Whitney stratifications and the continuity of local Lipschitz–Killing curvatures
Annales de l'Institut Fourier, Volume 68 (2018) no. 5, p. 2253-2276
In the paper we prove that the local Lipschitz–Killing curvatures of a definable set in a polynomially bounded o-minimal structure are continuous along the strata of a Whitney stratification. Moreover, if the stratification is (w)-regular the local Lipschitz–Killing curvatures are locally Lipschitz in any o-minimal structure.
On montre que les courbures Lipschitz–Killing locales d’un ensemble définissable dans une structure o-minimale polynomialement bornée sont continues le long des strates d’une stratification de Whitney. De plus, si la stratification est (w)-régulière les courbures Lipschitz–Killing locales sont localement lipschitziennes dans une structure o-minimale arbitraire.
Received : 2017-01-17
Revised : 2017-07-11
Accepted : 2017-11-07
Published online : 2018-11-23
DOI : https://doi.org/10.5802/aif.3208
Classification:  14B15,  14B10,  32B20,  57R45
Keywords: o-minimal structures, definable sets, stratifications, local Lipschitz–Killing curvatures
@article{AIF_2018__68_5_2253_0,
     author = {Nguyen, Nhan and Valette, Guillaume},
     title = {Whitney stratifications and the continuity of local Lipschitz--Killing curvatures},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {5},
     year = {2018},
     pages = {2253-2276},
     doi = {10.5802/aif.3208},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_5_2253_0}
}
Nguyen, Nhan; Valette, Guillaume. Whitney stratifications and the continuity of local Lipschitz–Killing curvatures. Annales de l'Institut Fourier, Volume 68 (2018) no. 5, pp. 2253-2276. doi : 10.5802/aif.3208. https://aif.centre-mersenne.org/item/AIF_2018__68_5_2253_0/

[1] Bernig, Andreas; Bröcker, Ludwig Courbures intrinsèques dans les catégories analytico-géométriques, Ann. Inst. Fourier, Tome 53 (2003) no. 6, pp. 1897-1924 http://aif.cedram.org/item?id=aif_2003__53_6_1897_0 | MR 2038783 | Zbl 1053.53053

[2] Bröcker, Ludwig; Kuppe, Martin Integral geometry of tame sets, Geom. Dedicata, Tome 82 (2000) no. 1-3, pp. 285-323 | Article | MR 1789065 | Zbl 10230.53057

[3] Brodersen, Hans; Trotman, David Whitney (b)-regularity is weaker than Kuo’s ratio test for real algebraic stratifications, Math. Scand., Tome 45 (1979) no. 1, pp. 27-34 | Article | MR 567430 | Zbl 0429.58001

[4] Comte, Georges Équisingularité réelle: nombres de Lelong et images polaires, Ann. Sci. Éc. Norm. Supér., Tome 33 (2000) no. 6, pp. 757-788 | Article | MR 1832990 | Zbl 0981.32018

[5] Comte, Georges An introduction to o-minimal geometry, Singularity theory and its applications, Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligraci Internazionali, Pisa (2000)

[6] Comte, Georges; Merle, Michel Équisingularité réelle. II. Invariants locaux et conditions de régularité, Ann. Sci. Éc. Norm. Supér., Tome 41 (2008) no. 2, pp. 221-269 | Article | MR 2468482 | Zbl 1163.32012

[7] Draper, Richard N. Intersection theory in analytic geometry, Math. Ann., Tome 180 (1969), pp. 175-204 | Article | MR 0247134 | Zbl 0157.40502

[8] Van Den Dries, Lou Tame topology and o-minimal structures, Cambridge University Press, London Mathematical Society Lecture Note Series, Tome 248 (1998), x+180 pages | Article | MR 1633348 | Zbl 0953.03045

[9] Van Den Dries, Lou; Miller, Chris Geometric categories and o-minimal structures, Duke Math. J., Tome 84 (1996) no. 2, pp. 497-540 | Article | MR 1404337 | Zbl 0889.03025

[10] Dutertre, Nicolas Euler characteristic and Lipschitz-Killing curvatures of closed semi-algebraic sets, Geom. Dedicata, Tome 158 (2012), pp. 167-189 | Article | MR 2922710 | Zbl 1256.14059

[11] Dutertre, Nicolas Stratified critical points on the real Milnor fibre and integral-geometric formulas, J. Singul., Tome 13 (2015), pp. 87-106 | Article | MR 3343616 | Zbl 1317.32009

[12] Dutertre, Nicolas Euler obstruction and Lipschitz-Killing curvatures, Isr. J. Math., Tome 213 (2016) no. 1, pp. 109-137 | Article | MR 3509470 | Zbl 1359.32005

[13] Dutertre, Nicolas Lipschitz-Killing curvatures and polar images (2017) (to appear in Adv. Geom.)

[14] Fu, Joseph H. G. Curvature measures of subanalytic sets, Am. J. Math., Tome 116 (1994) no. 4, pp. 819-880 | Article | MR 1287941 | Zbl 0818.53091

[15] Gibson, Christopher G.; Wirthmüller, Klaus; Du Plessis, Andrew A.; Looijenga, Eduard J. N. Topological stability of smooth mappings, Springer, Lecture Notes in Math., Tome 552 (1976), iv+155 pages | MR 0436203 | Zbl 0377.58006

[16] Juniati, Dwi; Noirel, Laurent; Trotman, David Whitney, Kuo-Verdier and Lipschitz stratifications for the surfaces y a =z b x c +x d , Topology Appl., Tome 234 (2018), pp. 335-347 | Zbl 1390.14107

[17] Juniati, Dwi; Trotman, David Determination of Lipschitz stratifications for the surfaces y a =z b x c +x d , Singularités Franco-Japonaises, Société Mathématique de France (Séminaires et Congrès) Tome 10 (2005), pp. 127-138 | MR 2145951 | Zbl 1083.14040

[18] Kurdyka, Krzysztof; Parusiński, Adam Quasi-convex decomposition in o-minimal structures. Application to the gradient conjecture, Singularity theory and its applications, Mathematical Society of Japan (Advanced Studies in Pure Mathematics) Tome 43 (2006), pp. 137-177 | MR 2325137 | Zbl 1132.32004

[19] Kurdyka, Krzysztof; Raby, Gilles Densité des ensembles sous-analytiques, Ann. Inst. Fourier, Tome 39 (1989) no. 3, pp. 753-771 | MR 1030848 | Zbl 0673.32015

[20] Loi, Ta Lê Verdier and strict Thom stratifications in o-minimal structures, Ill. J. Math., Tome 42 (1998) no. 2, pp. 347-356 http://projecteuclid.org/euclid.ijm/1256045049 | MR 1612771 | Zbl 0909.32008

[21] Loi, Ta Lê o-minimal structures, The Japanese-Australian Workshop on Real and Complex Singularities—JARCS III, Australian National University (Proc. Centre Math. Appl. Austral. Nat. Univ.) Tome 43 (2010), pp. 19-30 | MR 2763233 | Zbl 1247.14059

[22] Mather, John Notes on topological stability, Bull. Am. Math. Soc., Tome 49 (2012) no. 4, pp. 475-506 | Article | MR 2958928 | Zbl 1260.57049

[23] Nguyen, Xuan Viet Nhan Structure métrique et géométrie des ensembles définissables dans des structures o-minimales, Université Aix-Marseille (France) (2015) (Ph. D. Thesis)

[24] Orro, Patrice; Trotman, David Cône normal et régularités de Kuo-Verdier, Bull. Soc. Math. Fr., Tome 130 (2002) no. 1, pp. 71-85 | MR 1906193 | Zbl 1014.58004

[25] Du Plessis, Andrew A. Continuous controlled vector fields, Singularity theory (Liverpool, 1996), Cambridge University Press (London Mathematical Society Lecture Note Series) Tome 263 (1999), pp. 189-197 | MR 1709353 | Zbl 0935.57030

[26] Shiota, Masahiro Geometry of subanalytic and semialgebraic sets, Birkhäuser, Progress in Mathematics, Tome 150 (1997), xii+431 pages | Article | MR 1463945 | Zbl 0889.32006

[27] Teissier, Bernard Variétés polaires. II. Multiplicités polaires, sections planes, et conditions de Whitney, Algebraic geometry (La Rábida, 1981), Springer (Lecture Notes in Math.) Tome 961 (1982), pp. 314-491 | Article | MR 708342 | Zbl 0585.14008

[28] Trotman, David Whitney stratifications: faults and detectors, University of Warwick (UK) (1977) (Ph. D. Thesis)

[29] Trotman, David; Valette, Guillaume On the local geometry of definably stratified sets, Ordered algebraic structures and related topics, American Mathematical Society (Contemporary Mathematics) Tome 697 (2017), pp. 349-366 | Zbl 06860431

[30] Valette, Guillaume Volume, Whitney conditions and Lelong number, Ann. Pol. Math., Tome 93 (2008) no. 1, pp. 1-16 | Article | MR 2383338 | Zbl 1132.28307