## Tsukuba Journal of Mathematics

### Intrinsic and extrinsic structures of Lagrangian surfaces in complex space forms

Bang-Yen Chen

#### Abstract

Lagrangian $H$-umbilical submanifolds introduced in $[1, 2]$ can be regarded as the simplest Lagrangian submanifolds in Kaehler manifolds next to totally geodesic ones. It was proved in [1] that Lagrangian $H$-umbilical submanifolds of dimension $\geq 3$ in complex Euclidean spaces are complex extensors, Lagrangian pseudo-spheres, and flat Lagrangian $H$-umbilical submanifolds. Lagrangian $H$-umbilical submanifolds of dimension $\geq 3$ in non-flat complex space forms are classified in [2]. In this paper we deal with the remaining case; namely, non-totally geodesic Lagrangian $H$-umbilical surfaces in complex space forms. Such Lagrangian surfaces are characterized by a very simple property; namely, $JH$ is an eigenvector of the shape operator $A_{H}, where$H$is the mean curvature vector field. The main operator of this paper is to determine both the intrinsic and the extrinsic structures of Lagrangian$H\$-umbilical surfaces.

#### Article information

Source
Tsukuba J. Math., Volume 22, Number 3 (1998), 657-680.

Dates
First available in Project Euclid: 30 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.tkbjm/1496163671

Digital Object Identifier
doi:10.21099/tkbjm/1496163671

Mathematical Reviews number (MathSciNet)
MR1674103

Zentralblatt MATH identifier
0930.53022

#### Citation

Chen, Bang-Yen. Intrinsic and extrinsic structures of Lagrangian surfaces in complex space forms. Tsukuba J. Math. 22 (1998), no. 3, 657--680. doi:10.21099/tkbjm/1496163671. https://projecteuclid.org/euclid.tkbjm/1496163671