Geometric conditions on three-dimensional CR submanifolds in S⁶

Abstract

A six-dimensional unit sphere has an almost complex structure J defined by the vector cross product on the space of purely imaginary Cayley numbers, which makes S6 a nearly Khler manifold. In this paper, we study 3-dimensional CR submanifolds of S6(1), investigating certain geometric conditions. We show that if such a submanifold attains equality in Chen's inequality, it is always minimal. We recall that a classification of minimal 3-dimensional submanifolds was obtained in Djori, Vrancken, J. Geom. Phys. 56: 22792288, 2006. For 3-dimensional CR submanifolds, the restriction of the almost complex structure J to the tangent space automatically induces an almost contact structure on the submanifold. We prove that this structure is not Sasakian with respect to the induced metric. We also give an example from Hashimoto, Mashimo, J. Math. 28: 579591, 2005, see also Ejiri, Trans. Amer. Math. Soc. 297: 105124, 1986, of a tube around a superminimal almost complex curve in S6(1) for which this almost contact structure is Sasakian with respect to a constant scalar multiple of the induced metric. © 2010 de Gruyter.