On the Kähler angles of submanifolds. (Q1432744)

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 \textit{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 \[ \kappa= \sum_{1\leq\alpha\leq n}\log\Biggl({1+ \cos\theta_{\alpha}\over 1-\cos\theta_{\alpha}}\Biggr). \]