Comptes Rendus
no. 11-12
Differential Geometry
Singularities of Blaschke normal maps of convex surfaces
[Singularités des applications normales de Blaschke des surfaces convexes]

Présenté par : Jean-Pierre Demailly

Kentaro Saji1 ; Masaaki Umehara2 ; Kotaro Yamada3
1 Department of Mathematics, Faculty of Education, Gifu University, Yanagido 1-1, Gifu 501-1193, Japan
2 Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
3 Department of Mathematics, Tokyo Institute of Technology, O-okayama, Meguro, Tokyo 152-8551, Japan
Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 665-668.

Nous prouvons que la différence entre les nombres de queues d'aronde positives et queues d'aronde négatives de l'application normale de Blaschke, pour une surface convexe donnée dans l'espace d'affine, est égale au nombre d'Euler du sous-ensemble où l'opérateur de forme affine a un déterminant négatif.

We prove that the difference between the numbers of positive swallowtails and negative swallowtails of the Blaschke normal map for a given convex surface in affine space is equal to the Euler number of the subset where the affine shape operator has negative determinant.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.04.021
Kentaro Saji 1 ; Masaaki Umehara 2 ; Kotaro Yamada 3

1 Department of Mathematics, Faculty of Education, Gifu University, Yanagido 1-1, Gifu 501-1193, Japan
2 Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
3 Department of Mathematics, Tokyo Institute of Technology, O-okayama, Meguro, Tokyo 152-8551, Japan 
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     author = {Kentaro Saji and Masaaki Umehara and Kotaro Yamada},
     title = {Singularities of {Blaschke} normal maps of convex surfaces},
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     pages = {665--668},
     publisher = {Elsevier},
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     year = {2010},
     doi = {10.1016/j.crma.2010.04.021},
     language = {en},
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Kentaro Saji; Masaaki Umehara; Kotaro Yamada. Singularities of Blaschke normal maps of convex surfaces. Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 665-668. doi : 10.1016/j.crma.2010.04.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.04.021/

[1] V.I. Arnol'd; S.M. Gusein-Zade; A.N. Varchenko Singularities of Differentiable Maps, vol. 1, Monographs in Math., vol. 82, Birkhäuser, 1985

[2] D. Bleecker; L. Wilson Stability of Gauss maps, Illinois J. of Math., Volume 22 (1978), pp. 279-289

[3] S. Izumiya; W.L. Marar The Euler characteristic of a generic wavefront in a 3-manifold, Proc. Amer. Math. Soc., Volume 118 (1993), pp. 1347-1350

[4] K. Nomizu; T. Sasaki Affine Differential Geometry, Cambridge University Press, Cambridge, 1994

[5] K. Saji; M. Umehara; K. Yamada The geometry of fronts, Ann. of Math., Volume 169 (2009), pp. 491-529

[6] K. Saji; M. Umehara; K. Yamada Behavior of corank one singular points on wave fronts, Kyushu J. Math., Volume 62 (2008), pp. 259-280

[7] K. Saji; M. Umehara; K. Yamada The intrinsic duality of wave fronts (preprint) | arXiv

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