Vol. 60, No. 2, pp. 215-235 (2003)
On the Kähler Angles of Submanifolds
Isabel M. C. SalavessaCentro de Física das Interacçoes Fundamentais, Instituto Superior Técnico,
Edifício Ciência, Piso 3, 1049-001 Lisboa -- PORTUGAL
Abstract: We prove that under certain conditions on the mean curvature and on the Kähler angles, a compact submanifold $M$ of real dimension $2n$, immersed into a Kähler--Einstein manifold $N$ of complex dimension $2n$, must be either a complex or a Lagrangian submanifold of $N$, or have constant Kähler angle, depending on $n=1$, $n=2$, or $n\geq 3$, and the sign of the scalar curvature of $N$. These results generalize to non-minimal submanifolds some known results for minimal submanifolds. Our main tool is a Bochner-type technique involving a formula on the Laplacian of a symmetric function on the Kähler angles and the Weitzenböck formula for the Kähler form of $N$ restricted to $M$.
Keywords: Lagrangian submanifold; Kähler--Einstein manifold; Kähler angles.
Classification (MSC2000): 53C42.; 53C21, 53C55, 53C40.
Full text of the article:
Electronic version published on: 9 Feb 2006. This page was last modified: 27 Nov 2007.