CR-submanifolds of the nearly Kähler 6-sphere

Miroslava Antić (1) , Luc Vrancken (2, 3)

1 University of Belgrade [Belgrade]
2 LAMAV - Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015
3 KU Leuven - Catholic University of Leuven = Katholieke Universiteit Leuven

There is an almost complex structure J on the sphere S6(1) defined by multiplication of the Cayley numbers. This structure is nearly Kähler. A submanifold of a manifold with an almost complex structure is CR, by Bejancu, if it has a differentiable holomorphic distribution H such that its orthogonal complement H⊥⊂TM is a totally real distribution. A CR-submanifolds of S6(1) has to be at least three-dimensional, so with disregarding the hypersurfaces which are trivially CR in the focus of investigation are three and four dimensional submanifolds. We give examples of such submanifolds, show the existence and uniqueness theorem for the three dimensional case, and present the results concerning H and H⊥ totally geodesic submanifolds. We also give examples obtained from the almost contact manifolds. In the four dimensional case, we show the classification of CR minimal submanifolds that satisfy Chen’s basic equality and of those that are not linearly full in S6(1)

Mots clés: CR submanifold; D-geodesic submanifolds; Linearly full; Minimal submanifold; Nearly kähler six-sphere

Domaines: Mathématiques [math]

Miroslava Antić, Luc Vrancken. CR-submanifolds of the nearly Kähler 6-sphere. Sorin Dragomir; Mohammad Hasan Shahid. Geometry of Cauchy-Riemann Submanifolds, Springer Verlag, 390 p., 2016, 9789811009150. ⟨10.1007/978-981-10-0916-7_3⟩. ⟨hal-03149500⟩