报告题目：Lagrangian submanifolds of the 6 dimensional homogeneous nearly Kaehler space S^ 3 ×S^3
时 间：2026年5月18日（星期一）09：30
地 点：科研楼18号楼1102
主 办：数学与统计学院、分析数学及应用教育部重点实验室、福建省分析数学及应用重点实验室、福建省几何虚拟教研室
参加对象：相关专业师生
报告摘要: We show how the nearly Kaehler structure can be obtained from the Riemannian submersion of 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 classiffcations of Lagrangian submanifolds
(1) which are totally geodesic
(2) which have constant sectional curvature
(3) for which the angle functions are constant
(4) for which the projection on one of the two components does not have maximal rank (note that in that case the projection is necessarely either a constant map, or a minimal surface in S 3 )
Note that the above classes of Lagrangian submanifolds 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 (not even by using for example an existence and uniqueness theorem).
We also remark that the almost product structure can also be used to introduce a 1 parameter family of metrics on S^ 3 × S^ 3 including both the nearly Kaehler metric and the standard metric on S^ 3 × S^ 3 . All these metrics contain as subgroup of isometries S^ 3 × S^ 3 × S^ 3 .
We conclude the lecture by remarking that all of the previous examples are minimal with respect to any of the 1 parameter family of metrics and that these are the only Lagrangian submanifolds which are minimal with respect to all metrics.
报告人简介：Luc Vrancken教授是国际著名微分几何学家，现任上法兰西综合理工大学(Université Polytechnique des Hauts-de-France)及比利时天主教鲁汶大学 ( Katholieke Universiteit Leuven) 教授，德国洪堡基金获得者。