Characterizations of some homogeneous Hopf real hypersurfaces in a nonflat complex space form by extrinsic shapes of trajectories

We characterize some homogeneous Hopf real hypersurfaces in a nonflat complex space form by studying trajectories for Sasakian magnetic fields whose extrinsic shapes are tangentially of order 2.     

Ruled real hypersurfaces of a complex space form

The purpose of this paper is to define a ruled real hypersurface of a complex space formM _n (c), c≠0, and to give characterizations of this hypersurface by the infinitesimal affine transformation of the structure vector field induced on the hypersurface. Supported by Grant for the Institute of Mathematics, the University of Tsukuba, and TGRC-KOSEF (1993).     

Conformally flat real hypersurfaces in nonflat complex planes

In this paper, we prove that there are no conformally flat real hypersurfaces in nonflat complex space forms of complex dimension two provided that the structure vector field is an eigenvector field of the Ricci operator. This extends some recent results by Cho (Conformally flat normal almost contact 3-manifolds, Honam Math. J. 38 (2016) 59–69) and Kon (3-dimensional real hypersurfaces with η-harmonic curvature, in: Hermitian–Grassmannian Submanifolds, Springer, Singapore, 2017, pp. 155–164).     

Integral curves of characteristic vector fields of real hypersurfaces in nonflat complex space forms

In this paper, we study real hypersurfaces all of whose integral curves of characteristic vector fields are plane curves in a nonflat complex space form.      

Three real hypersurfaces some of whose geodesics are mapped to circles with the same curvature in a nonflat complex space form

We characterize all totally η-umbilic hypersurfaces and ruled real hypersurfaces in nonflat complex space forms and certain real hypersurfaces of type (A₂) in complex projective spaces by using the property that some of their geodesics are mapped to circles of the same curvature in these ambient spaces.     

Geodesics on real hypersurfaces of type A 2 in a complex space form

In this paper, we study geodesics with null structure torsions on real hypersurfaces of type A ₂ in a complex space form. These geodesics give a nice family of helices of order 3 generated by Killing vector fields on the ambient complex space form. Author’s address: Toshiaki Adachi, Department of Mathematics, Nagoya Institute of Technology, Nagoya 466-8555, Japan     

Semi-parallel hypersurfaces of a real space form

We show that a semi-parallel hypersurface of a sphere and a hyperbolic space is either flat, parallel or a rotation hypersurface whose profile curve is a helix. Supported by a research fellowship of the Research Council of the Katholieke Universiteit Leuven.     

The third fundamental form metric for hypersurfaces in nonflat space forms

For hypersurfaces with regular Weingarten operator in nonflat space forms we study the relations between the intrinsic geometry of the third fundamental form metric and the extrinsic geometry of the hypersurface. We prove a theorema-egregium-type result for this metric and, in particular, give a local classification of hypersurfaces in case of an Einstein structure of this metric.Partially supported by the project 19701003 of NSFC.The geometry groops at TU Berlin and KU Leuven cooperate within the GA DGET program.     

The structure vector field and structure Jacobi operator of real hypersurfaces in nonflat complex space forms

In this paper we determine the real hypersurfaces for which the structure Jacobi operator commutes over both the Ricci tensors and structure tensors (for a definition of the operator see Sect. 1). We prove that such hypersurfaces are homogneous real hypersurfaces of type (A) and are a special class of Hopf hypersurfaces.     

Real hypersurfaces with constant totally real sectional curvature in a complex space form

We characterize real hypersurfaces with constant holomorphic sectional curvature of a non flat complex space form as the ones which have constant totally real sectional curvature.     

Nonnegative sectional curvature hypersurfaces in a real space form

In this paper, we investigate the nonnegative sectional curvature hypersurfaces in a real space form M ⁿ⁺¹(c). We obtain some rigidity results of nonnegative sectional curvature hypersurfaces M ⁿ⁺¹(c) with constant mean curvature or with constant scalar curvature. In particular, we give a certain characterization of the Riemannian product S ^k (a) × S ^n-k (√1 ? a ²), 1 ≤ k ≤ n ? 1, in S ⁿ⁺¹(1) and the Riemannian product H ^k (tanh² r ? 1) × S ^n-k (coth² r ? 1), 1 ≤ k ≤ n ? 1, in H ⁿ⁺¹(?1).     

Real hypersurfaces of a complex space form

In this paper, we show that an n-dimensional connected non-compact Ricci soliton isometrically immersed in the flat complex space form ){(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle )}, with potential vector field of the Ricci soliton is the characteristic vector field of the real hypersurface is an Einstein manifold. We classify connected Hopf hypersurfaces in the flat complex space form (C ^{á ñ\fracn+12},J, á , ñ ){(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle )} and also obtain a characterization for the Hopf hypersurfaces in (C^\fracn+12,J, á , ñ ) {(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle ) }.     

Real hypersurfaces of a complex space form

In this paper, we show that an n-dimensional connected non-compact Ricci soliton isometrically immersed in the flat complex space form ${(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle )}$ , with potential vector field of the Ricci soliton is the characteristic vector field of the real hypersurface is an Einstein manifold. We classify connected Hopf hypersurfaces in the flat complex space form ${(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle )}$ and also obtain a characterization for the Hopf hypersurfaces in ${(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle ) }$ .     

Real hypersurfaces of a complex space form

In this paper we are interested in obtaining a condition under which a compact real hypersurface of a complex projective space CP ⁿ is a geodesic sphere. We also study the question as to whether the characteristic vector field of a real hypersurface of the complex projective space CP ⁿ is harmonic, and show that the answer is in negative.     

On some real hypersurfaces of a complex hyperbolic space

Given a real hypersurface of a complex hyperbolic space #x2102;?H ⁿ ,we construct a principal circle bundle over it which is a Lorentzian hypersurface of the anti-De Sitter space H ₁ ²ⁿ⁺¹ .Relations between the respective second fundamental forms are obtained permitting us to classify a remarkable family of real hypersurfaces of ?H ⁿ .     

Ricci η‐recurrent real hypersurfaces in 2‐dimensional nonflat complex space forms

Let M be a Hopf hypersurface in a nonflat complex space form $M^2(c)$, $c\ne 0$, of complex dimension two. In this paper, we prove that M has η‐recurrent Ricci operator if and only if it is locally congruent to a homogeneous real hypersurface of type (A) or (B) or a non‐homogeneous real hypersurface with vanishing Hopf principal curvature. This is an extension of main results in [17, 21] for real hypersurfaces of dimension three. By means of this result, we give some new characterizations of Hopf hypersurfaces of type (A) and (B) which generalize those in [14, 18, 26].