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The classification of homogeneous 2-spheres in \(\mathbb C P^n\). (English) Zbl 1025.53034

An immersion \(x:M\rightarrow \mathbb C P^{n}\) of a connected and orientable surface \(M\) in \(\mathbb C P^{n}\) is called homogeneous if for any two points \(p,q\in M\) there exists a holomorphic isometry \(T\) of \(\mathbb C P^{n}\) and a diffeomorphism \(\sigma :M\rightarrow M\) such that \(\sigma (p)=q\) and \(x\circ \sigma =T\circ x.\) Standard examples of homogeneous 2-spheres are the so-called Veronese sequence in \(\mathbb C P^{n},\) which are also minimal in \(\mathbb C P^{n}\). Using the Veronese sequence in \(\mathbb C P^{k}\), \(k=0,1,\dots\), the authors give a complete construction of homogeneous 2-spheres in \(\mathbb C P^{n}\). To prove the classification theorem, the authors use the operators \(\partial \) and \(\overline{\partial}\) which are nonlinear analogues of the operators introduced by E. Calabi [J. Differ. Geom. 1, 111-125 (1967; Zbl 0171.20504)] and R. Bryant [Trans. Am. Math. Soc. 290, 259-271 (1985; Zbl 0572.53002)].
Then the authors prove that any homogeneous surface in \(\mathbb C P^{n}\) generates a sequence of homogeneous surfaces in \(\mathbb C P^{n}\). First, the structure equations for submanifolds in \(\mathbb C P^{n}\) are derived. Then the properties of homogeneous sequences are studied. The construction of minimal homogeneous 2-spheres in \(\mathbb C P^{n}\) given by O. Bando and Y. Ohnita [J. Math. Soc. Japan 3, 477-487 (1987; Zbl 0609.53017)] is generalized in order to give examples of non-minimal homogeneous 2-spheres in \(\mathbb C P^{n}\). The classification result consists in showing that all homogeneous 2-spheres can be obtained by the construction mentioned earlier.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C43 Differential geometric aspects of harmonic maps
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