IJPAM: Volume 44, No. 3 (2008)

A CHARACTERIZATION OF MINIMAL LAGRANGIAN
SUBMANIFOLDS IN A COMPLEX PROJECTIVE SPACE

Yoshio Matsuyama
Department of Mathematics
Chuo University
1-13-27 Kasuga, Bunkyo-ku, Tokyo, 112-8551, JAPAN
e-mail: matuyama@math.chuo-u.ac.jp


Abstract.Let $M$ be a minimal Lagrangian submanifold in a complex projective space $CP^{n}(c)$ and $x$ any point of $M$. Then there exists a neighborhood $U$ of $x$ which a local field $\xi$ of any normal vector and the second fundamental form $A_{\xi}$ in the direction of $\xi$ are defined on $U$. In the present paper we will give a characterization of minimal Lagrangian submanifolds in $CP^{n}(c)$ which satisfy $(R(X, Y)A_{\xi}) Z = 0$ for all $X, Y$ and $Z$ tangent to $M$ and $A_{\xi}$ in the direction of any normal vector $\xi$.

Received: January 15, 2008

AMS Subject Classification: 53C40, 53B25

Key Words and Phrases: complex projective space, Lagrangina submanifold, minimal submanifold, parallel second fundamental submanifold

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 44
Issue: 3