Papers by Fernando Enrique Echaiz-Espinoza
Seja x uma imersao isometrica a A sua segunda forma fundamental. Os polinomios de Newton sao defi... more Seja x uma imersao isometrica a A sua segunda forma fundamental. Os polinomios de Newton sao definidos indutivamente po Po=I, Pk=Sk I-A Pk-1, onde Sk e a k-curvatura de imersao. Este trabalho define k-umbilicidade pela condicao de que o produto de A pelo polinomio de Newton de ondem k-1 e um multiplo de identidade. Como consequencias gerais do conceito de k-umbilicidade demonstra que, se uma imersao isometrica k-umbilica tem uma curvatura principal nula, entao tem n-k+1 curvaturas principais nulas e demonstra tambem que em toda imersao k-umbilica o perador Lk e eliptico sempre que Sk seja nao nulo, considerando Lk(f)=traco(Pk Hess(f)) um operador diferencial de segunda ordem. Para o caso k=2, demostra que toda imersao 2-umbilica em formas espaciais tem S2 constante. Classifica ainda parcialmente as hipersuperficies 2-umbilicas fechadas na esfera unitaria, exibindo uma familia enumeravel de tais imersoes.
Glasgow Mathematical Journal, 2009
We extend the concept of umbilicity to higher order umbilicity in Riemannian manifolds saying tha... more 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 $S^{k}(\frac{1}{\sqrt{2}})\times S^{k}(\frac{1}{\sqrt{2}})$ 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.
Glasgow Mathematical Journal, Apr 30, 2009
We extend the concept of umbilicity to higher order umbilicity in Riemannian manifolds saying tha... more We extend the concept of umbilicity to higher order umbilicity in Riemannian manifolds saying that an isometric immersion is k-umbilical when AP k−1 (A) is a multiple of the identity, where P k (A) is the kth Newton polynomial in the second fundamental form A with P 0 (A) being the identity. Thus, for k = 1, oneumbilical coincides with umbilical. We determine the principal curvatures of the twoumbilical 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 S k (1 √ 2) × S k (1 √ 2) embedded in the Euclidean sphere S 2k+1 of radius 1. We also introduce an operator φ k which measures how an isometric immersion fails to be k-umbilical, giving in particular that φ 1 ≡ 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.
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Papers by Fernando Enrique Echaiz-Espinoza