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Harmonic almost-complex structures. (English) Zbl 0834.53030
The author studies almost complex structures \(J\) on an oriented Riemannian manifold as harmonic sections of a Riemannian vector bundle. The harmonicity condition is given by \[ [J, \nabla^* \nabla J] = 0, \] a condition shown by G. Valli [J. Geom. Phys. 4, No. 3, 335-359 (1987; Zbl 0656.58010)] to be the equation for a geodesic in a certain gauge group. Integrability and conditions for \(J\) to be stably harmonic are studied as well as almost-Kähler structures.
Special attention is given to dimension 4 and to Calabi-Eckmann examples. This is a thorough and carefully written paper.

MSC:
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
58E20 Harmonic maps, etc.
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