanifolds of ℝn, such results were established by Langevin and Shifrin. Then we give infinitesimal versions of these results. As a corollary, we obtain a relation between the polar invariants of Comte and Merle and the densities of generic polar images.
Communicated by: T. Grundhöfer
Acknowledgements
The author thanks David Mond and Terry Gaffney for interesting discussions on double points of fold singularities.
References
[1] C. B. Allendoerfer, The Euler number of a Riemann manifold. Amer. J. Math. 62 (1940), 243–248. MR0002251 Zbl 0024.35101 JFM 66.0869.0410.2307/2371450Search in Google Scholar
[2] A. Bernig, L. Bröcker, Lipschitz-Killing invariants. Math. Nachr. 245 (2002), 5–25. MR1936341 Zbl 1074.5306410.1002/1522-2616(200211)245:1<5::AID-MANA5>3.0.CO;2-ESearch in Google Scholar
[3] A. Bernig, L. Bröcker, Courbures intrinsèques dans les catégories analytico-géométriques. Ann. Inst. Fourier (Grenoble) 53 (2003), 1897–1924. MR2038783 Zbl 1053.5305310.5802/aif.1995Search in Google Scholar
[4] J. Bochnak, M. Coste, M.-F. Roy, Real algebraic geometry. Springer 1998. MR1659509 Zbl 0912.1402310.1007/978-3-662-03718-8Search in Google Scholar
[5] L. Bröcker, M. Kuppe, Integral geometry of tame sets. Geom. Dedicata82 (2000), 285–323. MR1789065 Zbl 1023.5305710.1023/A:1005248711077Search in Google Scholar
[6] G. Comte, Équisingularité réelle: nombres de Lelong et images polaires. Ann. Sci. École Norm. Sup. (4) 33 (2000), 757–788. MR1832990 Zbl 0981.3201810.1016/S0012-9593(00)01052-1Search in Google Scholar
[7] G. Comte, M. Merle, Équisingularité réelle. II. Invariants locaux et conditions de régularité. Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), 221–269. MR2468482 Zbl 1163.3201210.24033/asens.2067Search in Google Scholar
[8] M. Coste, An introduction to o-minimal geometry. Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000.Search in Google Scholar
[9] N. Dutertre, Stratified critical points on the real Milnor fibre and integral-geometric formulas. J. Singul. 13 (2015), 87–106. MR3343616 Zbl 1317.3200910.5427/jsing.2015.13eSearch in Google Scholar
[10] H. Federer, The (φ,k) rectifiable subsets of n-space. Trans. Amer. Soc. 62 (1947), 114–192. MR0022594 Zbl 0032.1490210.2307/1990632Search in Google Scholar
[11] W. Fenchel, On total curvatures of Riemannian manifolds: I. J. London Math. Soc. 15 (1940), 15–22. MR0002252 Zbl 0026.26401 JFM 66.1338.0110.1112/jlms/s1-15.1.15Search in Google Scholar
[12] J. H. G. Fu, Curvature measures of subanalytic sets. Amer. J. Math. 116 (1994), 819–880. MR1287941 Zbl 0818.5309110.2307/2375003Search in Google Scholar
[13] M. Goresky, R. MacPherson, Stratified Morse theory. Springer 1988. MR932724 Zbl 0639.1401210.1007/978-3-642-71714-7Search in Google Scholar
[14] H. A. Hamm, On stratified Morse theory. Topology38 (1999), 427–438. MR1660321 Zbl 0936.5800710.1016/S0040-9383(98)00020-2Search in Google Scholar
[15] R. M. Hardt, Topological properties of subanalytic sets. Trans. Amer. Math. Soc. 211 (1975), 57–70. MR0379882 Zbl 0303.3200810.1090/S0002-9947-1975-0379882-8Search in Google Scholar
[16] R. Langevin, Courbures, feuilletages et surfaces, volume 3 of Publications Mathématiques ďOrsay 80. Université de Paris-Sud, Département de Mathématique, Orsay 1980. MR584793 Zbl 0466.53036Search in Google Scholar
[17] R. Langevin, T. Shifrin, Polar varieties and integral geometry. Amer. J. Math. 104 (1982), 553–605. MR658546 Zbl 0504.5304810.2307/2374154Search in Google Scholar
[18] D. T. Lê, B. Teissier, Variétés polaires locales et classes de Chern des variétés singulières. Ann. of Math. (2) 114 (1981), 457–491. MR634426 Zbl 0488.3200410.2307/1971299Search in Google Scholar
[19] D. T. Lê, B. Teissier, Cycles évanescents, sections planes et conditions de Whitney. II. In: Singularities, Part 2 (Arcata, Calif., 1981), volume 40 of Proc. Sympos. Pure Math., 65–103, Amer. Math. Soc. 1983. MR713238 Zbl 0532.3200310.1090/pspum/040.2/713238Search in Google Scholar
[20] T. L. Loi, Verdier and strict Thom stratifications in o-minimal structures. Illinois J. Math. 42 (1998), 347–356. MR1612771 Zbl 0909.3200810.1215/ijm/1256045049Search in Google Scholar
[21] T. L. Loi, Lecture 1: o-minimal structures. In: The Japanese-Australian Workshop on Real and Complex Singularities III, volume 43 of Proc. Centre Math. Appl. Austral. Nat. Univ., 19–30, Austral. Nat. Univ., Canberra 2010. MR2763233Search in Google Scholar
[22] J. N. Mather, Generic projections. Ann. of Math. (2) 98 (1973), 226–245. MR0362393 Zbl 0242.5800110.2307/1970783Search in Google Scholar
[23] N. Nguyen, S. Trivedi, D. Trotman, A geometric proof of the existence of definable Whitney stratifications. Illinois J. Math. 58 (2014), 381–389. MR3367654 Zbl 1347.0307610.1215/ijm/1436275489Search in Google Scholar
[24] X. V. N. Nguyen, Structure métrique et géométrie des ensembles définissables dans des structures o-minimales. Thèse de Doctorat, Aix-Marseille Université 2015.Search in Google Scholar
[25] X. V. N. Nguyen, G. Valette, Whitney stratifications and the continuity of local Lipschitz Killing curvatures. To appear in Ann. Inst. Fourier. arXiv:1507.0126210.5802/aif.3208Search in Google Scholar
[26] P. Orro, D. Trotman, Regularity of the transverse intersection of two regular stratifications. In: Real and complex singularities, volume 380 of London Math. Soc. Lecture Note Ser., 298–304, Cambridge Univ. Press 2010. MR2759067 Zbl 1218.5800410.1017/CBO9780511731983.022Search in Google Scholar
[27] L. A. Santaló, Integral geometry and geometric probability. Addison-Wesley 1976. MR0433364 Zbl 0342.53049Search in Google Scholar
[28] Z. Szafraniec, A formula for the Euler characteristic of a real algebraic manifold. Manuscripta Math. 85 (1994), 345–360. MR1305747 Zbl 0824.1404610.1007/BF02568203Search in Google Scholar
[29] B. Teissier, Variétés polaires. II. Multiplicités polaires, sections planes, et conditions de Whitney. In: Algebraic geometry (La Rábida, 1981), volume 961 of Lecture Notes in Math., 314–491, Springer 1982. MR708342 Zbl 0585.1400810.1007/BFb0071291Search in Google Scholar
[30] D. Trotman, Espaces stratifiés réels. In: Stratifications, singularities and differential equations, II (Marseille, 1990; Honolulu, HI, 1990), volume 55 of Travaux en Cours, 93–107, Hermann, Paris 1997. MR1473246 Zbl 0883.32025Search in Google Scholar
[31] G. Valette, Volume, Whitney conditions and Lelong number. Ann. Polon. Math. 93 (2008), 1–16. MR2383338 Zbl 1132.2830710.4064/ap93-1-1Search in Google Scholar
[32] L. van den Dries, Tame topology and o-minimal structures. Cambridge Univ. Press 1998. MR1633348 Zbl 0953.0304510.1017/CBO9780511525919Search in Google Scholar
[33] L. van den Dries, C. Miller, Geometric categories and o-minimal structures. Duke Math. J. 84 (1996), 497–540. MR1404337 Zbl 0889.0302510.1215/S0012-7094-96-08416-1Search in Google Scholar
[34] H. Weyl, On the volume of tubes. Amer. J. Math. 61 (1939), 461–472. MR1507388 Zbl 0021.35503 JFM 65.0796.0110.2307/2371513Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
