Seminar talk, 18 February 2009: Difference between revisions

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Speaker: Valentin Lychagin (В.В.Лычагин)
Speaker: [[Valentin Lychagin]]


Title: Geodesic webs of hypersurfaces (Геодезические ткани гиперповерхностей)
Title: Geodesic webs of hypersurfaces


Joint work with Vladislav Goldberg, [http://arxiv.org/abs/0812.2126 arXiv:0812.2126] (in Russian)
Joint work with Vladislav Goldberg, {{arXiv|0812.2126}} (in Russian)




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Geometric structures associated with foliations (or webs) of hypersurfaces are  studied.  We show that with any geodesic <math>(n+2)</math>-web on <math>n</math>-dimensional manifold there is a unique projective structure,  naturally associated with the web, and, provided that one of web foliations is pointed, there is also associated a unique affine structure.  The projective structure can be chosen by the claim that the leaves of all web foliations are totally geodesic, and the affine structure by an additional claim that one of web functions is affine.  These structures allow us to determine differential invariants of geodesic webs and give geometrically clear answers to some classical problems of the web theory such as the web linearization and the Gronwall theorem.
Geometric structures associated with foliations (or webs) of hypersurfaces are  studied.  We show that with any geodesic <math>(n+2)</math>-web on <math>n</math>-dimensional manifold there is a unique projective structure,  naturally associated with the web, and, provided that one of web foliations is pointed, there is also associated a unique affine structure.  The projective structure can be chosen by the claim that the leaves of all web foliations are totally geodesic, and the affine structure by an additional claim that one of web functions is affine.  These structures allow us to determine differential invariants of geodesic webs and give geometrically clear answers to some classical problems of the web theory such as the web linearization and the Gronwall theorem.


 
[[Category: Seminar|Seminar talk 7990-97-81]]
Изучаются геометрические структуры, связанные со слоениями (или тканями) гиперповерхностей. Мы показываем, что с каждой геодезической <math>(n+2)</math>-тканью гиперповерхностей на <math>n</math>-мерном многообразии естественным образом связаны единственная проективная структура и, при условии отмеченного слоения,- единственная аффинная структура. Проективная структура  выделяется требованием, чтобы слои всех слоений ткани были вполне геодезическими,  а аффинная структура - дополнительным требованием, чтобы одна из функций ткани была аффинной.  Эти структуры позволяют определить дифференциальные инварианты геодезических тканей, а также дать геометрически прозрачные ответы на классические вопросы теории тканей, такие как проблема линеаризации и теорема Гронвалла.
[[Category: Seminar abstracts|Seminar talk 7990-97-81]]
 
[[Category: Seminar|Seminar talk 2009-02-18]]
[[Category: Seminar abstracts|Seminar talk 2009-02-18]]

Latest revision as of 21:06, 30 November 2009

Speaker: Valentin Lychagin

Title: Geodesic webs of hypersurfaces

Joint work with Vladislav Goldberg, arXiv:0812.2126 (in Russian)


Absract:

Geometric structures associated with foliations (or webs) of hypersurfaces are studied. We show that with any geodesic -web on -dimensional manifold there is a unique projective structure, naturally associated with the web, and, provided that one of web foliations is pointed, there is also associated a unique affine structure. The projective structure can be chosen by the claim that the leaves of all web foliations are totally geodesic, and the affine structure by an additional claim that one of web functions is affine. These structures allow us to determine differential invariants of geodesic webs and give geometrically clear answers to some classical problems of the web theory such as the web linearization and the Gronwall theorem.

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