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In this paper we obtain some new characterizations of pseudo-Einstein real hypersurfaces in $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$. More precisely, we prove that a real hypersurface in $\mathbb{C}P^{2}$ or $\mathbb{C}H^{2}$ with constant mean curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein. We prove that a real hypersurface in $\mathbb{C}P^{2}$ with constant scalar curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein.

We first provide a necessary and sufficient condition for a ruled real hypersurface in a nonflat complex space form to have constant mean curvature in terms of integral curves of the characteristic vector field on it. This yields a characterization of minimal ruled real hypersurfaces by circles. We next characterize the homogeneous minimal ruled real hypersurface in a complex hyperbolic space by using the notion of strong congruency of curves.

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Generalized ${\mathcal{D}}$-Einstein Real Hypersurfaces in $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$

Generating Curves of Minimal Ruled Real Hypersurfaces in a Nonflat Complex Space Form

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2 results

Generalized ${\mathcal{D}}$-Einstein Real Hypersurfaces in $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$

Generating Curves of Minimal Ruled Real Hypersurfaces in a Nonflat Complex Space Form