Lagrangian submanifolds in complex projective space with parallel second fundamental form

Franki Dillen, Haizhong Li, Luc Vrancken and Xianfeng Wang

Vol. 255 (2012), No. 1, 79–115

DOI: 10.2140/pjm.2012.255.79

Abstract

From the Riemannian geometric point of view, one of the most fundamental problems in the study of Lagrangian submanifolds is the classification of Lagrangian submanifolds with parallel second fundamental form. In 1980’s, H. Naitoh completely classified the Lagrangian submanifolds with parallel second fundamental form and without Euclidean factor in complex projective space, by using the theory of Lie groups and symmetric spaces. He showed that such a submanifold is always locally symmetric and is one of the symmetric spaces: SO(k + 1)∕SO(k)(k ≥ 2), SU(k)∕SO(k)(k ≥ 3), SU(k)(k ≥ 3), SU(2k)∕Sp(k)(k ≥ 3), E₆∕F₄.

In this paper, we completely classify the Lagrangian submanifolds in complex projective space with parallel second fundamental form by an elementary geometrical method. We prove that such a Lagrangian submanifold is either totally geodesic, or the Calabi product of a point with a lower-dimensional Lagrangian submanifold with parallel second fundamental form, or the Calabi product of two lower-dimensional Lagrangian submanifolds with parallel second fundamental form, or one of the standard symmetric spaces: SU(k)∕SO(k), SU(k), SU(2k)∕Sp(k) (k ≥ 3), E₆∕F₄.

As the arguments are of a local nature, at the same time, due to the correspondence between C-parallel Lagrangian submanifolds in Sasakian space forms and parallel Lagrangian submanifolds in complex space forms, we can also give a complete classification of all C-parallel submanifolds of S²ⁿ⁺¹ equipped with its standard Sasakian structure.

Keywords

Lagrangian submanifolds, complex space forms, complex projective space, parallel second fundamental form

Mathematical Subject Classification 2010

Primary: 53B25

Secondary: 53C42

Milestones

Received: 10 January 2011

Revised: 31 May 2011

Accepted: 20 June 2011

Published: 14 March 2012