2-harmonic totally real submanifolds in a complex projective space. (English) Zbl 0960.53036

A map \(f: M\to N\) between Riemannian manifolds is a 2-harmonic map if it is a critical point of the 2-energy \(E_2(f)=(1/2)\int_M |\tau(f)|^2 *1\) where \(\tau(f)\) is the tension field of \(f\). Clearly, all harmonic maps are 2-harmonic, but the converse does not hold. In this paper, the authors consider an \(n\)-dimensional totally real submanifold \(M\) of the complex projective space \(\mathbb{C} P^n\) such that the isometric immersion \(i: M\to \mathbb{C} P^n\) is 2-harmonic. They discuss several sufficient conditions (on the second fundamental form or the Ricci tensor of \(M\)) under which \(M\) becomes a minimal submanifold of \(\mathbb{C} P^n\), or even a totally geodesic one.


53C43 Differential geometric aspects of harmonic maps
58E20 Harmonic maps, etc.
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds