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Les sous-variétés de type CR de la sphère de dimension 6

Abstract : Let M be a Riemannian submanifold of the six dimensional sphere equipped with its almost complex structure J. It is natural to investigate submanifolds M according to their relations to J. If the tangent bundle of M is invariant for J, then M is an almost complex submanifold, and if J maps the tangent bundle into the corresponding normal bundle M is a totally real submanifold. A natural generalization of almost complex and totally real submanifolds are CR submanifolds. They admit an almost complex distribution H such that its orthogonal complement is totally real. A CR submanifold is called proper if its neither totally real nor almost complex. A. Gray showed that there are no 4-dimensional almost complex submanifolds of the 6-sphere, so the dimension of the almost complex distribution, in the nontrivial case has to be 2, and therefore the dimension of the corresponding totally real distribution is less or equal 2. B. Y. Chen introduced a basic Riemannian invariant and proved an inequality realting this invariant with the length of the mean curvature vector. It is interesting to investigate when the submanifold attains equality in this inequality. In that case we say that M satisfies Chen's basic equality. In this thesis we investigate 3 and 4-dimensional minimal submanifolds. In the 4-dimensional case we classify such submanifolds that satisfy Chen's basic equality, and in both dimensions we investigate submanifolds that are contained in a totally geodesic 5-sphere. We also investigate the minimal immersions of a surface into the (2n+1)-dimensional sphere whose first (n-2) ellipses of curvature are circles. We construct a sequence of minimal immersions that satisfy the same conditions and characterize immersions whose sequences contain two immersions congruent over an orientation reversing isometry.
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Submitted on : Monday, December 14, 2020 - 6:10:00 PM
Last modification on : Tuesday, November 23, 2021 - 9:44:36 AM
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  • HAL Id : tel-03065345, version 1


Miroslava Ž Antić. Les sous-variétés de type CR de la sphère de dimension 6. Mathématiques [math]. Université de Valenciennes et du Hainaut-Cambrésis; Univerzitet u Beogradu, 2010. Français. ⟨NNT : 2010VALE0001⟩. ⟨tel-03065345⟩



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