On the structure vector field of a real hypersurface in complex two-plane Grassmannians

Considering the notion of Jacobi type vector fields for a real hypersurface in a complex two-plane Grassmannian, we prove that if a structure vector field is of Jacobi type it is Killing. As a consequence we classify real hypersurfaces whose structure vector field is of Jacobi type.     

Real hypersurfaces in complex projective space whose structure Jacobi operator is Lie ξ-parallel

We classify real hypersurfaces in complex projective spaces whose structure Jacobi operator is Lie parallel in the direction of the structure vector field.     

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 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.     

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 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 ) }.     

Minimal and Harmonic Unit Vector Fields in and Its Dual Space

 The complex two-plane Grassmannian carries a K?hler structure J and also a quaternionic K?hler structure ?. For we consider the classes of connected real hypersurfaces (M, g) with normal bundle such that and are invariant under the action of the shape operator. We prove that the corresponding unit Hopf vector fields on these hypersurfaces always define minimal immersions of (M, g), and harmonic maps from (M, g), into the unit tangent sphere bundle with Sasaki metric . The radial unit vector fields corresponding to the tubular hypersurfaces are also minimal and harmonic. Similar results hold for the dual space .     

Minimal and Harmonic Unit Vector Fields in and Its Dual Space

 The complex two-plane Grassmannian carries a K?hler structure J and also a quaternionic K?hler structure ?. For we consider the classes of connected real hypersurfaces (M, g) with normal bundle such that and are invariant under the action of the shape operator. We prove that the corresponding unit Hopf vector fields on these hypersurfaces always define minimal immersions of (M, g), and harmonic maps from (M, g), into the unit tangent sphere bundle with Sasaki metric . The radial unit vector fields corresponding to the tubular hypersurfaces are also minimal and harmonic. Similar results hold for the dual space . (Received 27 August 1999; in revised form 18 November 1999)     

Congruence of hypersurfaces inS 6 and inC P n

The invariants needed to decide when a pair of hypersurfaces ofS ⁶ orCP ⁿ are respectivelyG ₂-congruent or holomorpic congruent are determined and this result is used to characterize the hypersurfaces of these spaces whose Hopf vector fields are also Killing fields.     

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.      

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     

Liouville theorem,conformally invariant cones and umbilical surfaces for Grushin-type metrics

We prove a classification theorem for conformal maps with respect to the control distance generated by a system of diagonal vector fields in ℝ ⁿ . It turns out that in many cases all such maps can be obtained as compositions of suitable dilations, inversions and isometries. Our methods involve a study of the singular Riemannian metric associated with the vector fields. In particular, we identify some conformally invariant cones related to the Weyl tensor. The knowledge of such cones enables us to classify all umbilical hypersurfaces.     

An Optimal Inequality and Extremal Classes of Affine Spheres in Centroaffine Geometry

A hypersurface f : M → Rⁿ⁺¹ in an affine (n+1)-space is called centroaffine if its position vector is always transversal to f_*(TM) in Rⁿ⁺¹. In this paper, we establish a general optimal inequality for definite centroaffine hypersurfaces in Rⁿ⁺¹ involving the Tchebychev vector field. We also completely classify the hypersurfaces which verify the equality case of the inequality.     

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.     

On the structure Jacobi operator of a real hypersurface in complex projective space

We prove the non-existence of a certain family of real hypersurfaces in complex projective space. From this result we classify real hypersurfaces whose structure Jacobi operator satisfies a condition that generalizes parallelness.     

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.     

An extension of Takahashi's theorem

A classical result of T. Takahashi [8] is generalized to the case of hypersurfaces in the Euclidean space E ^m . More concretely, we classify Euclidean hypersurfaces whose coordinate functions in E ^m are eigenfunctions of their Laplacian.Partially supported by a CAICYT Grant PR84-1242-C02-02 Spain.     

Weakly Convex Biharmonic Hypersurfaces in Nonpositive Curvature Space Forms are Minimal

A submanifold M ^m of a Euclidean space R ^m+p is said to have harmonic mean curvature vector field if ${\Delta \vec{H}=0}$ , where ${\vec{H}}$ is the mean curvature vector field of ${M\hookrightarrow R^{m+p}}$ and Δ is the rough Laplacian on M. There is a famous conjecture named after Bangyen Chen which states that submanifolds of Euclidean spaces with harmonic mean curvature vector fields are minimal. In this paper we prove that weakly convex hypersurfaces (i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic mean curvature vector fields in Euclidean spaces are minimal. Furthermore we prove that weakly convex biharmonic hypersurfaces in nonpositively curved space forms are minimal.     

Curvature surfaces of Hopf hypersurfaces in complex space forms

We study the principal curvatures of a Hopf hypersurfaceM in ℂP ⁿ or ℂH ⁿ . The respective eigenspaces of the shape operator often turn out to induce totally real foliations ofM, whose leaves are spherical in the ambient space. Finally we classify the Hopf hypersurfaces with three distinct principal curvatures satisfying a certain non-degeneracy condition.