Lagrangian submanifolds in the homogeneous nearly Kähler \({\mathbb {S}}^3 \times {\mathbb {S}}^3\)
In this paper, the authors investigate Lagrangian submanifolds \(M\) in the nearly Kähler manifold \(\mathbf{S}^{3}\times \mathbf{S}^{3}\). Since \(M\) is Lagrangian, we can consider two endomorphisms \(A,B:TM\rightarrow TM\) such that the restriction of the almost product structure \(P\) to \(M\) is \(A+JB\). At each point \(p\) in \(M\), there is an orthonormal basis \(e_{1},e_{2},e_{3}\in T_{p}M\) such that \[ Pe_{i}=\cos 2\theta_{i}e_{i}+\sin2\theta_{i}Je_{i}, i=1,2,3 \] According to several properties of the angle functions \(\theta_{i}\), the following results are provided. Corollary 3.6: Let \(M\) be a Lagrangian submanifold of the nearly Kähler \(\mathbf{S}^{3}\times \mathbf{S}^{3}\). If \(M\) is totally geodesic, then the angles \(\theta_{1},\theta_{2},\theta_{3}\) are constant. Conversely, if the angles are constant and \(h_{12}^{3}=0\), then \(M\) is totally geodesic. \newline Lemma 3.9 states that for a totally geodesic Lagrangian submanifolds in \(\mathbf{S}^{3}\times \mathbf{S}^{3}\), there exist 6 possible cases for angle functions after a possible permutation of the angles. According to these 6 possible angle functions, 6 examples of totally geodesic submanifolds were obtained and there are two more non totally geodesic Lagrangian submanifolds are provided. In the last section, it is proved that these are congruent to one of the examples of constant sectional curvature. As a result, the radius of a round Lagrangian sphere in \(\mathbf{S}^{3}\times \mathbf{S}^{3}\) can be either \(\frac{2}{\sqrt{3}}\) or \(\frac{4}{\sqrt{3}}\).
scientific article; zbMATH DE number 6829770
scientific article