Lagrangian Submanifolds with Constant Angle Functions of the nearly Kähler \mathbbS³×\mathbbS³

2016/11/03 by Bektas, Burcu, Moruz, Marilena, Van der Veken, Joeri +1
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1611.00958

Abstract

We study Lagrangian submanifolds of the nearly Kähler \mathbbS³×\mathbbS³ with respect to their, so called, angle functions. We show that if all angle functions are constant, then the submanifold is either totally geodesic or has constant sectional curvature and there is a classification theorem that follows from a recent paper of B. Dioos, L. Vrancken and X. Wang (arXiv:1604.05060). Moreover, we show that if precisely one angle function is constant, then it must be equal to 0,\fracπ3 or (2π)/(3). Using then two remarkable constructions together with the classification of Lagrangian submanifolds of which the first component has nowhere maximal rank, we obtain a classification of such Lagrangian submanifolds.

Related

• Lagrangian submanifolds of the nearly Kähler \mathbbS³ × \mathbbS³ from minimal surfaces in \mathbbS³
• Lagrangian submanifolds in the nearly kaehler s³ × s³
• Minimal Lagrangian submanifolds of the complex hyperquadric
• Lagrangian submanifolds of the nearly Kähler full flag manifold F_1,2(ℂ³)
• Totally geodesic Lagrangian submanifolds of the pseudo-nearly Kähler SL(2,ℝ)\timesSL(2,ℝ)