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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.

MSC:

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
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