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Keywords:
ruled real hypersurface; nonflat complex space form; real hypersurfaces of type $({\rm A}_2)$ in a complex projective space; geodesics; structure torsion; Hopf manifold
ruled real hypersurface; nonflat complex space form; real hypersurfaces of type $({\rm A}_2)$ in a complex projective space; geodesics; structure torsion; Hopf manifold
Summary:
In the class of real hypersurfaces $M^{2n-1}$ isometrically immersed into a nonflat complex space form $\widetilde {M}_n(c)$ of constant holomorphic sectional curvature $c$ $(\ne 0)$ which is either a complex projective space $\mathbb {C}P^n(c)$ or a complex hyperbolic space $\mathbb {C}H^n(c)$ according as $c > 0$ or $c < 0$, there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds. In this paper, inspired by a simple characterization of all ruled real hypersurfaces in $\widetilde {M}_n(c)$, we consider a certain real hypersurface of type $({\rm A}_2)$ in $\mathbb {C}P^n(c)$ and give a geometric characterization of this Hopf manifold.
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