Critical points of higher order for the normal map of immersions in R^ d

• Autor(a) principal: Monera, M. G.
• Data de Publicação: 2012
• Outros Autores: Montesinos-Amilibia, A., Moraes, S. M., Sanabria-Codesal, E.
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: LOCUS Repositório Institucional da UFV
• Texto Completo: https://doi.org/10.1016/j.topol.2011.09.029; http://www.locus.ufv.br/handle/123456789/21909
• topic: Normal map; Critical points; Focal set; Strong principal directions; Veronese of curvature; Ellipse of curvature

We study the critical points of the normal map ν : N M → R k + n , where M is an immersed k-dimensional submanifold of R k + n , N M is the normal bundle of M and ν ( m , u ) = m + u if u ∈ N m M. Usually, the image of these critical points is called the focal set. However, in that set there is a subset where the focusing is highest, as happens in the case of curves in R 3 with the curve of the centers of spheres with contact of third order with the curve. We give a definition of r-critical points of a smooth map between manifolds, and apply it to study the 2 and 3-critical points of the normal map in general and the 2-critical points for the case k = n = 2 in detail. In the later case we analyze the relation with the strong principal directions of Montaldi (1986) [2].