## On the Kähler angles of submanifolds.(English)Zbl 1073.53077

Let $$(N,J,g)$$ be a Kähler-Einstein manifold of complex dimensions $$2n$$, and $$F:M^{2n}\to N^{2n}$$ be an immersed submanifold $$M$$ of real dimensions $$2n$$. Denote by $$\omega(X, Y)= g(JX,Y)$$ the Kähler form. The cosines of the Kähler angles $$\{\theta_{\alpha}, 1\leq\alpha\leq n\}$$ are the eigenvalues of $$F^*\omega$$, these are important quantities of the submanifold. The author and G. Valli considered the case that $$F$$ is a minimal immersion with equal Kähler angles and $$n\geq 2$$ [Pac. J. Math. 205, No. 1, 197–235 (2002; Zbl 1055.53046)] and claim, under some assumptions on the scalar curvature, $$F$$ should be either a complex submanifold (the Kähler angles are $$0$$), or a Lagrangian submanifold (the Kähler angles are $$\pi/2$$). Now the author considers the case without assuming minimality of $$M$$, the main results are as follows:
Theorem 1.2: Assume $$n=2$$ and $$M$$ is closed, $$N$$ is non-Ricci flat, and $$F: M\to N$$ is an immersion with equal Kähler angles, $$\theta_{\alpha}=\theta,\, \forall \alpha$$. If
$R\cdot F^*\omega((JH)^{T},\nabla \sin^2\theta)\leq 0,$
where $$H$$ is the mean curvature vector field, then $$F$$ is either a complex submanifold or a Lagrangian submanifold.
The condition in this theorem is valid when $$R<0$$, $$F$$ has parallel mean curvature vector field, and $$\| H\|^2\geq- {\text{R8}\over\sin^2\theta}$$.
Theorem 1.3: Assume $$M$$ is closed, $$n\geq 3$$ and $$F:M\to N$$ is an immersion with equal Kähler angles.
(A) If $$R<0$$, and if $$\delta F^*\omega((JH)^{T})\geq 0$$, then $$F$$ is either a complex submanifold or a Lagrangian submanifold.
(B) If $$R=0$$, and if $$\delta F^*\omega((JH)^{T})\geq 0$$, then the Kähler angles are constant.
(C) If $$F$$ has constant Kähler angles and $$R\neq 0$$, then $$F$$ is either a complex submanifold or a Lagrangian submanifold.
The main tool of this paper is a Weitzenböck formula for $$\Delta\kappa$$, where $\kappa= \sum_{1\leq\alpha\leq n}\log\Biggl({1+ \cos\theta_{\alpha}\over 1-\cos\theta_{\alpha}}\Biggr).$

### MSC:

 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Zbl 1055.53046
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