Geodesic
CONSTANT SCALAR CURVATURE HYPERSURFACES WITH SECOND-ORDER UMBILICITY
Glasgow mathematical journal, Tome 51 (2009) no. 2, pp. 219-241

Voir la notice de l'article provenant de la source Cambridge University Press

We extend the concept of umbilicity to higher order umbilicity in Riemannian manifolds saying that an isometric immersion is k-umbilical when APk−1(A) is a multiple of the identity, where Pk(A) is the kth Newton polynomial in the second fundamental form A with P0(A) being the identity. Thus, for k=1, one-umbilical coincides with umbilical. We determine the principal curvatures of the two-umbilical isometric immersions in terms of the mean curvatures. We give a description of the two-umbilical isometric immersions in space forms which includes the product of spheres embedded in the Euclidean sphere S2k+1 of radius 1. We also introduce an operator φk which measures how an isometric immersion fails to be k-umbilical, giving in particular that φ1 ≡ 0 if and only if the immersion is totally umbilical. We characterize the two-umbilical hypersurfaces of a space form as images of isometric immersions of Einstein manifolds.
DOI : 10.1017/S0017089508004643
Mots-clés : Primary 53C42, 53C40, Secondary 53B20, 53B25 
COLARES, ANTONIO GERVASIO; ECHAIZ-ESPINOZA, FERNANDO ENRIQUE. CONSTANT SCALAR CURVATURE HYPERSURFACES WITH SECOND-ORDER UMBILICITY. Glasgow mathematical journal, Tome 51 (2009) no. 2, pp. 219-241. doi: 10.1017/S0017089508004643
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