Hopf hypersurfaces in pseudo-Riemannian complex and para-complex space forms

The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denoting by J be the complex or para-complex structure of a pseudo-complex or para-complex space form respectively, a non-degenerate hypersurface of such space with unit normal vector field N is said to be Hopf if the tangent vector field JN is a principal direction. It is proved that if a hypersurface is Hopf, then the corresponding principal curvature (the Hopf curvature) is constant. It is also observed that in some cases a Hopf hypersurface must be, locally, a tube over a complex (or para-complex) submanifold, thus generalizing previous results of Cecil, Ryan and Montiel.     

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 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.     

On conformally flat hypersurfaces,curved flats and cyclic systems

It will be shown that suitable “Gauß maps” associated to a conformally flat hypersurface inS ⁿ⁺¹ (n≥3) yield normal congruences of circles having a whole 1-parameter family of conformally flat orthogonal hypersurfaces. However such a “cyclic system” is not uniquely associated to a conformally flat hypersurface. The key idea is to show that these Gauß maps are “curved flats” in a pseudo Riemannian symmetric space. Additionally, in this context some characterizations of 3-dimensional conformally flat hypersurfaces arise with a new flavour. The curved flat approach allows us to handle conformally flat hypersurfaces in the context of integrable system theory.     

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 in a complex space form with a condition on the structure Jacobi operator

In this paper we classify the real hypersurfaces in a non-flat complex space form with its structure Jacobi operator R _ξ satisfying (? _X R _ξ )ξ = 0, for all vector fields X in the maximal holomorphic distribution D. With this result, we prove the non-existence of real hypersurfaces with D-parallel as well as D-recurrent structure Jacobi operator in complex projective and hyperbolic spaces. We can also prove the non-existence of real hypersurfaces with recurrent structure Jacobi operator in a non-flat complex space form as a corollary.     

Real hypersurfaces of a complex projective space

In this paper, we classify real hypersurfaces in the complex projective space C P^\fracn+12C P^{\frac{n+1}{2}} whose structure vector field is a φ-analytic vector field (a notion similar to analytic vector fields on complex manifolds). We also define Jacobi-type vector fields on a Riemannian manifold and classify real hypersurfaces whose structure vector field is a Jacobi-type vector field.     

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.     

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 with constant totally real bisectional curvature in complex space forms

In this paper we classify real hypersurfaces with constant totally real bisectional curvature in a non flat complex space form M _m (c), c ≠ 0 as those which have constant holomorphic sectional curvature given in [6] and [13] or constant totally real sectional curvature given in [11].     

Classification of hypersurfaces with two distinct principal curvatures and closed M?bius form in \mathbb{S}^{m + 1}

Let x be an m-dimensional umbilic-free hypersurface in an (m + 1)-dimensional unit sphere Sm+1 (m≥3). In this paper, we classify and explicitly express the hypersurfaces with two distinct principal curvatures and closed Mbius form, and then we characterize and classify conformally flat hypersurfaces of dimension larger than 3.     

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.     

On a family of lie algebras related to homogeneous surfaces

Real affine homogeneous hypersurfaces of general position in three-dimensional complex space ?³ are studied. The general position is defined in terms of the Taylor coefficients of the surface equation and implies, first of all, that the isotropy groups of the homogeneous manifolds under consideration are discrete. It is this case that has remained unstudied after the author’s works on the holomorphic (in particular, affine) homogeneity of real hypersurfaces in three-dimensional complex manifolds. The actions of affine subgroups G ? Aff(3, ?) in the complex tangent space T ^? _p M of a homogeneous surface are considered. The situation with homogeneity can be described in terms of the dimensions of the corresponding Lie algebras. The main result of the paper eliminates “almost trivial” actions of the groups G on the spaces T _p ^? M for affine homogeneous strictly pseudoconvex surfaces of general position in ?³ that are different from quadrics.     

Real hypersurfaces in a complex space form with ��-parallel shape operator

In this paper, we classify the real hypersurfaces in a non-flat complex space form with ??-parallel shape operator.     

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.     

Completeness of curvature surfaces of submanifolds in complex space forms

We assume that on an open subset of a submanifold M of an arbitrary Riemannian ambient space N the eigenspaces of the shape operator of M induce a foliation L whose leaves are spherical submanifolds of N. In this situation we derive a condition which characterizes when the leaves of L are complete Riemannian submanifolds of M (see Theorem 2.4). We apply this result to real hypersurfaces of complex space forms, in particular Hopf hypersurfaces (see Theorem 3.2 and Proposition 3.3).     

Biconservative surfaces in BCV‐spaces

Biconservative hypersurfaces are hypersurfaces with conservative stress‐energy tensor with respect to the bienergy functional, and form a geometrically interesting family which includes that of biharmonic hypersurfaces. In this paper we study biconservative surfaces in the 3‐dimensional Bianchi–Cartan–Vranceanu spaces, obtaining their characterization in the following cases: when they form a constant angle with the Hopf vector field; when they are SO(2)‐invariant.     

The Hartogs phenomenon on weakly pseudoconcave hypersurfaces

In 2002, Henkin and Michel proved a local Hartogs phenomenon for real analytic CR functions on real analytic weakly pseudoconcave CR manifolds. The aim of the present article is to remove the assumptions on real analyticity in the case of weakly pseudoconcave hypersurfaces ${M\subset\mathbb{C}^n}$ . If M is a graph of class ${\mathcal{C}^2}$ and n??? 3, a global theorem is proved for the extension of holomorphic germs along M. If the appearing domains have nicely shaped boundary, a Hartogs theorem even holds for continuous CR functions, where the difference to the case of holomorphic germs relies on the possible presence of lower-dimensional CR orbits. Levi flat hypersurfaces in ${\mathbb{C}^2}$ require a separate treatment. Here an affirmative answer is given to the question of Tomassini, whether 2-spheres bound 3-balls in M.