A d’Alembert Formula for Hopf Hypersurfaces

A Hopf hypersurface in complex hyperbolic space ${\mathbb{C}{\rm H}^n}$ is one for which the complex structure applied to the normal vector is a principal direction at each point. In this paper, Hopf hypersurfaces for which the corresponding principal curvature is small (relative to ambient curvature) are studied by means of a generalized Gauss map into a product of spheres, and it is shown that the hypersurface may be recovered from the image of this map, via an explicit parametrization.     

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].     

Real hypersurfaces with constant principal curvatures in complex space forms

We present the motivation and current state of the classification problem of real hypersurfaces with constant principal curvatures in complex space forms. In particular, we explain the classification result of real hypersurfaces with constant principal curvatures in nonflat complex space forms and whose Hopf vector field has nontrivial projection onto two eigenspaces of the shape operator. This constitutes the following natural step after Kimura and Berndt?s classifications of Hopf real hypersurfaces with constant principal curvatures in complex space forms.     

The Ricci tensor and structure Jacobi operator of real hypersurfaces in a complex projective space

Let M be a real hypersurface with almost contact metric structure ${(\phi, \xi, \eta, g)}$ in a complex projective space ${P_{n}\mathbb{C}}$ . A Real hypersurface M is said to be a Hopf hypersurface if ξ is principal. In this paper we investigate real hypersurfaces of ${P_{n}\mathbb{C}}$ whose Ricci tensors S satisfy ${\nabla_{\phi\nabla_{\xi}\xi}S = 0}$ . Under some further conditions we characterize Hopf hypersurfaces of ${P_{n}\mathbb{C}}$ .     

On the Riemannian curvature tensor of a real hypersurface in a complex projective space

We consider real hypersurfaces M in complex projective space equipped with both the Levi–Civita and generalized Tanaka–Webster connections and classify them when the covariant derivatives associated with both connections, either in the direction of the structure vector field or any direction of the maximal holomorphic distribution, coincide when applying to the Riemannian curvature tensor of the real hypersurface.     

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 .     

Submanifolds of generalized Hopf manifolds,type numbers and the first Chern class of the normal bundle

Summary Any orientable real hypersurface M of a complex Hopf manifold (carrying the locally conformal Kaehler (l.c.K.) metric discovered by I.Vaisman [33]) has a natural f-structure P as a generic Cauchy-Riemann submanifold; we show (cf. our § 5) that if P anti-commutes with the Weingarten operator, then the type number of the hypersurface is less equal than 1. Moreover, M carries the natural almost contact metrical structure observed by Y.Tashiro [30]; if its almost contact vector is an eigenvector of the Weingarten operator corresponding to a nowhere vanishing eigenfunction and the holomorphic distribution is involutive, then M is foliated with globally conformai Kaehler manifolds (cf. our § 5), provided that some restrictions on the type number of M are imposed. We derive (cf. our § 6) a «Simons type» formula and apply it to compact orientable hypersurfaces with non-negative sectional curvature (in a complex Hopf manifold) and parallel mean curvature vector. Several examples of submanifolds of l.c.K. manifolds are exhibited in § 3. Our § 7 studies complex submanifolds of generalized Hopf manifolds; for instance, we show that the first Chern class of the normal bundle of a complex submanifold having a flat normal connection is vanishing.     

Semi-parallel real hypersurfaces in complex two-plane Grassmannians

We prove that there does not exist any semi-parallel real hypersurface in complex two-plane Grassmannians. With this result, the nonexistence of recurrent real hypersurfaces in complex two-plane Grassmannians can also be proved.     

Real hypersurfaces in complex space forms concerned with the local symmetry

This paper consists of two parts. In the first, we find some geometric conditions derived from the local symmetry of the inverse image by the Hopf fibration of a real hypersurface M in complex space form M _m(4ε). In the second, we give a complete classification of real hypersurfaces in M _m(4ε) which satisfy the above geometric facts. The second author was supported by DGICYT research project BFM 2001-2871-C04-01 and the first and the third authors were supported by grant Proj. No. R14-2002-003-01001-0 from Korea Research Foundation, Korea 2006.     

On the complete linear Weingarten spacelike hypersurfaces with two distinct principal curvatures in Lorentzian space forms

We deal with complete linear Weingarten spacelike hypersurfaces immersed in a Lorentzian space form, having two distinct principal curvatures. In this setting, we show that such a spacelike hypersurface must be isometric to a certain isoparametric hypersurface of the ambient space, under suitable restrictions on the values of the mean curvature and of the norm of the traceless part of its second fundamental form. Our approach is based on the use of a Simons type formula related to an appropriated Cheng–Yau modified operator jointly with some generalized maximum principles.     

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 ${(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle ) }$ .     

A characterization of a certain real hypersurface of type (A<Subscript>2</Subscript>) in a complex projective space

In the class of real hypersurfaces M ^2n?1 isometrically immersed into a nonflat complex space form \(\widetilde {{M_n}}\left( c \right)\) of constant holomorphic sectional curvature c (≠ 0) which is either a complex projective space ?P ⁿ (c) or a complex hyperbolic space ?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 \(\widetilde {{M_n}}\left( c \right)\), we consider a certain real hypersurface of type (A₂) in ?P ⁿ (c) and give a geometric characterization of this Hopf manifold.     

Real hypersurfaces foliated by totally real totally geodesic submanifolds

We bridge between submanifold geometry and curve theory. In the first half of this paper we classify real hypersurfaces in a complex projective plane and a complex hyperbolic plane all of whose integral curves γ of the characteristic vector field are totally real circles of the same curvature which is independent of the choice of γ in these planes. In the latter half, we construct real hypersurfaces which are foliated by totally real (Lagrangian) totally geodesic submanifolds in a complex hyperbolic plane, which provide one of the examples obtained in the classification.     

Mean curvature flow of space-like Lagrangian submanifolds in almost para-K?hler manifolds

Given an almost para-K?hler manifold equipped with a metric and para-complex connection, we define a generalized second fundamental form and generalized mean curvature vector of space-like Lagrangian submanifolds. We then show that the deformation induced by this variant of the mean curvature vector field preserves the Lagrangian condition, if the connection satisfies also some Einstein condition. In case the almost para-K?hler structure is integrable, the flow coincides with the classical mean curvature flow in the pseudo-Riemannian context.     

Ricci Curvature of Real Hypersurfaces in Non-flat Complex Space Forms

We establish an inequality among the Ricci curvature, the squared mean curvature, and the normal curvature for real hypersurfaces in complex space forms. We classify real hypersurfaces in two-dimensional non-flat complex space forms which admit a unit vector field satisfying identically the equality case of the inequality.     

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).