The Center Map of an Affine Immersion |
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Authors: | Hitoshi Furuhata Luc Vrancken |
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Institution: | (1) Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan;(2) Laboratoire de Mathématiques, Université de Valenciennes, 59313 Valenciennes, Cedex 9, France |
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Abstract: | We study the center map of an equiaffine immersion which is introduced using the equiaffine support function. The center map
is a constant map if and only if the hypersurface is an equiaffine sphere. We investigate those immersions for which the center
map is affine congruent with the original hypersurface. In terms of centroaffine geometry, we show that such hypersurfaces
provide examples of hypersurfaces with vanishing centroaffine Tchebychev operator. We also characterize them in equiaffine
differential geometry using a curvature condition involving the covariant derivative of the shape operator. From both approaches,
assuming the dimension is 2 and the surface is definite, a complete classification follows.
Received: May 24, 2006. Revised: July 26, 2006. Accepted: July 28, 2006. |
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Keywords: | 53A15 |
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