"Una caracterización de las Hipersuperficies de Cartan por medio del laplaciano Infinito"

JULIO BARROS; CRISTIÁN U. SÁNCHEZ

San Miguel del Tucumán

Congreso; LXI Reunión de Comunicaciones Científicas de la UMA; 2011

Unión Matemática Argentina

PLANAR NORMAL SECTIONS ON ISOPARAMETRICHYPERSURFACESJULIO C. BARROS AND CRISTIAN U. SANCHEZLet M be a compact connected n-dimensional Rimannian manifold. Let p be apoint in M and considering, in the tangent space Tp (M), a unit vector X. We maydene an ane subspace of Rn+k by Sec (p;X) = p + SpanfX; T?p (M)g. If U is asmall enough neighborhood of p in M, then the intersection U \ Sec (p;X) can beconsidered a C1 regular curve  (s), parametrized by arclength, such that  (0) = p,0(0) = X. This curve is called a normal section of M at p in the direction of X. Wesay that the normal section  of M at p in the direction of X is planar at p if its rstthree derivatives 0(0), 00(0) and 000(0) are linearly dependent. If M is a compactspherical submanifold in Rn+k, given a point p inM, then we shall denote by cXp [M]the algebraic set dened by, cXp [M] = fX 2 Tp (M) : kXk = 1; (rX) (X;X) = 0g.In order to study the normal sections at p, it is convenient to consider certainhomogeneous polynomials of degree three. In this work we show an application ofthe normal section studies, for some particular families of isoparametric submanifolsof rank two or equivalently of isoparametric hypersurfaces in the sphere. Moreover,the polynomials that govern the behaviour of the planar normal sections on theisoparametric hypersurfaces homogeneous, were calculated. Finally, we show thecharacterization of hypersurfaces of Cartan in terms of the Innity Laplacian of thepolynomials that dene planar normal sections.References[1] Sanchez C. Algebraic Sets Associated to Isoparametric Submanifols, New developments inLie Theory and Geometry. Contemporary Mathematics, Vol.491.A M S, 2009.[2] Sanchez C. Normal sections of R-spaces I,Preprint, 2010[3] Sanchez C. U., Garca A., Dal Lago W. Planar normal sections on the natural imbedding ofa real ag manifold. Beitrage zur Algebra und Geometrie 41, 513-530 2000.(author 1) Universidad Nacional de Ro Cuarto - ArgentinaE-mail address: jbarros@exa.unrc.edu.ar(author 2) Facultad de Matematica Asatronoma y Fsica - CIEM - Universidad Na-cional de Cordoba - ArgentinaE-mail address: csanchez@mate.uncor1