Lagrangian submanifolds in the homogeneous nearly Kähler $${\mathbb {S}}^3 \times {\mathbb {S}}^3$$ S 3 × S 3

Abstract

In this paper, we investigate Lagrangian submanifolds in the nearly Kähler S 3 × S 3 . We construct a new example which is a flat Lagrangian torus. We give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature in the nearly Kähler S 3 × S 3 . As a corollary, we obtain that the radius of a round Lagrangian sphere in the nearly Kähler S 3 × S 3 can only be 2.

H-Umbilical Lagrangian Submanifolds of the Nearly Kähler \( {\mathbb{S}^3\times\mathbb{S}^3} \)

“…Lemma 2. [17] Let M be a Lagrangian submanifold of S 3 × S 3 and define the local angle functions θ 1 , θ 2 and θ 3 as above. Then, the sum of these functions vanishes modulo π.…”

“…Finally, we recall the equation of Codazzi for a Lagrangian submanifold M of S 3 × S 3 from [17]. It states that…”

“…The study of Lagrangian submanifolds originates from symplectic geometry and classical mechanics. Lagrangian submanifolds of the nearly Kähler S 6 were for example studied in [13,14], while various results on and examples of Lagrangian submanifolds of the nearly Kähler S 3 × S 3 were obtained in [15][16][17][18]. In general, Schäfer and Smozcyk [19] proved that Lagrangian submanifolds of nearly Kähler manifolds of dimension six and twistor spaces are always minimal and orientable.…”

“…[17] Let M be a Lagrangian submanifold of S 3 × S 3 and take a local orthonormal frame {E 1 , E 2 , E 3 } and local functions θ 1 , θ 2 and θ 3 as above. Denote by h k ij the components of the second fundamental form of the immersion and by ω k ij the components of the Levi-Civita connection of M with respect to {E 1 , E 2 , E 3 }, i.e., h k…”

Lagrangian submanifolds of the nearly Kähler full flag manifold F1,2(ℂ3)

“…More recently, Lagrangian submanifolds on the nearly Kähler manifold S 3 × S 3 were constructed from generalized Killing spinors on S 3 by Moroianu and Semmelmann in [20]. In [9] all Lagrangian submanifolds with constant sectional curvature are classified by means of a triple of angle functions the authors introduce. This method is further studied in [2,26].…”