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

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DOI

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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     title = {CONSTANT {SCALAR} {CURVATURE} {HYPERSURFACES} {WITH} {SECOND-ORDER} {UMBILICITY}},
     journal = {Glasgow mathematical journal},
     pages = {219--241},
     year = {2009},
     volume = {51},
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     doi = {10.1017/S0017089508004643},
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