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Escola de altos estudos CAPES

Course:

Lie sphere geometry and Dupin hypersurfaces

Professor:

Thomas E. Cecil (Holy Cross College- USA)

Organized by:

Prof. Martha P. Dussan Angulo and Prof. Paolo Piccione.

Date:

January 09 to 20 /2012

Additional information:

 See the new site:http://www.ime.usp.br/~geometriaescola
 
Abstract: In his doctoral dissertation, published in Math. Ann. in 1872, Lie introduced his geometry of oriented hyperspheres in Euclidean space R^n in the context of his work on contact transformations. Lie established a bijective correspondence between the set of all oriented hyperspheres, oriented hyperplanes and point spheres  in R^nU\{infty} and the set of all points on the quadric hypersurface Q^{n+1} in the real projective space P^{n+2} given by the equation <x,x>=0, where <,> is an indefinite scalar product with signature (n+1,2) on R^{n+3}. Equivalently, one can study the space of all oriented hyperspheres and points in S^n.

In this short-course we give Lie's construction in detail, and discuss its applications to the modern study of Dupin hypersurfaces. A hypersurface M in R^n (or S^n) is said to be proper Dupin if the number of g of distinct principal curvatures is constant on M and each distinct principal curvature is constant along each leaf of its corresponding principal foliation. Examples of proper Dupin hyersurfaces in R^n are the images under steoreographic projection of isoparametric (constant principal curvatures) hypersurfaces in the sphere S^n, including the cyclides of Dupin in R^3.

The classifications of compact proper Dupin hypersurfaces with g=4 or 6 principal curvatures have not  yet been completed, although Stolz (g=4) and Grove and Halperin (g=6) proved that the multiplicities of the principal curvatures must be the same as for an isoparametric hypersurface and partial classifications have been found. In this short-course, we will discuss various local and global classification results for proper Dupin hypersurfaces in S^n that have been obtained in the context of Lie shere geometry. The course is based primarily on the author's book, Lie sphere Geometry with Applications to Submanifolds, Second Edition, Springer, New York, 2008.