Indefinite Complex Space Form Research Articles

Complex submanifolds of indefinite complex space forms

In this short paper, we derive a new result on Umehara algebra. As a consequence, we prove that an indefinite complex hyperbolic space and an indefinite complex projective space do not share a common complex submanifold with induced metrics, answering a question raised in Cheng et al.

Umehara algebra and complex submanifolds of indefinite complex space forms

The Umehara algebra is studied with motivation on the problem of the non-existence of common complex submanifolds. In this paper, we prove some new results in Umehara algebra and obtain some applications. In particular, if a complex manifolds admits a holomorphic polynomial isometric immersion to one indefinite complex space form, then it cannot admits a holomorphic isometric immersion to another indefinite complex space form of different type. Other consequences include the non-existence of the common complex submanifolds for indefinite complex projective space or hyperbolic space and a complex manifold with a distinguished metric, such as homogeneous domains, the Hartogs triangle, the minimal ball, the symmetrized polydisc, etc, equipped with their intrinsic Bergman metrics, which generalizes more or less all existing results.

Generic lightlike submanifolds of an indefinite Kaehler manifold with an $(\ell, m)$-type metric connection

We study generic lightlike submanifolds $M$ of an indefinite Kaehler manifold $\bar{M}$ or an indefinite complex space form $\bar{M}(c)$ with an $(\ell, m)$-type metric connection subject such that the characteristic vector field $\zeta$ of $\bar{M}$ belongs to our screen distribution $S(TM)$ of $M$.

Kähler immersions of Kähler–Ricci solitons into definite or indefinite complex space forms

Let $(g, X)$ be a K\"ahler-Ricci soliton on a complex manifold $M$. We prove that if the K\"ahler manifold $(M, g)$ can be K\"ahler immersed into a definite or indefinite complex space form of constant holomorphic sectional curvature $2c$, then $g$ is Einstein. Moreover, its Einstein constant is a rational multiple of $c$.

Lightlike Hypersurfaces of an Indefinite Kaehler Manifold with an (<img src=image/13415329_01.gif>)-type Connection

Jin [1] defined an ( )-type connection on semi-Riemannian manifolds. Semi-symmetric nonmetric connection and non-metric ∅-symmetric connection are two important examples of this connection such that ( ) = (1; 0) and ( ) = (0; 1), respectively. In semi-Riemannian geometry, there are few literatures for the lightlike geometry, so we expose new theories for non-degenerate submanifolds in semi-Riemannian geometry. The goal of this paper is to study a characterization of a (Lie) recurrent lightlike hypersurface M of an indefinite Kaehler manifold with an ( )-type connection when the charateristic vector field is tangnet to M. In the special case that an indefinite Kaehler manifold of constant holomorphic sectional curvature is an indefinite complex space form, we investigate a lightlike hypersurface of an indefinite complex space form with an ( )-type connection when the charateristic vector field is tangnet to M. Moreover, we show that the total space, the complex space form, is characterized by the screen conformal lightlike hypersurface with an ( )-type connection. With a semi-symmetric non-metric connection, we show that an indefinite complex space form is flat.

Kähler submanifolds and the Umehara algebra

We show that the indefinite complex space form [Formula: see text] is not a relative to the indefinite complex space form [Formula: see text] or [Formula: see text]. We further study whether two Fubini–Study spaces are relatives or not.

HALF LIGHTLIKE SUBMANIFOLDS OF AN INDEFINITE KAEHLER MANIFOLD WITH A SEMI-SYMMETRIC NON-METRIC CONNECTION

In this paper, we study half lightlike submanifolds of an indefinite Kaehler manifold with a semi-symmetric non-metric connection. First, we characterize the geometry of two types of half lightlike submanifolds of such an indefinite Kaehler manifold. Next, we investigate the geometry of half lightlike submanifolds of an indefinite complex space form with a semi-symmetric non-metric connection.

Axioms of spheres in lightlike geometry of submanifolds

We prove that if an indefinite Kaehler manifold $\bar {M}$ with lightlike submanifolds satisfies the axioms of holomorphic 2r-spheres, axioms of holomorphic 2r-planes, axioms of transversal r-spheres and axioms of transversal r-planes, then it is an indefinite complex space form.

Lorentzian isotropic Lagrangian immersions

In this note we are interested in isotropic totally real Lorentzian submanifolds of indefinite complex space forms. We show that such submanifolds are always H-umbilical warped product immersions and we determine also the warping function.

Minimal Lagrangian isotropic immersions in indefinite complex space forms

Let Mn be an n-dimensional (n≥3) minimal Lagrangian isotropic submanifold in an indefinite complex space form. We show that the dimension of Mn satisfies n=3r+2 with r, a positive integer. When n<14, we give a classification of such submanifolds.

Characterization of Holomorphic Bisectional Curvature ofGCR-Lightlike Submanifolds

We obtain the expressions for sectional curvature, holomorphic sectional curvature, and holomorphic bisectional curvature of aGCR-lightlike submanifold of an indefinite Kaehler manifold. We discuss the boundedness of holomorphic sectional curvature ofGCR-lightlike submanifolds of an indefinite complex space form. We establish a condition for aGCR-lightlike submanifold of an indefinite complex space form to be null holomorphically flat. We also obtain some characterization theorems for holomorphic sectional and holomorphic bisectional curvature.

On CR‐Lightlike Product of an Indefinite Kaehler Manifold

We have studied mixed foliate CR‐lightlike submanifolds and CR‐lightlike product of an indefinite Kaehler manifold and also obtained relationship between them. Mixed foliate CR‐lightlike submanifold of indefinite complex space form has also been discussed and showed that the indefinite Kaehler manifold becomes the complex semi‐Euclidean space.

LIGHTLIKE REAL HYPERSURFACES WITH TOTALLY UMBILICAL SCREEN DISTRIBUTIONS

In this paper, we study the geometry of lightlike real hyper-surfaces of an indefinite Kaehler manifold. The main result is a characterization theorem for lightlike real hypersurfaces M of an indefinite complex space form <TEX>$\bar{M}(c)$</TEX> such that the screen distribution is totally umbilic.

GEOMETRY OF SCREEN CONFORMAL REAL HALF LIGHTLIKE SUBMANIFOLDS

In this paper, we study the geometry of real half lightlike submanifolds of an indefinite Kaehler manifold. The main result is a characterization theorem for screen conformal real half lightlike submanifolds of an indefinite complex space form.

SCREEN CONFORMAL LIGHTLIKE REAL HYPERSURFACES OF AN INDEFINITE COMPLEX SPACE FORM

In this paper, we study the geometry of screen conformal lightlike real hypersurfaces of an indefinite Kaehler manifold. The main result is a characterization theorem for screen conformal lightlike real hypersurfaces of an indefinite complex space form.

Minimal Lagrangian submanifolds with constant sectional curvature in indefinite complex space forms

We study minimal Lagrangian immersions from an indefinite real space form M s n ( c ) M^n_s(c) into an indefinite complex space form M ~ s n ( 4 c ~ ) {\tilde {\mathbb {M}}}^n_s(4\tilde c) . Provided that c ≠ c ~ c \ne \tilde c , we show that M s n ( c ) M^n_s(c) has to be flat and we obtain an explicit description of the immersion. In the case when the metric is positive definite or Lorentzian, this result was respectively obtained by Ejiri (1982) and by Kriele and the author (1999). In the case that c = c ~ c = \tilde c , this theorem is no longer true; see for instance the examples discovered by Chen and the author (accepted for publication in the Tôhoku Mathematical Journal).

Real hypersurfaces of indefinite Kaehler manifolds

We show the existence of ( ϵ )-almost contact metric structures and give examples of ( ϵ )-Sasakian manifolds. Then we get a classification theorem for real hypersurfaces of indefinite complex space-forms with parallel structure vector field. We prove that ( ϵ )-Sasakian real hypersurfaces of a semi-Euclidean space are either open sets of the pseudosphereS2S2n+1(1) or of the pseudohyperbolic spaceH2S−12n+1(1). Finally, we get the causal character of ( ϵ ) cosymplectic real hypersurfaccs of indefinite complex space-forms.

Product complex submanifolds on of indefinite complex space forms

Product complex submanifolds on of indefinite complex space forms

Semi-Kaehlerian submanifolds of an indefinite complex space form

The purpose of this paper is to study several classes of semi-Kaehlerian submanifolds of an indefinite complex space form. Introduction. An indefinite Kaehlerian manifold of constant holomorphic sectional curvature is called an indefinite complex space form. Montiel and Romero [9] investigated indefinite complex Einstein hypersurfaces of an indefinite complex space forms and showed that totally geodesic indefinite complex hypersurfaces Pί(c'), C?, H(—c) and an indefinite complex quadric Qf are those examples. Ikawa, Romero and one of present authors [6] have also shown that by using an indefinite Segre imbedding there exists a product of complex hyperbolic spaces which becomes an example of space-like Einstein-Kaehlerian submanifolds of an indefinite complex hyperbolic space. Recently, concerning with the study of Calabi's classification [5] for Kaehlerian imbeddings of complex space forms into complex space forms, Romero [18] and Umehara [21] have independently found that there exists a strongly full holomorphic isometric immersion of indefinite complex space forms into indefinite complex space forms. From this point of view the purpose of this paper is to study several classes of complete semi-Kaehlerian submanifolds of an indefinite complex space form M#nc' In the first section, the brief summary of indefinite complex submanifolds of an indefinite Kaehlerian manifold are recalled. The examples of space-like complex Einstein submanifolds of indefinite complex space forms are given in § 2. § 3 is devoted to the study of the spacelike complex submanifolds with constant scalar curvature of M%(c'). In particular, by estimating the scalar curvature and by using Nishikawa's theorem [12], we shall characterize space-like Einstein submanifolds in the case of c'<0. Received December 8, 1987

Mixed foliateCR-submanifolds of indefinite complex space-forms

In this paper we have studied a class of mixed foliate CR-submanifolds of a complex space-form carrying an indefinite metric. Further study has been carried out by assuming flat normal connection and parallelizable totally real distribution.