Narrow Results

We classify real hypersurfaces with isometric Reeb flow in the complex quadrics Q^m = SO_m+2/SO_mSO₂, m ≥ 3. We show that m is even, say m = 2k, and any such hypersurface is an open part of a tube around a k-dimensional complex projective space ℂP^k which is embedded canonically in Q^2k as a totally geodesic complex submanifold. As a consequence, we get the non-existence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics Q^2k+1, k ≥ 1. To our knowledge the odd-dimensional complex quadrics are the first examples of homogeneous Kähler manifolds which do not admit a real hypersurface with isometric Reeb flow.

On a real hypersurface M in complex projective space we can consider the Levi-Civita connection and for any nonnull constant k the kth g-Tanaka–Webster connection. We classify real hypersurfaces such that both the Lie derivative associated to the Levi-Civita connection and the kth g-Tanaka–Webster derivative in the direction of the structure vector field ξ coincide when we apply them to either the shape operator or the structure Jacobi operator of M.

We classify real hypersurfaces with isometric Reeb flow in the complex hyperbolic quadrics ${Q^{\ast }}^m=S{O}_{2,m}^o/S{O}_{m}S{O}_2$, $m\ge 3$. We show that $m$ is even, say $m=2k$, and any such hypersurface becomes an open part of a tube around a $k$-dimensional complex hyperbolic space $ℂH^k$ which is embedded canonically in ${Q^{\ast }}^{2k}$ as a totally geodesic complex submanifold or a horosphere whose center at infinity is $𝔄$-isotropic singular. As a consequence of the result, we get the nonexistence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics ${Q^{\ast }}^{2k+1}$, $k\ge 1$.

Let $M$ be a three-dimensional real hypersurface in a nonflat complex space form of complex dimension two. In this paper, we prove that $M$ is $\eta$-parallel with two distinct principal curvatures at each point if and only if it is locally congruent to a geodesic sphere in $ℂP^2$ or a horosphere, a geodesic sphere or a tube over totally geodesic complex hyperbolic plane in $ℂH^2$. Moreover, $\eta$-parallel real hypersurfaces in $ℂP^2$ and $ℂH^2$ under some other conditions are classified and these results extend Suh’s in [Characterizations of real hypersurfaces in complex space forms in terms of Weingarten map, Nihonkai Math. J. 6 (1995) 63–79] and Kon–Loo’s in [On characterizations of real hypersurfaces in a complex space form with $\eta$-parallel shape operator, Canad. Math. Bull. 55 (2012) 114–126].

Any real hypersurface of a Kaehler manifold carries a natural almost contact metric structure. There are four basic classes of real hypersurfaces of a Kaehler manifold with respect to the induced almost contact metric structure. In this paper we study the basic classes of real hypersurfaces of a complex space form in terms of their Hermitian-like curvatures.

We give a simple argument for the nonexistence of Einstein hypersurfaces in the complex space forms CPⁿ and CHⁿ which is valid for all dimensions n ≥ 2. In addition, we survey classification results that are stated in terms of intrinsic geometrical properties of the hypersurface. Many questions that have been settled for Hopf hypersurfaces and/or for dimensions n ≥ 3 still remain open in the general case.

The study of real hypersurfaces in complex projective space has been an active field of study over the past decade. M. Okumura classified the real hypersurfaces of type (A) in complex projective space [7]. And η-parallelism of real hypersurfaces are discussed in M. Kimura and S. Maeda [4]. Maeda proved the nonexistence of semi-parallel real hypersurfaces [8]. In this paper, we study real hypersurfaces of type (A) in complex projective space with parallelism of some symmetric tensor.