On the Kähler Angles of Submanifolds

Isabel M. C. Salavessa

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.

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