1. 报告人: Luc Vrancken (KU Leuven)

时间: 2026年5月26日（周二 ）下午2:00 – 3:00

地点: 双清综合楼 302

题目: Lagrangian submanifolds of the 6 dimensional homogeneous nearly Kaehler space S^3 × S^3

摘要:

We show how the nearly Kaehler structure can be obtained from the Riemannian submersion S^3 × S^3 × S^3 → S^3 × S^3, leading also to the introduction of an almost product structure on S^3 × S^3.

We then investigate Lagrangian submanifolds of this space. The existence of the almost product structure leads naturally to the introduction of angle functions in order to describe these submanifolds. We present in particular the classifications of Lagrangian submanifolds

(1) which are totally geodesic,

(2) which have constant sectional curvature,

(3) for which the angle functions are constant.

We also remark that the almost product structure can be used to introduce a 2-parameter family of metrics on S^3 × S^3 including both the nearly Kaehler metric and the standard metric. All these metrics contain as subgroup of isometries S^3 × S^3 × S^3.

We conclude by looking at Lagrangian submanifolds for which one angle function is constant. There are exactly 3 special cases:

- θ = π /3 corresponds with the first component not being an immersion,

- θ = 2π /3 corresponds to the second component not being an immersion,

- θ = 0 corresponds to the Lagrangian submanifold being minimal in S^7.

These are the only explicitly known examples of Lagrangian submanifolds of S^3 × S^3. Naturally more examples should exist, but as yet there is no description of them.

2.

报告人: Joeri Van der Veken (KU Leuven)

时间: 2026年5月26日（周二 ）下午 3:30 – 4:30

地点: 双清综合楼 302

题目: On minimal homogeneous submanifolds of the hyperbolic space

摘要:

A fundamental question in submanifold theory is: “How does the intrinsic geometry of a Riemannian manifold influence its possible isometric immersions?” Of particular interest is the case where the ambient space has constant sectional curvature. We consider Riemannian manifolds that are (intrinsically) homogeneous. It is conjectured that a minimal isometric immersion of such a homogeneous manifold into a Euclidean or hyperbolic space must be totally geodesic. For hypersurfaces, this follows from classification results by Nagano and Takahashi, but few results exist in higher codimension. In this joint work with Felippe Guimarães (Federal University of Rio de Janeiro), we prove the conjecture for minimal isometric immersions of homogeneous Riemannian manifolds of dimension at least 5 and codimension 2 in hyperbolic spaces.

邀请人：李海中、马辉、陈大广