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Characterizations of three homogeneous real hypersurfaces in a complex projective space. (English) Zbl 1396.53037
In [Differ. Geom. Appl. 29, S246–S250 (2011; Zbl 1225.53056)], the second author presented a characterization of horospheres in the complex hyperbolic space \(\mathbb{C}H^n(c)\) of constant holomorphic sectional curvature \(c<0\). Inspired by this characterization, the authors of the paper under review establish four characterization theorems on real hypersurfaces in complex projective space \(\mathbb{C}P^n(c)\) of constant holomorphic sectional curvature \(c>0\). Moreover, they show that a real hypersurface \(M\) isometrically immersed in \(\mathbb{C}P^n(4)\) is locally congruent to a geodesic hypersphere \(G(r)\) of radius \(r\in(\pi/4,\pi/2)\) if and only if there exists \(\alpha\in(0,\pi)\), \(\alpha\neq\pi/2\) such that for each point \(p\in M\) and each unit tangent vector \(X_p\in T_pM\) with \(g(X_p,\xi_p)=\cos\alpha\), where \(\xi\) is the characteristic vector field, the geodesic \(\gamma\) of \(M\) satisfying \(\gamma(0)=p\) and \(\dot{\gamma}(0)=X\) is an extrinsic geodesic.
MSC:
53B25 Local submanifolds
53C40 Global submanifolds
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