Indefinite Space Form Research Articles

Some remarks on invariant lightlike submanifolds of indefinite Sasakian manifold

PurposeThe author considers an invariant lightlike submanifold M, whose transversal bundle tr(TM) is flat, in an indefinite Sasakian manifold M¯(c) of constant φ¯-sectional curvature c. Under some geometric conditions, the author demonstrates that c=1, that is, M¯ is a space of constant curvature 1. Moreover, M and any leaf M′ of its screen distribution S(TM) are, also, spaces of constant curvature 1.Design/methodology/approachThe author has employed the techniques developed by K. L. Duggal and A. Bejancu of reference number 7.FindingsThe author has discovered that any totally umbilic invariant ligtlike submanifold, whose transversal bundle is flat, in an indefinite Sasakian space form is, in fact, a space of constant curvature 1 (see Theorem 4.4).Originality/valueTo the best of the author’s findings, at the time of submission of this paper, the results reported are new and interesting as far as lightlike geometry is concerned.

On pseudo-umbilical spacelike submanifolds in indefinite space form $$\mathcal {M}_p^{n+p}(c)$$

An intrinsic condition for Pseudo-umbilical spacelike submanifold immersed in indefinite space form is derived, which is used to show that such submanifold is totally geodesic. Next, using a result of Aiyama (Tokyo J Math 18:81–90, 1995), it is proved that Pseudo-umbilical spacelike submanifold is totally umbilical.

Characterizations of the Total Space (Indefinite Trans-Sasakian Manifolds) Admitting a Semi-Symmetric Metric Connection

We investigate recurrent, Lie-recurrent, and Hopf lightlike hypersurfaces of an indefinite trans-Sasakian manifold with a semi-symmetric metric connection. In these hypersurfaces, we obtain several new results. Moreover, we characterize that the total space (an indefinite generalized Sasakian space form) with a semi-symmetric metric connection is an indefinite Kenmotsu space form under various lightlike hypersurfaces.

Willmore spacelike submanifolds in an indefinite space form Nn+p q(c)

Let Nn+p q(c) be an (n+p)-dimensional connected indefinite space form of index q(1 ? q ? p) and of constant curvature c. Denote by ? : M ? Nn+p q (c) the n-dimensional spacelike submanifold in Nn+p q (c), ? : M ? Nn+p q(c) is called a Willmore spacelike submanifold in Nn+p q(c) if it is a critical submanifold to the Willmore functional W(?) = ?q M ?n dv =?M (S-nH2)n/2 dv, where S and H denote the norm square of the second fundamental form and the mean curvature of M and ?2 = S ? nH2. If q = p, in [14], we proved some integral inequalities of Simons? type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in a Lorentzian space form Nn+p q(c). In this paper, we continue to study this topic and prove some integral inequalities of Simons? type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in an indefinite space form Nn+p q(c) (1 ? q ? p).

Spacelike submanifolds with parallel mean curvature vector in S q n+p (1)

Let Mn be a complete spacelike submanifold in an indefinite space form Sqn+p(1). When p = q, there are numerous rigidity results concerning submanifolds with parallel mean curvature vector. However, there are few results concerning the case q < p. In this paper, we will focus on this kind of submanifold and give some classifications of submanifolds with parallel mean curvature vector according to the squared norm of the second fundamental form.

Classification of Lorentz surfaces with parallel mean curvature vector in non-flat pseudo-Riemannian space forms $S_2^4(1)$ and $H_2^4(-1)$

Lorentz surfaces with parallel mean curvature vector in $\mathbb{E}_2^4$ have been classified in [14]. In this paper, we continue to classify Lorentz surfaces with parallel mean curvature vector in pseudo-Riemannian space forms $S_2^4(1)$ and $H_2^4(-1)$. Consequently, we achieve the complete classification of Lorentz surfaces with parallel mean curvature vector in 4-dimensional neutral indefinite space form with index 2.

On Parallelism of Half-Lightlike Submanifolds of Indefinite Kenmotsu Manifolds

We mainly investigate the parallelism of half-lightlike submanifolds of indefinite Kenmotsu manifolds. It is proved that a tangential half-lightlike submanifold M of an indefinite Kenmotsu space form M-c,g- with semiparallel second fundamental form h either satisfies c=-1 or is J-RadTM,TM-mixed geodesic.

Nonexistence of Totally Contact Umbilical Slant Lightlike Submanifolds of Indefinite Cosymplectic Manifolds

We study totally contact umbilical slant lightlike submanifolds of indefinite cosymplectic manifolds. We prove that there do not exist totally contact umbilical proper slant lightlike submanifolds in indefinite cosymplectic manifolds other than totally contact geodesic proper slant lightlike submanifolds. We also prove that there do not exist totally contact umbilical proper slant lightlike submanifolds of indefinite cosymplectic space forms. Finally we give characterization theorems on minimal slant lightlike submanifolds.

Non Existence of Totally Contact Umbilical GCR-Lightlike Submanifolds of Indefinite Cosymplectic Manifolds

We introduce GCR-lightlike submanifolds of indefinite Cosymplectic manifolds and give an example. We also study the existence of this class in indefinite Cosymplectic space form. Then we prove that there does not exist a totally contact umbilical GCR-lightlike submanifold of an indefinite Cosymplectic manifold other than totally contact geodesic GCR-lightlike submanifold and moreover it is a totally geodesic GCR-lightlike submanifold.

On spacelike submanifolds with parallel mean curvature in an indefinite space form

In this work we use a Simon’s type inequality for a suitable tensor and apply it to obtain sharp estimates for the supremum of the scalar curvature for complete spacelike submanifold Mn with parallel mean curvature vector in an indefinite space form \({N^{n+p}_p(c)}\).

On the Geometry of Lightlike Submanifolds

We give some characterizations of non-existence of lightlike hypersurfaces of an indefinite space form. Some geometric objects for the induced Ricci tensor to be symmetric are studied.

Wintgen Ideal Surfaces in Four-dimensional Neutral Indefinite Space Form $${R^4_2(c)}$$

For an oriented space-like surface M in a four-dimensional indefinite space form\({R^4_2(c)}\), there is a Wintgen type inequality; namely, the Gauss curvature K, the normal curvature KD and mean curvature vector H of M in \({R^4_2(c)}\) satisfy the general inequality: \({K+K^D \geq \langle H,H \rangle+c}\). An oriented space-like surface in \({R^4_2(c)}\) is called Wintgen ideal if it satisfies the equality case of the inequality identically. In this paper, we study Wintgen ideal surfaces in \({R^4_2(c)}\) . In particular, we classify Wintgen ideal surfaces in \({R^4_2(c)}\) with constant Gauss and normal curvatures. We also completely classify Wintgen ideal surfaces in \({\mathbb E^4_2}\) satisfying |K| = |KD| identically.

Classification of minimal Lorentz surfaces in indefinite space forms with arbitrary codimension and arbitrary index

Since J. L. Lagrange initiated in 1760 the study of minimal surfaces of Euclidean 3-space, minimal surfaces in real space forms have been studied extensively by many mathematicians during the last two and half centuries. In contrast, so far very few results on minimal Lorentz surfaces in indefinite space forms are known. Hence, in this paper we investigate minimal Lorentz surfaces in arbitrary indefinite space forms. As a consequence, we obtain several classification results for minimal Lorentz surfaces in indefinite space forms. In particular, we completely classify all minimal Lorentz surfaces in a pseudo-Euclidean space $\mathbb E^m_s$ with arbitrary dimension $m$ and arbitrary index $s$.

Complete classification of parallel Lorentz surfaces in four-dimensional neutral pseudosphere

A Lorentz surface of an indefinite space form is called parallel if its second fundamental form is parallel with respect to the Van der Waerden–Bortolotti connection. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in general relativity since extrinsic invariants of such surfaces do not change from point to point. Parallel Lorentz surfaces in four-dimensional (4D) Lorentzian space forms are classified by Chen and Van der Veken [“Complete classification of parallel surfaces in 4-dimensional Lorentz space forms,” Tohoku Math. J. 61, 1 (2009)]. Recently, explicit classification of parallel Lorentz surfaces in the pseudo-Euclidean 4-space E24 and in the pseudohyperbolic 4-space H24(−1) are obtained recently by Chen et al. [“Complete classification of parallel Lorentzian surfaces in Lorentzian complex space forms,” Int. J. Math. 21, 665 (2010); “Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space,” Cent. Eur. J. Math. 8, 706 (2010)], respectively. In this article, we completely classify the remaining case; namely, parallel Lorentz surfaces in 4D neutral pseudosphere S24(1). Our result states that there are 24 families of such surfaces in S24(1). Conversely, every parallel Lorentz surface in S24(1) is obtained from one of the 24 families. The main result indicates that there are major differences between Lorentz surfaces in the de Sitter 4-space dS4 and in the neutral pseudo 4-sphere S24.

Complete explicit classification of parallel Lorentz surfaces in arbitrary pseudo-Euclidean spaces

A Lorentz surface of an indefinite space form is called parallel if its second fundamental form is parallel. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in physics since extrinsic invariants of such surfaces do not change from point to point. Recently, parallel spacelike surfaces in an arbitrary indefinite space form are classified in Chen (2010) [20] . Moreover, parallel Lorentz surfaces in 4D indefinite space forms are completely classified in a series of recent articles Chen (submitted for publication) [16] , Chen (submitted for publication) [17] , Chen (in press) [18] , Chen (2010) [19] , Chen and Van der Veken (2009) [15] (see also Graves (1979) [12] , Graves (1979) [13] and Magid (1984) [14] for some partial results). In this paper, we achieve the complete classification of parallel Lorentz surfaces in a pseudo-Euclidean space with arbitrary codimension and arbitrary index.

Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space

AbstractA Lorentz surface of an indefinite space form is called a parallel surface if its second fundamental form is parallel with respect to the Van der Waerden-Bortolotti connection. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in general relativity since extrinsic invariants of such surfaces do not change from point to point. Recently, parallel Lorentz surfaces in 4D neutral pseudo Euclidean 4-space $$ \mathbb{E}_2^4 $$ and in neutral pseudo 4-sphere S 24 (1) were classified in [14] and in [10], respectively. In this paper, we completely classify parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space H 24 (−1). Our main result states that there are 53 families of parallel Lorentz surfaces in H 24 (−1). Conversely, every parallel Lorentz surface in H 24 (−1) is obtained from the 53 families. As an immediate by-product, we achieve the complete classification of all parallel Lorentz surfaces in 4D neutral indefinite space forms.

COMPLETE CLASSIFICATION OF LORENTZ SURFACES WITH PARALLEL MEAN CURVATURE VECTOR IN ARBITRARY PSEUDO-EUCLIDEAN SPACE

Surfaces with parallel mean curvature vector play important roles in the theory of harmonic maps, differential geometry as well as in physics. Surfaces with parallel mean curvature vector in Riemannian space forms were classified in the early 1970s by Chen and Yau. Recently, space-like surfaces with parallel mean curvature vector in arbitrary indefinite space forms were completely classified by Chen in two papers in 2009. In this paper, we completely classify Lorentz surfaces with parallel mean curvature vector in a pseudo-Euclidean space Ems with arbitrary dimension m and arbitrary index s. Our main result states that there are 23 families of Lorentz surfaces with parallel mean curvature vector in a pseudo-Euclidean m-space Ems . Conversely, every Lorentz surface with parallel mean curvature vector in Ems is obtained from the 23 families.

Generalized Cauchy--Riemann lightlike submanifolds of Kaehler manifolds

We introduce a class of submanifolds, namely, Generalized Cauchy--Riemann (GCR) lightlike submanifolds of indefinite Kaehler manifolds. We show that this new class is an umbrella of invariant (complex), screen real [8] and CR lightlike [6] submanifolds. We study the existence (or non-existence) of this new class in an indefinite space form. Then, we prove characterization theorems on the existence of totally umbilical, irrotational screen real, complex and CR minimal lightlike submanifolds. We also give one example each of a non totally geodesic proper minimal GCR and CR lightlike submanifolds.

Conformally flat Lorentz hypersurfaces

We shall investigate conformally flat Lorentz hypersurfaces in indefinite space forms. Some particular classes of such hypersurfaces are explicitly described and classified.

Lightlike real hypersurfaces of indefinite quaternion Kaehler manifold

In this paper, we introduce real lightlike hypersurfaces of indefinite quaternion Kaehler manifold. Fundamental properties of real lightlike hypersurfaces of an indefinite quaternion Kaehler manifold are investigated. We prove the non existence of real lightlike hypersurfaces in indefinite qaternionic space form under some conditions.