Homogeneous Real Hypersurfaces Research Articles

Ruled real hypersurfaces in the complex hyperbolic quadric

Abstract In this article, we introduce a new family of real hypersurfaces in the complex hyperbolic quadric Q n ∗ = S O 2 , n o ∕ S O 2 S O n {{Q}^{n}}^{\ast }=S{O}_{2,n}^{o}/S{O}_{2}S{O}_{n} , namely, the ruled real hypersurfaces foliated by complex hypersurfaces. Berndt described an example of such a real hypersurface in Q n ∗ {{Q}^{n}}^{\ast } as a homogeneous real hypersurface generated by a A {\mathfrak{A}} -principal horocycle in a real form R H n {\mathbb{R}}{H}^{n} . So, in this article, we compute a detailed expression of the shape operator for ruled real hypersurfaces in Q n ∗ {{Q}^{n}}^{\ast } and investigate their characterizations in terms of the shape operator and the integrable distribution C = { X ∈ T M ∣ X ⊥ ξ } {\mathcal{C}}=\left\{X\in TM| X\perp \xi \right\} . Then, by using these observations, we give two kinds of classifications of real hypersurfaces in Q n ∗ {{Q}^{n}}^{\ast } satisfying η \eta -parallelism under either η \eta -commutativity of the shape operator or integrability of the distribution C {\mathcal{C}} . Moreover, we prove that the unit normal vector field of a real hypersurface with η \eta -parallel shape operator in Q n ∗ {{Q}^{n}}^{\ast } is A {\mathfrak{A}} -principal. On the other hand, it is known that all contact real hypersurfaces in Q n ∗ {{Q}^{n}}^{\ast } have a A {\mathfrak{A}} -principal normal vector field. Motivated by these results, we give a characterization of contact real hypersurfaces in Q n ∗ {{Q}^{n}}^{\ast } in terms of η \eta -parallel shape operator.

$$\phi $$-Sectional curvatures of homogeneous real hypersurfaces in a complex projective space

$$\phi $$-Sectional curvatures of homogeneous real hypersurfaces in a complex projective space

О 7-мерных алгебрах голоморфных векторных полей в $ \Bbb C^4 $, имеющих 5-мерный абелев идеал

In connection with the problem of describing holomorphically homogeneous real hypersurfaces in $ \Bbb C^4 $ we study in this article the 7-dimensional orbits of real Lie algebras in this space. By the well-known Morozov theorem, any nilpotent 7-dimensional Lie algebra has at least a 4-dimensional Abelian ideal. The article considers nilpotent indecomposable 7-dimensional Lie algebras containing a 5-dimensional Abelian ideal. It is proved that in the space $ \Bbb C^4 $ all the orbits of such algebras are Levi degenerate. This statement covers 73 algebras from the complete list of 149 indecomposable 7-dimensional nilpotent Lie algebras.

Decomposable Five-Dimensional Lie Algebras in the Problem on Holomorphic Homogeneity in ℂ3

In connection with the problem of describing holomorphically homogeneous real hypersurfaces in the space ℂ3, we study five-dimensional real Lie algebras realized as algebras of holomorphic vector fields on such manifolds. We prove the following assertion: If on a holomorphically homogeneous real hypersurface M of the space ℂ3, there is a decomposable, solvable, five-dimensional Lie algebra of holomorphic vector fields having a full rank near some point P ∈ M, then this surface is either degenerate near P in the sense of Levy or is a holomorphic image of an affine-homogeneous surface.

Sectional curvatures of homogeneous real hypersurfaces of types (A) and (B) in a complex projective space

Sectional curvatures of homogeneous real hypersurfaces of types (A) and (B) in a complex projective space

Holomorphically homogeneous real hypersurfaces in $\mathbb {C}^3$

We give a complete description and classification of locally homogeneous real hypersurfaces in $\mathbb C^3$. Various groups of mathematicians have been studying this problem in the last 25 years, and several significant classes of hypersurfaces under consideration have been studied and classified. The final results in the classification problem presented in this paper are obtained by using the classification of abstract 5-dimensional real Lie algebras, and by studying their representations by algebras of holomorphic vector fields in complex 3-space. The complete list of pairwise inequivalent hypersurfaces that we obtain contains 47 types of homogeneous hypersurfaces; some of the types are 1- or 2-parametric families, and each of the others is single hypersurface or a finite family of hypersurface.

On the Problem of Describing Holomorphically Homogeneous Real Hypersurfaces of Four-Dimensional Complex Spaces

We discuss two fragments of a large problem that extends the author’s recently completed similar studies in the space $$\mathbb C^3$$ to the next dimension. One of the fragments is related to the local description of nonspherical holomorphically homogeneous strictly pseudoconvex hypersurfaces in $$\mathbb C^4$$ with stabilizers of submaximal dimension. Using the Moser normal form technique and the properties of subgroups of the unitary group $$\mathrm U(3)$$ , we show that up to holomorphic equivalence there exist only two such surfaces. Both of them are natural generalizations of known homogeneous hypersurfaces in the space $$\mathbb C^3$$ . In the second part of the paper, we consider a technique of holomorphic realization in $$\mathbb C^4$$ of abstract seven-dimensional Lie algebras that correspond, in particular, to homogeneous hypersurfaces with trivial stabilizer. Some sufficient conditions for the Lie algebras are obtained under which the orbits of all realizations of such algebras are Levi degenerate. The schemes of studying holomorphically homogeneous hypersurfaces that were used in the two-dimensional (E. Cartan) and three-dimensional (Doubrov, Medvedev, and The; Fels and Kaup; Beloshapka and Kossovskiy; Loboda) situations and resulted in full descriptions of such hypersurfaces turn out to be quite efficient in the case of greater dimension of the ambient space as well.

On the Problem of Describing Holomorphically Homogeneous Real Hypersurfaces of Four-Dimensional Complex Spaces

Обсуждаются два фрагмента объемной задачи, являющейся продолжением по размерности недавно завершенных аналогичных исследований автора в пространстве $\mathbb C^3$. Один из фрагментов связан с локальным описанием несферических голоморфно однородных строго псевдовыпуклых гиперповерхностей в $\mathbb C^4$ с субмаксимальными по размерности стабилизаторами. С использованием техники нормальных форм Мозера и свойств подгрупп унитарной группы $\mathrm U(3)$ показано, что с точностью до голоморфной эквивалентности существуют всего две такие поверхности. Обе они являются естественными обобщениями известных однородных гиперповерхностей из пространства $\mathbb C^3$. Во второй части работы рассматривается техника голоморфной реализации в $\mathbb C^4$ абстрактных алгебр Ли размерности $7$, соответствующих, в частности, однородным гиперповерхностям, стабилизатор которых тривиален. Получены некоторые достаточные условия на обсуждаемые алгебры Ли, при которых орбиты всех реализаций таких алгебр являются вырожденными по Леви. Схемы изучения голоморфно однородных гиперповерхностей, позволившие получить их описания в двумерном (Э. Картан) и трехмерном (Дубров, Медведев, Ти; Фелс, Кауп; Белошапка, Коссовский; Лобода) случаях, оказываются вполне продуктивными при повышении размерности объемлющего пространства.

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

AbstractLet M be a Hopf hypersurface in a nonflat complex space form , , 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].

On orbits of action of 5-dimensional non-solvable Lie algebras in three-dimensional complex space

After the description by E. Cartan in 1932 holomorphically homogeneous real hypersurfaces of two-dimensional complex spaces, a similar study in the 3-dimensional case remains incomplete. In a series of works performed by several international teams of authors, the problem is reduced to describing homogeneous surfaces that are non-degenerate in Levi sense and have exactly 5-dimensional Lie algebras of holomorphic vector fields. In this paper, precisely such homogeneous surfaces are investigated. At the same time, a significant part of the extensive list of abstract 5-dimensional Lie algebras does not provide new examples of homogeneity. A complete description of the orbits of 5-dimensional non-solvable Lie algebras in a three-dimensional complex space, given in the paper, includes examples of new homogeneous hypersurfaces. The presented results bring to finish a large-scale scientific study of interest to various branches of mathematics.

On Holomorphic Realizations of Nilpotent Lie Algebras

Realizations of five-dimensional Lie algebras as algebras of holomorphic vector fields on homogeneous real hypersurfaces of a three-dimensional complex space are studied. In view of already known results, in the problem of describing such varieties only Levy nondegenerate hypersurfaces with exactly five-dimensional Lie algebras are of interest. It is shown that only two of the nine existing distinct nilpotent Lie algebras admit realizations associated with such varieties, and the varieties corresponding to these exceptional algebras are standard quadrics.

English

We study homogeneous real hypersurfaces having no focal submanifolds in a complex hyperbolic space. They are called Lie hypersurfaces in this space. We clarify the geometry of Lie hypersurfaces in terms of their sectional curvatures, the behavior of the characteristic vector field and their holomorphic distributions.

Curvature properties of homogeneous real hypersurfaces in nonflat complex space forms

In this paper, we study curvature properties of all homogeneous real hypersurfaces in nonflat complex space forms, and determine their minimalities and the signs of their sectional curvatures completely. These properties reflect the sign of the constant holomorphic sectional curvature $c$ of the ambient space. Among others, for the case of $c$ < 0 there exist homogeneous real hypersurfaces with positive sectional curvature and also ones with negative sectional curvature, whereas for the case of $c$ > 0 there do not exist any homogeneous real hypersurfaces with nonpositive sectional curvature.

Characterizations of three homogeneous real hypersurfaces in a complex projective space

In an $n$-dimensional complex hyperbolic space $\mathbb{C}H^n(c)$ of constant holomorphic sectional curvature $c (\lt 0)$, the horosphere HS, which is defined by ${\rm HS} = \lim_{r\to\infty}G(r)$, is one of nice examples in the class of real hypersurfaces. Here, $G(r)$ is a geodesic sphere of radius $r$ $(0 \lt r \lt \infty)$ in $\mathbb{C}H^n(c)$. The second author ([14]) gave a geometric characterization of HS. In this paper, motivated by this result, we study real hypersurfaces $M^{2n-1}$ isometrically immersed into an $n$-dimensional complex projective space $\mathbb{C}P^n(c)$ of constant holomorphic sectional curvature $c(\gt 0)$.

Realizations of some contact metric manifolds as Ricci soliton real hypersurfaces

Ricci soliton contact metric manifolds with certain nullity conditions have recently been studied by Ghosh and Sharma. Whereas the gradient case is well-understood, they provided a list of candidates for the nongradient case.These candidates can be realized as Lie groups, but one only knows the structures of the underlying Lie algebras, which are hard to be analyzed apart from the three-dimensional case. In this paper, we study these Lie groups with dimension greater than three, and prove that the connected, simply-connected, and complete ones can be realized as homogeneous real hypersurfaces in noncompact real two-plane Grassmannians. These realizations enable us to prove, in a Lie-theoretic way, that all of them are actually Ricci soliton.

Об аффинно-однородных вещественных гиперповерхностях общего положения в C3

Об аффинно-однородных вещественных гиперповерхностях общего положения в C3

A characterization of homogeneous real hypersurfaces of types (C), (D) and (E) in a complex projective space

We give a necessary and sufficient condition that a Riemannian manifold M2n−1 isometrically immersed into a complex projective space CPn(c) is locally congruent to one of homogeneous real hypersurfaces of types (C), (D) and (E) in this ambient space in terms of its restricted principal curvature distributions.

A characterization of the homogeneous ruled real hypersurface in a complex hyperbolic space in terms of the first curvature of some integral curves

Due to the works of Berndt and others (J Reine Angew Math 395:132–141, 1989; Geom Dedicata 138:129–150, 2009; Trans Am Math Soc 359:3425–3438, 2007), we find that the class of all homogeneous real hypersurfaces M in \({\mathbb{C}H^n(c)}\) has just one example which is minimal in this space. Furthermore, in this ambient space, a homogeneous real hypersurface M is minimal if and only if it is ruled. This fact implies that it is interesting to study this minimal homogeneous real hypersurface from the viewpoint of the geometry of ruled real hypersurfaces. It is known that the shape operator A of every ruled real hypersurface M is given by the characteristic field \({\xi}\) and the vector field U. The purpose of this paper is to characterize this minimal homogeneous real hypersurface in \({\mathbb{C}H^n(c)}\) in the class of all ruled real hypersurfaces M by investigating the first curvature of all integral curves of the vector fields \({\xi}\) and U. Note that there exist minimal non-homogeneous ruled real hypersurfaces in \({\mathbb{C}H^n(c)}\) (see Geom Dedicata 79:267–286, 1999; Hokkaido Math J 43:1–14, 2014).

Reeb parallel Ricci tensor for homogeneous real hypersurfaces in complex hyperbolic two‐plane Grassmannians

In this paper, we introduce the notion of Reeb parallel Ricci tensor for homogeneous real hypersurfaces in complex hyperbolic two‐plane Grassmannians which has a remarkable geometric structure as a Hermitian symmetric space of rank 2. By using a new method of simultaneous diagonalizations, we give a complete classification for real hypersurfaces in complex hyperbolic two‐plane Grassmannians with the Reeb parallel Ricci tensor.

Naturally reductive homogeneous real hypersurfaces in a nonflat complex space form

In this paper, in the class of homogeneous real hypersurfaces in a nonflat complex space form we study naturally reductive Riemannian homogeneous spaces having nonnegative sectional curvature.