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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 636782, 18 pages
Research Article

Lightlike Submanifolds of a Semi-Riemannian Manifold of Quasi-Constant Curvature

1Department of Mathematics, Dongguk University, Kyongju 780-714, Republic of Korea
2Department of Mathematics, Sogang University, Sinsu-dong, Mapo-gu, Seoul 121-742, Republic of Korea

Received 19 January 2012; Revised 29 February 2012; Accepted 14 March 2012

Academic Editor: Chein-ShanΒ Liu

Copyright Β© 2012 D. H. Jin and J. W. Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study the geometry of lightlike submanifolds (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(π‘‡π‘€βŸ‚)) of a semi-Riemannian manifold (𝑀,̃𝑔) of quasiconstant curvature subject to the following conditions: (1) the curvature vector field ΞΆ of 𝑀 is tangent to 𝑀, (2) the screen distribution 𝑆(𝑇𝑀) of 𝑀 is totally geodesic in 𝑀, and (3) the coscreen distribution 𝑆(π‘‡π‘€βŸ‚) of 𝑀 is a conformal Killing distribution.

1. Introduction

In the generalization from the theory of submanifolds in Riemannian to the theory of submanifolds in semi-Riemannian manifolds, the induced metric on submanifolds may be degenerate (lightlike). Therefore, there is a natural existence of lightlike submanifolds and for which the local and global geometry is completely different than nondegenerate case. In lightlike case, the standard text book definitions do not make sense, and one fails to use the theory of nondegenerate geometry in the usual way. The primary difference between the lightlike submanifolds and nondegenerate submanifolds is that in the first case, the normal vector bundle intersects with the tangent bundle. Thus, the study of lightlike submanifolds becomes more difficult and different from the study of nondegenerate submanifolds. Moreover, the geometry of lightlike submanifolds is used in mathematical physics, in particular, in general relativity since lightlike submanifolds produce models of different types of horizons (event horizons, Cauchy’s horizons, and Kruskal’s horizons). The universe can be represented as a four-dimensional submanifold embedded in a (4+𝑛)-dimensional spacetime manifold. Lightlike hypersurfaces are also studied in the theory of electromagnetism [1]. Thus, large number of applications but limited information available motivated us to do research on this subject matter. Kupeli [2] and Duggal and Bejancu [1] developed the general theory of degenerate (lightlike) submanifolds. They constructed a transversal vector bundle of lightlike submanifold and investigated various properties of these manifolds.

In the study of Riemannian geometry, Chen and Yano [3] introduced the notion of a Riemannian manifold of a quasiconstant curvature as a Riemannian manifold (𝑀,̃𝑔) with the curvature tensor 𝑅 satisfying the condition̃𝑔𝑅(𝑋,π‘Œ)𝑍,π‘Š=𝛼{̃𝑔(π‘Œ,𝑍)̃𝑔(𝑋,π‘Š)βˆ’Μƒπ‘”(𝑋,𝑍)̃𝑔(π‘Œ,π‘Š)}+𝛽{̃𝑔(𝑋,π‘Š)πœƒ(π‘Œ)πœƒ(𝑍)βˆ’Μƒπ‘”(𝑋,𝑍)πœƒ(π‘Œ)πœƒ(π‘Š)+̃𝑔(π‘Œ,𝑍)πœƒ(𝑋)πœƒ(π‘Š)βˆ’Μƒπ‘”(π‘Œ,π‘Š)πœƒ(𝑋)πœƒ(𝑍)},(1.1)for any vector fields 𝑋,π‘Œ,𝑍, and π‘Š on 𝑀, where 𝛼,𝛽 are scalar functions and πœƒ is a 1-form defined byπœƒ(𝑋)=̃𝑔(𝑋,𝜁),(1.2) where 𝜁 is a unit vector field on 𝑀 which called the curvature vector field. It is well known that if the curvature tensor 𝑅 is of the form (1.1), then the manifold is conformally flat. If 𝛽=0, then the manifold reduces to a space of constant curvature.

A nonflat Riemannian manifold of dimension 𝑛(>2) is defined to be a quasi-Einstein manifold [4] if its Ricci tensor satisfies the conditionξ‚‹Ric(𝑋,π‘Œ)=π‘ŽΜƒπ‘”(𝑋,π‘Œ)+π‘πœ™(𝑋)πœ™(π‘Œ),(1.3)

where π‘Ž,𝑏 are scalar functions such that 𝑏≠0, and πœ™ is a nonvanishing 1-form such that ̃𝑔(𝑋,π‘ˆ)=πœ™(𝑋) for any vector field 𝑋, where π‘ˆ is a unit vector field. If 𝑏=0, then the manifold reduces to an Einstein manifold. It can be easily seen that every Riemannian manifold of quasiconstant curvature is a quasi-Einstein manifold.

The subject of this paper is to study the geometry of lightlike submanifolds of a semi-Riemannian manifold (𝑀,̃𝑔) of quasiconstant curvature. We prove two characterization theorems for such a lightlike submanifold (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(π‘‡π‘€βŸ‚)) as follows.

Theorem 1.1. Let 𝑀 be an π‘Ÿ-lightlike submanifold of a semi-Riemannian manifold (𝑀,̃𝑔) of quasiconstant curvature. If the curvature vector field 𝜁 of 𝑀 is tangent to 𝑀 and 𝑆(𝑇𝑀) is totally geodesic in 𝑀, then one has the following results: (1)if 𝑆(π‘‡π‘€βŸ‚) is a Killing distribution, then the functions 𝛼 and 𝛽, defined by (1.1), vanish identically. Furthermore, 𝑀, 𝑀, and the leaf π‘€βˆ— of 𝑆(𝑇𝑀) are flat manifolds;(2)if 𝑆(π‘‡π‘€βŸ‚) is a conformal Killing distribution, then the function 𝛽 vanishes identically. Furthermore, 𝑀 and π‘€βˆ— are space of constant curvatures, and 𝑀 is an Einstein manifold such that Ric=(π‘Ÿ/(π‘šβˆ’π‘Ÿ))𝑔, where π‘Ÿ is the induced scalar curvature of 𝑀.

Theorem 1.2. Let 𝑀 be an irrotational π‘Ÿ-lightlike submanifold of a semi-Riemannian manifold (𝑀,̃𝑔) of quasiconstant curvature. If 𝜁 is tangent to 𝑀, 𝑆(𝑇𝑀) is totally umbilical in 𝑀, and 𝑆(π‘‡π‘€βŸ‚) is a conformal Killing distribution with a nonconstant conformal factor, then the function 𝛽 vanishes identically. Moreover, 𝑀 and π‘€βˆ— are space of constant curvatures, and 𝑀 is a totally umbilical Einstein manifold such that Ric=(𝑐/(π‘šβˆ’π‘Ÿ))𝑔, where 𝑐 is the scalar quantity of 𝑀.

2. Lightlike Submanifolds

Let (𝑀,𝑔) be an π‘š-dimensional lightlike submanifold of an (π‘š+𝑛)-dimensional semi-Riemannian manifold (𝑀,̃𝑔). We follow Duggal and Bejancu [1] for notations and results used in this paper. The radical distribution Rad(𝑇𝑀)=π‘‡π‘€βˆ©π‘‡π‘€βŸ‚ is a vector subbundle of the tangent bundle 𝑇𝑀 and the normal bundle π‘‡π‘€βŸ‚, of rank π‘Ÿ(1β‰€π‘Ÿβ‰€min{π‘š,𝑛}). Then, in general, there exist two complementary nondegenerate distributions 𝑆(𝑇𝑀) and 𝑆(π‘‡π‘€βŸ‚) of Rad(𝑇𝑀) in 𝑇𝑀 and π‘‡π‘€βŸ‚, respectively, called the screen and coscreen distribution on 𝑀, and we have the following decompositions:𝑇𝑀=Rad(𝑇𝑀)βŠ•orth𝑆(𝑇𝑀);π‘‡π‘€βŸ‚=Rad(𝑇𝑀)βŠ•orthπ‘†ξ€·π‘‡π‘€βŸ‚ξ€Έ,(2.1) where the symbol βŠ•orth denotes the orthogonal direct sum. We denote such a lightlike submanifold by 𝑀=(𝑀,𝑔,𝑆(𝑇𝑀),𝑆(π‘‡π‘€βŸ‚)). Let tr(𝑇𝑀) and ltr(𝑇𝑀) be complementary (but not orthogonal) vector bundles to 𝑇𝑀 in 𝑇𝑀|𝑀 and π‘‡π‘€βŸ‚ in 𝑆(𝑇𝑀)βŸ‚, respectively, and let {𝑁𝑖} be a lightlike basis of Ξ“(ltr(𝑇𝑀)|𝒰) consisting of smooth sections of 𝑆(𝑇𝑀)βŸ‚|𝒰, where 𝒰 is a coordinate neighborhood of 𝑀, such that𝑁̃𝑔𝑖,πœ‰π‘—ξ€Έ=𝛿𝑖𝑗𝑁,̃𝑔𝑖,𝑁𝑗=0,(2.2)

where {πœ‰1,…,πœ‰π‘Ÿ} is a lightlike basis of Ξ“(Rad(𝑇𝑀)). Then,𝑇𝑀=π‘‡π‘€βŠ•tr(𝑇𝑀)={Rad(𝑇𝑀)βŠ•tr(𝑇𝑀)}βŠ•orth𝑆(𝑇𝑀)={Rad(𝑇𝑀)βŠ•ltr(𝑇𝑀)}βŠ•orth𝑆(𝑇𝑀)βŠ•orthπ‘†ξ€·π‘‡π‘€βŸ‚ξ€Έ.(2.3) We say that a lightlike submanifold (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(π‘‡π‘€βŸ‚)) of 𝑀 is

(1) π‘Ÿ-lightlike submanifold if 1β‰€π‘Ÿ<π‘šπ‘–π‘›{π‘š,𝑛},

(2) coisotropic submanifold if 1β‰€π‘Ÿ=𝑛<π‘š,

(3) isotropic submanifold if 1β‰€π‘Ÿ=π‘š<𝑛,

(4) totally lightlike submanifold if 1β‰€π‘Ÿ=π‘š=𝑛.

The above three classes (2)~(4) are particular cases of the class (1) as follows: 𝑆(π‘‡π‘€βŸ‚)={0},𝑆(𝑇𝑀)={0}, and 𝑆(𝑇𝑀)=𝑆(π‘‡π‘€βŸ‚)={0}, respectively.

Example 2.1. Consider in ℝ42 the 1-lightlike submanifold 𝑀 given by equations π‘₯3=1√2ξ€·π‘₯1+π‘₯2ξ€Έ,π‘₯4=12ξ‚€ξ€·π‘₯log1+1βˆ’π‘₯2ξ€Έ2,(2.4) then we have 𝑇𝑀=span{π‘ˆ1,π‘ˆ2} and π‘‡π‘€βŸ‚={𝐻1,𝐻2}, where we set π‘ˆ1=√2ξ‚€ξ€·π‘₯1+1βˆ’π‘₯2ξ€Έ2ξ‚πœ•π‘₯1+ξ‚€ξ€·π‘₯1+1βˆ’π‘₯2ξ€Έ2ξ‚πœ•π‘₯3+√2ξ€·π‘₯1βˆ’π‘₯2ξ€Έπœ•π‘₯4,π‘ˆ2=√2ξ‚€ξ€·π‘₯1+1βˆ’π‘₯2ξ€Έ2ξ‚πœ•π‘₯2+ξ‚€ξ€·π‘₯1+1βˆ’π‘₯2ξ€Έ2ξ‚πœ•π‘₯3+√2ξ€·π‘₯1βˆ’π‘₯2ξ€Έπœ•π‘₯4,𝐻1=πœ•π‘₯1+πœ•π‘₯2+√2πœ•π‘₯3,𝐻2ξ‚€ξ€·π‘₯=21+2βˆ’π‘₯1ξ€Έ2ξ‚πœ•π‘₯2+√2ξ€·π‘₯1βˆ’π‘₯2ξ€Έπœ•π‘₯3+ξ‚€ξ€·π‘₯1+1βˆ’π‘₯2ξ€Έ2ξ‚πœ•π‘₯4.(2.5) It follows that Rad(𝑇𝑀) is a distribution on 𝑀 of rank 1 spanned by πœ‰=𝐻1. Choose 𝑆(𝑇𝑀) and 𝑆(π‘‡π‘€βŸ‚) spanned by π‘ˆ2 and 𝐻2 where are timelike and spacelike, respectively. Finally, the lightlike transversal vector bundle ξƒ―1ltr(𝑇𝑀)=Span𝑁=2πœ•π‘₯1+12πœ•π‘₯2+1√2πœ•π‘₯3ξƒ°(2.6) and the transversal vector bundle ξ€½tr(𝑇𝑀)=Span𝑁,𝐻2ξ€Ύ(2.7) are obtained.
Let ξ‚βˆ‡ be the Levi-Civita connection of 𝑀 and 𝑃 the projection morphism of Ξ“(𝑇𝑀) on Ξ“(𝑆(𝑇𝑀)) with respect to the decomposition (2.1). For an π‘Ÿ-lightlike submanifold, the local Gauss-Weingartan formulas are given byξ‚βˆ‡π‘‹π‘Œ=βˆ‡π‘‹π‘Œ+π‘Ÿξ“π‘–=1β„Žβ„“π‘–(𝑋,π‘Œ)𝑁𝑖+𝑛𝛼=π‘Ÿ+1β„Žπ‘ π›Ό(𝑋,π‘Œ)π‘Šπ›Ό,ξ‚βˆ‡(2.8)𝑋𝑁𝑖=βˆ’π΄π‘π‘–π‘‹+π‘Ÿξ“π‘—=1πœπ‘–π‘—(𝑋)𝑁𝑗+𝑛𝛼=π‘Ÿ+1πœŒπ‘–π›Ό(𝑋)π‘Šπ›Όξ‚βˆ‡,(2.9)π‘‹π‘Šπ›Ό=βˆ’π΄π‘Šπ›Όπ‘‹+π‘Ÿξ“π‘–=1πœ™π›Όπ‘–(𝑋)𝑁𝑖+𝑛𝛽=π‘Ÿ+1πœƒπ›Όπ›½(𝑋)π‘Šπ›½βˆ‡,(2.10)π‘‹π‘ƒπ‘Œ=βˆ‡βˆ—π‘‹π‘ƒπ‘Œ+π‘Ÿξ“π‘–=1β„Žβˆ—π‘–(𝑋,π‘ƒπ‘Œ)πœ‰π‘–βˆ‡,(2.11)π‘‹πœ‰π‘–=βˆ’π΄βˆ—πœ‰π‘–π‘‹βˆ’π‘Ÿξ“π‘—=1πœπ‘—π‘–(𝑋)πœ‰π‘—,(2.12) for any 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀), where βˆ‡ and βˆ‡βˆ— are induced linear connections on 𝑇𝑀 and 𝑆(𝑇𝑀), respectively, the bilinear forms β„Žβ„“π‘– and β„Žπ‘ π›Ό on 𝑀 are called the local lightlike second fundamental form and local screen second fundamental form on 𝑇𝑀, respectively, and β„Žβˆ—π‘– is called the local radical second fundamental form on 𝑆(𝑇𝑀). 𝐴𝑁𝑖,π΄βˆ—πœ‰π‘–, and π΄π‘Šπ›Ό are linear operators on Ξ“(𝑇𝑀), and πœπ‘–π‘—,πœŒπ‘–π›Ό,πœ™π›Όπ‘–, and πœƒπ›Όπ›½ are 1-forms on 𝑇𝑀.
Since ξ‚βˆ‡ is torsion-free, βˆ‡ is also torsion-free and both β„Žβ„“π‘– and β„Žπ‘ π›Ό are symmetric. From the fact that β„Žβ„“π‘–ξ‚βˆ‡(𝑋,π‘Œ)=̃𝑔(π‘‹π‘Œ,πœ‰π‘–), we know that β„Žβ„“π‘– are independent of the choice of a screen distribution. Note that β„Žβ„“π‘–,πœπ‘–π‘—, and πœŒπ‘–π›Ό depend on the section πœ‰βˆˆΞ“(Rad(𝑇𝑀)|𝒰). Indeed, take πœ‰π‘–=βˆ‘π‘Ÿπ‘—=1π‘Žπ‘–π‘—πœ‰π‘—, then we have 𝑑(tr(πœπ‘–π‘—))=𝑑(tr(Μƒπœπ‘–π‘—)) [5].
The induced connection βˆ‡ on 𝑇𝑀 is not metric and satisfies ξ€·βˆ‡π‘‹π‘”ξ€Έ(π‘Œ,𝑍)=π‘Ÿξ“π‘–=1ξ€½β„Žβ„“π‘–(𝑋,π‘Œ)πœ‚π‘–(𝑍)+β„Žβ„“π‘–(𝑋,𝑍)πœ‚π‘–ξ€Ύ,(π‘Œ)(2.13) where πœ‚π‘– is the 1-form such that πœ‚π‘–ξ€·(𝑋)=̃𝑔𝑋,𝑁𝑖,βˆ€π‘‹βˆˆΞ“(𝑇𝑀),π‘–βˆˆ{1,…,π‘Ÿ}.(2.14) But the connection βˆ‡βˆ— on 𝑆(𝑇𝑀) is metric. The above three local second fundamental forms of 𝑀 and 𝑆(𝑇𝑀) are related to their shape operators by β„Žβ„“π‘–ξ‚€π΄(𝑋,π‘Œ)=π‘”βˆ—πœ‰π‘–ξ‚βˆ’π‘‹,π‘Œπ‘Ÿξ“π‘˜=1β„Žβ„“π‘˜ξ€·π‘‹,πœ‰π‘–ξ€Έπœ‚π‘˜β„Ž(π‘Œ),(2.15)ℓ𝑖𝐴(𝑋,π‘ƒπ‘Œ)=π‘”βˆ—πœ‰π‘–ξ‚ξ‚€π΄π‘‹,π‘ƒπ‘Œ,Μƒπ‘”βˆ—πœ‰π‘–π‘‹,π‘π‘—ξ‚πœ–=0,(2.16)π›Όβ„Žπ‘ π›Όξ€·π΄(𝑋,π‘Œ)=π‘”π‘Šπ›Όξ€Έβˆ’π‘‹,π‘Œπ‘Ÿξ“π‘–=1πœ™π›Όπ‘–(𝑋)πœ‚π‘–πœ–(π‘Œ),(2.17)π›Όβ„Žπ‘ π›Ό(𝐴𝑋,π‘ƒπ‘Œ)=π‘”π‘Šπ›Όξ€Έξ€·π΄π‘‹,π‘ƒπ‘Œ,Μƒπ‘”π‘Šπ›Όπ‘‹,𝑁𝑖=πœ–π›ΌπœŒπ‘–π›Ό(β„Žπ‘‹),(2.18)βˆ—π‘–ξ€·π΄(𝑋,π‘ƒπ‘Œ)=𝑔𝑁𝑖𝑋,π‘ƒπ‘Œ,πœ‚π‘—ξ€·π΄π‘π‘–π‘‹ξ€Έ+πœ‚π‘–ξ‚€π΄π‘π‘—π‘‹ξ‚=0,(2.19) and πœ–π›½πœƒπ›Όπ›½=βˆ’πœ–π›Όπœƒπ›½π›Ό, where 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀). From (2.19), we know that the operators 𝐴𝑁𝑖 are shape operators related to β„Žβˆ—π‘– for each 𝑖, called the radical shape operators on 𝑆(𝑇𝑀). From (2.16), we know that the operators π΄βˆ—πœ‰π‘– are Ξ“(𝑆(𝑇𝑀)) valued. Replace π‘Œ by πœ‰π‘— in (2.15), then we have β„Žβ„“π‘–(𝑋,πœ‰π‘—)+β„Žβ„“π‘—(𝑋,πœ‰π‘–)=0 for all π‘‹βˆˆΞ“(𝑇𝑀). It follows that β„Žβ„“π‘–ξ€·π‘‹,πœ‰π‘–ξ€Έ=0,β„Žβ„“π‘–ξ€·πœ‰π‘—,πœ‰π‘˜ξ€Έ=0.(2.20) Also, replace 𝑋 by πœ‰π‘— in (2.15) and use (2.20), then we have β„Žβ„“π‘–ξ€·π‘‹,πœ‰π‘—ξ€Έξ‚€=𝑔𝑋,π΄βˆ—πœ‰π‘–πœ‰π‘—ξ‚,π΄βˆ—πœ‰π‘–πœ‰π‘—+π΄βˆ—πœ‰π‘—πœ‰π‘–=0,π΄βˆ—πœ‰π‘–πœ‰π‘–=0.(2.21) Thus πœ‰π‘– is an eigenvector field of π΄βˆ—πœ‰π‘– corresponding to the eigenvalue 0. For an π‘Ÿ-lightlike submanifold, replace π‘Œ by πœ‰π‘– in (2.17), then we have β„Žπ‘ π›Όξ€·π‘‹,πœ‰π‘–ξ€Έ=βˆ’πœ–π›Όπœ™π›Όπ‘–(𝑋).(2.22)
From (2.15)~(2.18), we show that the operators π΄βˆ—πœ‰π‘– and π΄π‘Šπ›Ό are not self-adjoint on Ξ“(𝑇𝑀) but self-adjoint on Ξ“(𝑆(𝑇𝑀)).

Theorem 2.2. Let (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(π‘‡π‘€βŸ‚)) be an π‘Ÿ-lightlike submanifold of a semi-Riemannian manifold (𝑀,̃𝑔), then the following assertions are equivalent: (i)π΄βˆ—πœ‰π‘– are self-adjoint on Ξ“(𝑇𝑀) with respect to 𝑔,  for all 𝑖,(ii)β„Žβ„“π‘– satisfy β„Žβ„“π‘–(𝑋,πœ‰π‘—)=0 for all π‘‹βˆˆΞ“(𝑇𝑀),𝑖 and 𝑗,(iii)π΄βˆ—πœ‰π‘–πœ‰π‘—=0 for all 𝑖 and 𝑗, that is, the image of Rad(𝑇𝑀) with respect to π΄βˆ—πœ‰π‘– for each 𝑖 is a trivial vector bundle,(iv)β„Žβ„“π‘–(𝑋,π‘Œ)=𝑔(π΄βˆ—πœ‰π‘–π‘‹,π‘Œ) for all 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀) and 𝑖, that is, π΄βˆ—πœ‰π‘– is a shape operator on 𝑀, related by the second fundamental form β„Žβ„“π‘–.

Proof. From (2.15) and the fact that β„Žβ„“π‘– are symmetric, we have π‘”ξ‚€π΄βˆ—πœ‰π‘–ξ‚ξ‚€π‘‹,π‘Œβˆ’π‘”π‘‹,π΄βˆ—πœ‰π‘–π‘Œξ‚=π‘Ÿξ“π‘—=1ξ€½β„Žβ„“π‘˜ξ€·π‘‹,πœ‰π‘–ξ€Έπœ‚π‘˜(π‘Œ)βˆ’β„Žβ„“π‘˜ξ€·π‘Œ,πœ‰π‘–ξ€Έπœ‚π‘˜ξ€Ύ(𝑋).(2.23)
(i)⇔(ii). If β„Žβ„“π‘–(𝑋,πœ‰π‘—)=0 for all π‘‹βˆˆΞ“(𝑇𝑀),𝑖 and 𝑗, then we have 𝑔(π΄βˆ—πœ‰π‘–π‘‹,π‘Œ)=𝑔(π΄βˆ—πœ‰π‘–π‘Œ,𝑋) for all 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀), that is, π΄βˆ—πœ‰π‘– are self-adjoint on Ξ“(𝑇𝑀) with respect to 𝑔. Conversely, if π΄βˆ—πœ‰π‘– are self-adjoint on Ξ“(𝑇𝑀) with respect to 𝑔, then we have β„Žβ„“π‘˜ξ€·π‘‹,πœ‰π‘–ξ€Έπœ‚π‘˜(π‘Œ)=β„Žβ„“π‘˜ξ€·π‘Œ,πœ‰π‘–ξ€Έπœ‚π‘˜(𝑋),(2.24) for all 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀). Replace π‘Œ by πœ‰π‘— in this equation and use the second equation of (2.20), then we have β„Žβ„“π‘—(𝑋,πœ‰π‘–)=0 for all π‘‹βˆˆΞ“(𝑇𝑀),𝑖 and 𝑗.
(ii)⇔(iii). Since 𝑆(𝑇𝑀) is nondegenerate, from the first equation of (2.21), we have β„Žβ„“π‘–(𝑋,πœ‰π‘—)=0β‡”π΄βˆ—πœ‰π‘–πœ‰π‘—=0, for all 𝑖 and 𝑗.
(ii)⇔(iv). From (2.16), we have β„Žβ„“π‘–(𝑋,π‘Œ)=𝑔(π΄βˆ—πœ‰π‘–π‘‹,π‘Œ)β‡”β„Žβ„“π‘—(𝑋,πœ‰π‘–)=0 for any 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀) and for all 𝑖 and 𝑗.

Corollary 2.3. Let (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(π‘‡π‘€βŸ‚)) be a 1-lightlike submanifold of a semi-Riemannian manifold (𝑀,̃𝑔), then the operators π΄βˆ—πœ‰π‘– are self-adjoint on Ξ“(𝑇𝑀) with respect to 𝑔.

Definition 2.4. An π‘Ÿ-lightlike submanifold (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(π‘‡π‘€βŸ‚)) of a semi-Riemannian manifold (𝑀,̃𝑔) is said to be irrotational if ξ‚βˆ‡π‘‹πœ‰π‘–βˆˆΞ“(𝑇𝑀) for any π‘‹βˆˆΞ“(𝑇𝑀) and 𝑖.

For an π‘Ÿ-lightlike submanifold 𝑀 of 𝑀, the above definition is equivalent to β„Žβ„“π‘—(𝑋,πœ‰π‘–)=0 and β„Žπ‘ π›Ό(𝑋,πœ‰π‘–)=0 for any π‘‹βˆˆΞ“(𝑇𝑀). In this case, π΄βˆ—πœ‰π‘– are self-adjoint on Ξ“(𝑇𝑀) with respect to 𝑔, for all 𝑖.

We need the following Gauss-Codazzi equations (for a full set of these equations see [1, chapter 5]) for M and 𝑆(𝑇𝑀). Denote by 𝑅,𝑅, and π‘…βˆ— the curvature tensors of the Levi-Civita connection ξ‚βˆ‡ of 𝑀, the induced connection βˆ‡ of 𝑀, and the induced connection βˆ‡βˆ— on 𝑆(𝑇𝑀), respectively:+̃𝑔𝑅(𝑋,π‘Œ)𝑍,π‘ƒπ‘Š=𝑔(𝑅(𝑋,π‘Œ)𝑍,π‘ƒπ‘Š)π‘Ÿξ“π‘–=1ξ€½β„Žβ„“π‘–(𝑋,𝑍)β„Žβˆ—π‘–(π‘Œ,π‘ƒπ‘Š)βˆ’β„Žβ„“π‘–(π‘Œ,𝑍)β„Žβˆ—π‘–(ξ€Ύ+𝑋,π‘ƒπ‘Š)𝑛𝛼=π‘Ÿ+1πœ–π›Όξ€½β„Žπ‘ π›Ό(𝑋,𝑍)β„Žπ‘ π›Ό(π‘Œ,π‘ƒπ‘Š)βˆ’β„Žπ‘ π›Ό(π‘Œ,𝑍)β„Žπ‘ π›Όξ€Ύ,πœ–(𝑋,π‘ƒπ‘Š)(2.25)𝛼̃𝑔𝑅(𝑋,π‘Œ)𝑍,π‘Šπ›Όξ‚=ξ€·βˆ‡π‘‹β„Žπ‘ π›Όξ€Έξ€·βˆ‡(π‘Œ,𝑍)βˆ’π‘Œβ„Žπ‘ π›Όξ€Έ+(𝑋,𝑍)π‘Ÿξ“π‘–=1ξ€½β„Žβ„“π‘–(π‘Œ,𝑍)πœŒπ‘–π›Ό(𝑋)βˆ’β„Žβ„“π‘–(𝑋,𝑍)πœŒπ‘–π›Όξ€Ύ+(π‘Œ)𝑛𝛽=π‘Ÿ+1ξ‚†β„Žπ‘ π›½(π‘Œ,𝑍)πœƒπ›½π›Ό(𝑋)βˆ’β„Žπ‘ π›½(𝑋,𝑍)πœƒπ›½π›Όξ‚‡,(π‘Œ)(2.26)̃𝑔𝑅(𝑋,π‘Œ)𝑍,𝑁𝑖=̃𝑔𝑅(𝑋,π‘Œ)𝑍,𝑁𝑖+π‘Ÿξ“π‘—=1ξ‚†β„Žβ„“π‘—(𝑋,𝑍)πœ‚π‘–ξ‚€π΄π‘π‘—π‘Œξ‚βˆ’β„Žβ„“π‘—(π‘Œ,𝑍)πœ‚π‘–ξ‚€π΄π‘π‘—π‘‹+𝑛𝛼=π‘Ÿ+1πœ–π›Όξ€½β„Žπ‘ π›Ό(𝑋,𝑍)πœŒπ‘–π›Ό(π‘Œ)βˆ’β„Žπ‘ π›Ό(π‘Œ,𝑍)πœŒπ‘–π›Όξ€Ύ,(𝑋)(2.27)̃𝑔𝑅(𝑋,π‘Œ)πœ‰π‘–,𝑁𝑗=̃𝑔𝑅(𝑋,π‘Œ)πœ‰π‘–,𝑁𝑗+π‘Ÿξ“π‘˜=1ξ€½β„Žβ„“π‘˜ξ€·π‘‹,πœ‰π‘–ξ€Έπœ‚π‘—ξ€·π΄π‘π‘˜π‘Œξ€Έβˆ’β„Žβ„“π‘˜ξ€·π‘Œ,πœ‰π‘–ξ€Έπœ‚π‘—ξ€·π΄π‘π‘˜π‘‹+𝑛𝛼=π‘Ÿ+1ξ€½πœŒπ‘—π›Ό(𝑋)πœ™π›Όπ‘–(π‘Œ)βˆ’πœŒπ‘—π›Ό(π‘Œ)πœ™π›Όπ‘–ξ€Ύξ‚€π΄(𝑋)=π‘”βˆ—πœ‰π‘–π‘‹,π΄π‘π‘—π‘Œξ‚ξ‚€π΄βˆ’π‘”βˆ—πœ‰π‘–π‘Œ,π΄π‘π‘—π‘‹ξ‚βˆ’2π‘‘πœπ‘—π‘–+(𝑋,π‘Œ)π‘Ÿξ“π‘˜=1ξ€½β„Žβ„“π‘˜ξ€·π‘‹,πœ‰π‘–ξ€Έπœ‚π‘—ξ€·π΄π‘π‘˜π‘Œξ€Έβˆ’β„Žβ„“π‘˜ξ€·π‘Œ,πœ‰π‘–ξ€Έπœ‚π‘—ξ€·π΄π‘π‘˜π‘‹+ξ€Έξ€Ύπ‘Ÿξ“π‘˜=1ξ€½πœπ‘—π‘˜(𝑋)πœπ‘˜π‘–(π‘Œ)βˆ’πœπ‘—π‘˜(π‘Œ)πœπ‘˜π‘–ξ€Ύ+(𝑋)𝑛𝛼=π‘Ÿ+1ξ€½πœŒπ‘—π›Ό(𝑋)πœ™π›Όπ‘–(π‘Œ)βˆ’πœŒπ‘—π›Ό(π‘Œ)πœ™π›Όπ‘–(ξ€Ύ,𝑅𝑋)(2.28)̃𝑔(𝑅(𝑋,π‘Œ)𝑃𝑍,π‘ƒπ‘Š)=π‘”βˆ—ξ€Έ+(𝑋,π‘Œ)𝑃𝑍,π‘ƒπ‘Šπ‘Ÿξ“π‘–=1ξ€½β„Žβˆ—π‘–(𝑋,𝑃𝑍)β„Žβ„“π‘–(π‘Œ,π‘ƒπ‘Š)βˆ’β„Žβˆ—π‘–(π‘Œ,𝑃𝑍)β„Žβ„“π‘–ξ€Ύ,𝑔(𝑋,π‘ƒπ‘Š)(2.29)𝑅(𝑋,π‘Œ)𝑃𝑍,𝑁𝑖=ξ€·βˆ‡π‘‹β„Žβˆ—π‘–ξ€Έξ€·βˆ‡(π‘Œ,𝑃𝑍)βˆ’π‘Œβ„Žβˆ—π‘–ξ€Έ+(𝑋,𝑃𝑍)π‘Ÿξ“π‘—=1ξ€½β„Žβˆ—π‘—(𝑋,𝑃𝑍)πœπ‘–π‘—(π‘Œ)βˆ’β„Žβˆ—π‘—(π‘Œ,𝑃𝑍)πœπ‘–π‘—ξ€Ύ.(𝑋)(2.30)

The Ricci tensor of 𝑀 is given by𝑇𝑀Ric(𝑋,π‘Œ)=traceπ‘βŸΆπ‘…(𝑍,𝑋)π‘Œ,βˆ€π‘‹,π‘ŒβˆˆΞ“,(2.31) for any 𝑋,ξ‚‹π‘ŒβˆˆΞ“(𝑇𝑀). Let 𝑀dim = π‘š+𝑛. Locally, ξ‚‹Ric is given byξ‚‹Ric(𝑋,π‘Œ)=π‘š+𝑛𝑖=1πœ–π‘–ξ‚€ξ‚π‘…ξ€·πΈΜƒπ‘”π‘–ξ€Έ,π‘‹π‘Œ,𝐸𝑖,(2.32) where {𝐸1,…,πΈπ‘š+𝑛} is an orthonormal frame field of 𝑇𝑀. If ξ‚‹dim(𝑀)>2 andξ‚‹Ric=ξ‚πœ…Μƒπ‘”,ξ‚πœ…isaconstant,(2.33) then 𝑀 is an Einstein manifold. If ξ‚‹dim(𝑀)=2, any 𝑀 is Einstein, but ξ‚πœ… in (2.33) is not necessarily constant. The scalar curvature Μƒπ‘Ÿ is defined byΜƒπ‘Ÿ=π‘š+𝑛𝑖=1πœ–π‘–ξ‚‹ξ€·πΈRic𝑖,𝐸𝑖.(2.34) Putting (2.33) in (2.34) implies that 𝑀 is Einstein if and only ifξ‚‹Ric=Μƒπ‘Ÿπ‘š+𝑛̃𝑔.(2.35)

3. The Tangential Curvature Vector Field

Let 𝑅(0,2) denote the induced Ricci tensor of type (0,2) on 𝑀, given by𝑅(0,2)𝑇𝑀(𝑋,π‘Œ)=trace{π‘βŸΆπ‘…(𝑍,𝑋)π‘Œ},βˆ€π‘‹,π‘ŒβˆˆΞ“.(3.1) Consider an induced quasiorthonormal frame fieldξ€½πœ‰1,…,πœ‰π‘Ÿ;𝑁1,…,π‘π‘Ÿ;π‘‹π‘Ÿ+1,…,π‘‹π‘š;π‘Šπ‘Ÿ+1,…,π‘Šπ‘›ξ€Ύ,(3.2)

where {𝑁𝑖,π‘Šπ›Ό} is a basis of Ξ“(tr(𝑇𝑀)|𝒰) on a coordinate neighborhood 𝒰 of 𝑀 such that π‘π‘–βˆˆΞ“(ltr(𝑇𝑀)|𝒰) and π‘Šπ›ΌβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)|𝒰). By using (2.29) and (3.1), we obtain the following local expression for the Ricci tensor:ξ‚‹Ric(𝑋,π‘Œ)=π‘›ξ“π‘Ž=π‘Ÿ+1πœ–π‘Žξ‚€ξ‚π‘…ξ€·π‘ŠΜƒπ‘”π‘Žξ€Έ,π‘‹π‘Œ,π‘Šπ‘Žξ‚+π‘Ÿξ“π‘–=1ξ‚€ξ‚π‘…ξ€·πœ‰Μƒπ‘”π‘–ξ€Έ,π‘‹π‘Œ,𝑁𝑖+π‘šξ“π‘=π‘Ÿ+1πœ–π‘ξ‚€ξ‚π‘…ξ€·π‘‹Μƒπ‘”π‘ξ€Έ,π‘‹π‘Œ,𝑋𝑏+π‘Ÿξ“π‘–=1𝑅𝑁̃𝑔𝑖,π‘‹π‘Œ,πœ‰π‘–ξ‚,𝑅(3.3)(0,2)(𝑋,π‘Œ)=π‘šξ“π‘Ž=π‘Ÿ+1πœ–π‘Žπ‘”ξ€·π‘…ξ€·π‘‹π‘Žξ€Έ,π‘‹π‘Œ,π‘‹π‘Žξ€Έ+π‘Ÿξ“π‘–=1ξ€·π‘…ξ€·πœ‰Μƒπ‘”π‘–ξ€Έ,π‘‹π‘Œ,𝑁𝑖.(3.4) Substituting (2.25) and (2.27) in (3.3) and using (2.15)~(2.18) and (3.4), we obtain𝑅(0,2)ξ‚‹(𝑋,π‘Œ)=Ric(𝑋,π‘Œ)+π‘Ÿξ“π‘–=1β„Žβ„“π‘–(𝑋,π‘Œ)tr𝐴𝑁𝑖+𝑛𝛼=π‘Ÿ+1β„Žπ‘ π›Ό(𝑋,π‘Œ)trπ΄π‘Šπ›Όβˆ’π‘Ÿξ“π‘–=1𝑔𝐴𝑁𝑖𝑋,π΄βˆ—πœ‰π‘–π‘Œξ‚βˆ’π‘›ξ“π›Ό=π‘Ÿ+1πœ–π›Όπ‘”ξ€·π΄π‘Šπ›Όπ‘‹,π΄π‘Šπ›Όπ‘Œξ€Έβˆ’π‘Ÿξ“π‘–,𝑗=1β„Žβ„“π‘—ξ€·πœ‰π‘–ξ€Έπœ‚,π‘Œπ‘–ξ‚€π΄π‘π‘—π‘‹ξ‚+π‘Ÿξ“π‘›π‘–=1𝛼=π‘Ÿ+1πœŒπ‘–π›Ό(𝑋)πœ™π›Όπ‘–βˆ’(π‘Œ)𝑛𝛼=π‘Ÿ+1πœ–π›Όξ‚€ξ‚π‘…ξ€·π‘ŠΜƒπ‘”π›Όξ€Έ,π‘‹π‘Œ,π‘Šπ›Όξ‚βˆ’π‘Ÿξ“π‘–=1ξ‚€ξ‚π‘…ξ€·πœ‰Μƒπ‘”π‘–ξ€Έ,π‘Œπ‘‹,𝑁𝑖,(3.5)

for any 𝑋, π‘ŒβˆˆΞ“(𝑇𝑀). This shows that 𝑅(0,2) is not symmetric. A tensor field 𝑅(0,2) of 𝑀, given by (3.1), is called its induced Ricci tensor if it is symmetric. From now and in the sequel, a symmetric 𝑅(0,2) tensor will be denoted by Ric.

Using (2.28), (3.5), and the first Bianchi identity, we obtain𝑅(0,2)(𝑋,π‘Œ)βˆ’π‘…(0,2)(π‘Œ,𝑋)=π‘Ÿξ“π‘–=1ξ‚†π‘”ξ‚€π΄βˆ—πœ‰π‘–π‘‹,π΄π‘π‘–π‘Œξ‚ξ‚€π΄βˆ’π‘”βˆ—πœ‰π‘–π‘Œ,𝐴𝑁𝑖𝑋s+ξ‚ξ‚‡π‘Ÿξ“π‘–,𝑗=1ξ‚†β„Žβ„“π‘—ξ€·π‘‹,πœ‰π‘–ξ€Έπœ‚π‘–ξ‚€π΄π‘π‘—π‘Œξ‚βˆ’β„Žβ„“π‘—ξ€·π‘Œ,πœ‰π‘–ξ€Έπœ‚π‘–ξ‚€π΄π‘π‘—π‘Œ+ξ‚ξ‚‡π‘Ÿξ“π‘›π‘–=1𝛼=π‘Ÿ+1ξ€½πœŒπ‘–π›Ό(𝑋)πœ™π›Όπ‘–(π‘Œ)βˆ’πœŒπ‘–π›Ό(π‘Œ)πœ™π›Όπ‘–ξ€Ύβˆ’(𝑋)π‘Ÿξ“π‘–=1̃𝑔𝑅(𝑋,π‘Œ)πœ‰π‘–,𝑁𝑖.(3.6)

From this equation and (2.28), we have𝑅(0,2)(𝑋,π‘Œ)βˆ’π‘…(0,2)ξ€·ξ€·πœ(π‘Œ,𝑋)=2𝑑tr𝑖𝑗(𝑋,π‘Œ).(3.7)

Theorem 3.1 (see[5]). Let 𝑀 be a lightlike submanifold of a semi-Riemannian manifold (𝑀,̃𝑔), then the tensor field 𝑅(0,2) is a symmetric Ricci tensor Ric if and only if each 1-form tr(πœπ‘–π‘—) is closed, that is, 𝑑(tr(πœπ‘–π‘—))=0, on any π’°βŠ‚π‘€.

Note 1. Suppose that the tensor 𝑅(0,2) is symmetric Ricci tensor Ric, then the 1-form tr(πœπ‘–π‘—) is closed by Theorem 3.1. Thus, there exist a smooth function 𝑓 on 𝒰 such that tr(πœπ‘–π‘—)=df. Consequently, we get tr(πœπ‘–π‘—)(𝑋)=𝑋(𝑓). If we take Μƒπœ‰π‘–=βˆ‘π‘Ÿπ‘—=1π›Όπ‘–π‘—πœ‰π‘—, it follows that tr(πœπ‘–π‘—)(𝑋)=tr(Μƒπœπ‘–π‘—)(𝑋)+𝑋(lnΞ”). Setting Ξ”=exp(𝑓) in this equation, we get tr(Μƒπœπ‘–π‘—)(𝑋)=0 for any π‘‹βˆˆΞ“(𝑇𝑀|𝒰). We call the pair {πœ‰π‘–,𝑁𝑖}𝑖 on 𝒰 such that the corresponding 1-form tr(πœπ‘–π‘—) vanishes the canonical null pair of 𝑀.

For the rest of this paper, let 𝑀 be a lightlike submanifold of a semi-Riemannian manifold 𝑀 of quasiconstant curvature. We may assume that the curvature vector field 𝜁 of 𝑀 is a unit spacelike tangent vector field of 𝑀 and 𝑀dim>4,ξ‚‹ξ‚€ξ‚π‘…ξ€·πœ‰Ric(𝑋,π‘Œ)={(𝑛+π‘šβˆ’1)𝛼+𝛽}𝑔(𝑋,π‘Œ)+(𝑛+π‘šβˆ’2)π›½πœƒ(𝑋)πœƒ(π‘Œ),(3.8)̃𝑔𝑖,π‘Œπ‘‹,π‘π‘–ξ‚πœ–=𝛼𝑔(𝑋,π‘Œ)+π›½πœƒ(𝑋)πœƒ(π‘Œ),(3.9)π›Όξ‚€ξ‚π‘…ξ€·π‘ŠΜƒπ‘”π›Όξ€Έ,π‘Œπ‘‹,π‘Šπ›Όξ‚=𝛼𝑔(𝑋,π‘Œ)+π›½πœƒ(𝑋)πœƒ(π‘Œ),(3.10) for all 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀). Substituting (3.8)~(3.10) into (3.5), we have𝑅(0,2)+(𝑋,π‘Œ)={(π‘šβˆ’1)𝛼+𝛽}𝑔(𝑋,π‘Œ)+(π‘šβˆ’2)π›½πœƒ(𝑋)πœƒ(π‘Œ)π‘Ÿξ“π‘–=1β„Žβ„“π‘–(𝑋,π‘Œ)tr𝐴𝑁𝑖+𝑛𝛼=π‘Ÿ+1β„Žπ‘ π›Ό(𝑋,π‘Œ)trπ΄π‘Šπ›Όβˆ’π‘Ÿξ“π‘–=1𝑔𝐴𝑁𝑖𝑋,π΄βˆ—πœ‰π‘–π‘Œξ‚βˆ’π‘›ξ“π›Ό=π‘Ÿ+1πœ–π›Όπ‘”ξ€·π΄π‘Šπ›Όπ‘‹,π΄π‘Šπ›Όπ‘Œξ€Έβˆ’π‘Ÿξ“π‘–,𝑗=1β„Žβ„“π‘—ξ€·πœ‰π‘–ξ€Έπœ‚,π‘Œπ‘–ξ‚€π΄π‘π‘—π‘‹ξ‚+π‘Ÿξ“π‘›π‘–=1𝛼=π‘Ÿ+1πœŒπ‘–π›Ό(𝑋)πœ™π›Όπ‘–(π‘Œ).(3.11)

Definition 3.2. We say that the screen distribution 𝑆(𝑇𝑀) of 𝑀 is totally umbilical [1] in 𝑀 if, on any coordinate neighborhood π’°βŠ‚π‘€, there is a smooth function 𝛾𝑖 such that 𝐴𝑁𝑖𝑋=𝛾𝑖𝑃X for any π‘‹βˆˆΞ“(𝑇𝑀), or equivalently,

β„Žβˆ—π‘–(𝑋,π‘ƒπ‘Œ)=𝛾𝑖𝑔(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀).(3.12) In case 𝛾𝑖=0 on 𝒰, we say that 𝑆(𝑇𝑀) is totally geodesic in 𝑀.

A vector field 𝑋 on 𝑀 is said to be a conformal Killing vector field [6] if ℒ𝑋̃𝑔=βˆ’2𝛿̃𝑔 for any smooth function 𝛿, where ℒ𝑋 denotes the Lie derivative with respect to 𝑋, that is,ℒ𝑋[][]𝑇𝑀̃𝑔(π‘Œ,𝑍)=𝑋(̃𝑔(π‘Œ,𝑍))βˆ’Μƒπ‘”(𝑋,π‘Œ,𝑍)βˆ’Μƒπ‘”(π‘Œ,𝑋,𝑍),βˆ€π‘‹,π‘Œ,π‘βˆˆΞ“.(3.13)

In particular, if 𝛿=0, then 𝑋 is called a Killing vector field [7]. A distribution 𝒒 on 𝑀 is called a conformal Killing (resp., Killing) distribution on 𝑀 if each vector field belonging to 𝒒 is a conformal Killing (resp., Killing) vector field on 𝑀. If the coscreen distribution 𝑆(π‘‡π‘€βŸ‚) is a Killing distribution, using (2.10) and (2.17), we haveξ‚€ξ‚βˆ‡Μƒπ‘”π‘‹π‘Šπ›Όξ‚ξ€·π΄,π‘Œ=βˆ’π‘”π‘Šπ›Όξ€Έ+𝑋,π‘Œπ‘Ÿξ“π‘–=1πœ™π›Όπ‘–(𝑋)πœ‚π‘–(π‘Œ)=βˆ’πœ–π›Όβ„Žπ‘ π›Ό(𝑋,π‘Œ).(3.14)

Therefore, since β„Žπ‘ π›Ό are symmetric, we obtainξ‚€ξ‚β„’π‘Šπ›Όξ‚Μƒπ‘”(π‘Œ,𝑍)=βˆ’2πœ–π›Όβ„Žπ‘ π›Ό(𝑋,π‘Œ).(3.15)

Theorem 3.3. Let 𝑀 be an π‘Ÿ-lightlike submanifold of a semi-Riemannian manifold (𝑀,̃𝑔), then the coscreen distribution 𝑆(π‘‡π‘€βŸ‚) is a conformal Killing (resp., Killing) distribution if and only if there exists a smooth function 𝛿𝛼 such that β„Žπ‘ π›Ό(𝑋,π‘Œ)=πœ–π›Όπ›Ώπ›Όπ‘”ξ€½(𝑋,π‘Œ),resp.β„Žπ‘ π›Όξ€Ύ(𝑋,π‘Œ)=0,βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀).(3.16)

Theorem 3.4. Let 𝑀 be an irrotational π‘Ÿ-lightlike submanifold of a semi-Riemannian manifold (𝑀,̃𝑔) of quasiconstant curvature. If the curvature vector field 𝜁 of 𝑀 is tangent to 𝑀, 𝑆(𝑇𝑀) is totally umbilical in 𝑀, and 𝑆(π‘‡π‘€βŸ‚) is a conformal Killing distribution, then the tensor field 𝑅(0,2) is an induced symmetric Ricci tensor of 𝑀.

Proof. From (2.17)~(2.20), (2.22), (3.16), and (3.11), we have β„Žπ‘ π›Ό(𝑋,π‘Œ)=πœ–π›Όπ›Ώπ›Όπ‘”(𝑋,π‘Œ),πœ™π›Όπ‘–(𝑋)=0,π΄π‘Šπ›Όπ‘‹=𝛿𝛼𝑃𝑋+π‘Ÿξ“π‘–=1πœ–π›ΌπœŒπ‘–π›Ό(𝑋)πœ‰π‘–,(3.17)𝑅(0,2)ξƒ―(𝑋,π‘Œ)=(π‘šβˆ’1)𝛼+𝛽+(π‘šβˆ’π‘Ÿβˆ’1)𝑛𝛼=π‘Ÿ+1πœ–π›Όπ›Ώ2𝛼+π‘›ξ“π‘Ÿπ›Ό=π‘Ÿ+1𝑖=1π›Ώπ›ΌπœŒπ‘–π›Όξ€·πœ‰π‘–ξ€Έξƒ°π‘”(𝑋,π‘Œ)+(π‘šβˆ’2)π›½πœƒ(𝑋)πœƒ(π‘Œ)+(π‘šβˆ’π‘Ÿβˆ’1)π‘Ÿξ“π‘–=1π›Ύπ‘–π‘”ξ‚€π΄βˆ—πœ‰π‘–ξ‚π‘‹,π‘Œ,βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀).(3.18) Using (3.17), we show that 𝑅(0,2) is symmetric.

4. Proof of Theorem 1.1

As β„Žβˆ—π‘–=0, we get ̃𝑔(𝑅(𝑋,π‘Œ)𝑃𝑍,𝑁𝑖)=0 by (2.30). From (2.27) and (3.16), we have̃𝑔𝑅(𝑋,π‘Œ)𝑃𝑍,𝑁𝑖=𝑛𝛼=π‘Ÿ+1𝛿𝛼𝑔(𝑋,𝑃𝑍)πœŒπ‘–π›Ό(π‘Œ)βˆ’π‘”(π‘Œ,𝑃𝑍)πœŒπ‘–π›Όξ€Ύ.(𝑋)(4.1)

By Theorems 3.1 and 3.4, we get π‘‘πœ=0 on 𝑇𝑀. Thus, we have ̃𝑔(𝑅(𝑋,π‘Œ)πœ‰π‘–,𝑁𝑖)=0 due to (2.28). From the above results, we deduce the following equation:̃𝑔𝑅(𝑋,π‘Œ)𝑍,𝑁𝑖=𝑛𝛼=π‘Ÿ+1𝛿𝛼𝑔(𝑋,𝑃𝑍)πœŒπ‘–π›Ό(π‘Œ)βˆ’π‘”(π‘Œ,𝑃𝑍)πœŒπ‘–π›Όξ€Ύ.(𝑋)(4.2) Taking 𝑋=πœ‰π‘– and 𝑍=𝑋 to (4.2) and then comparing with (3.9), we haveξƒ―π›½πœƒ(𝑋)πœƒ(π‘Œ)=βˆ’π›Ό+𝑛𝛼=π‘Ÿ+1π›Ώπ›ΌπœŒπ‘–π›Όξ€·πœ‰π‘–ξ€Έξƒ°π‘”(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀).(4.3)

Case 1. If 𝑆(𝑇MβŸ‚) is a Killing distribution, that is, 𝛿𝛼=0, then we have π›½πœƒ(𝑋)πœƒ(π‘Œ)=βˆ’π›Όπ‘”(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀).(4.4) Substituting (4.3) into (1.1) and using (2.25) and the facts ̃𝑔(𝑅(𝑋,π‘Œ)𝑍,πœ‰π‘–)=0 and ̃𝑔(𝑅(𝑋,π‘Œ)𝑍,𝑁𝑖)=0 due to (1.1), we have 𝑅(𝑋,π‘Œ)𝑍=βˆ’π›Ό{𝑔(π‘Œ,𝑍)π‘‹βˆ’π‘”(𝑋,𝑍)π‘Œ},βˆ€π‘‹,π‘Œ,π‘βˆˆΞ“(𝑇𝑀).(4.5) Thus, 𝑀 is a space of constant curvature βˆ’π›Ό. Taking 𝑋=π‘Œ=𝜁 to (4.3), we have 𝛽=βˆ’π›Ό. Substituting (4.3) into (3.18) with 𝛿𝛼=𝛾𝑖=0, we have Ric(𝑋,π‘Œ)=0,βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀).(4.6) On the other hand, substituting (4.5) and 𝑔(𝑅(πœ‰π‘–,π‘Œ)𝑋,𝑁𝑖)=0 into (3.4), we have Ric(𝑋,π‘Œ)=βˆ’(π‘šβˆ’1)𝛼𝑔(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀).(4.7) From the last two equations, we get 𝛼=0 as π‘š>1. Thus, 𝛽=0, and 𝑀 and 𝑀 are flat manifolds by (1.1) and (4.5). From this result and (2.29), we show that π‘€βˆ— is also flat.

Case 2. If 𝑆(π‘‡π‘€βŸ‚) is a conformal Killing distribution, assume that 𝛽≠0. Taking 𝑋=π‘Œ=𝜁 to (4.3), we have βˆ‘π›½=βˆ’{𝛼+𝑛𝛼=π‘Ÿ+1π›Ώπ›ΌπœŒπ‘–π›Ό(πœ‰π‘–)}. From this and (4.3), we show that 𝑔(𝑋,π‘Œ)=πœƒ(𝑋)πœƒ(π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀).(4.8) Substituting (4.8) into (1.1) and using (2.25) with β„Žβˆ—π‘–=0 and (3.16), we have 𝑔(𝑅(𝑋,π‘Œ)𝑍,π‘Š)=𝛼+2𝛽+𝑛𝛼=π‘Ÿ+1πœ–π›Όπ›Ώ2𝛼ξƒͺ{𝑔(π‘Œ,𝑍)𝑔(𝑋,π‘Š)βˆ’π‘”(𝑋,𝑍)𝑔(π‘Œ,π‘Š)},(4.9) for all 𝑋,π‘Œ,𝑍,π‘ŠβˆˆΞ“(𝑇𝑀). Substituting (4.8) into (3.18) with 𝛾𝑖=0, we have ξƒ―Ric(𝑋,π‘Œ)=(π‘šβˆ’π‘Ÿβˆ’1)𝛼+𝛽+𝑛𝛼=π‘Ÿ+1πœ–π›Όπ›Ώ2𝛼𝑔(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀),(4.10) by the fact that βˆ‘π‘›π›Ό=π‘Ÿ+1π›Ώπ›ΌπœŒπ‘–π›Ό(πœ‰π‘–)=βˆ’(𝛼+𝛽). On the other hand, from (2.27), (3.9), and (4.3), we have 𝑔(𝑅(πœ‰π‘–,π‘Œ)𝑋,𝑁𝑖)=0. Substituting this result and (4.9) into (3.4), we have ξƒ―Ric(𝑋,π‘Œ)=(π‘šβˆ’π‘Ÿβˆ’1)𝛼+2𝛽+𝑛𝛼=π‘Ÿ+1πœ–π›Όπ›Ώ2𝛼𝑔(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀).(4.11) The last two equations imply 𝛽=0 as π‘šβˆ’π‘Ÿ>1. It is a contradiction. Thus, 𝛽=0 and 𝑀 is a space of constant curvature 𝛼. From (2.29) and (4.9), we show that π‘€βˆ— is a space of constant curvature βˆ‘(𝛼+𝑛𝛼=π‘Ÿ+1πœ–π›Όπ›Ώ2𝛼). But 𝑀 is not a space of constant curvature by (3.17)3. Let βˆ‘πœ…=(π‘šβˆ’π‘Ÿβˆ’1)(𝛼+𝑛𝛼=π‘Ÿ+1πœ–π›Όπ›Ώ2𝛼), then the last two equations reduce to 𝑅(0,2)(𝑋,π‘Œ)=Ric(𝑋,π‘Œ)=πœ…π‘”(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀).(4.12) Thus 𝑀 is an Einstein manifold. The scalar quantity π‘Ÿ of 𝑀 [8], obtained from 𝑅(0,2) by the method of (2.34), is given by π‘Ÿ=π‘Ÿξ“π‘–=1𝑅(0,2)ξ€·πœ‰π‘–,πœ‰π‘–ξ€Έ+π‘šξ“π‘Ž=π‘Ÿ+1πœ–π‘Žπ‘…(0,2)ξ€·π‘‹π‘Ž,π‘‹π‘Žξ€Έ.(4.13) Since 𝑀 is an Einstein manifold satisfying (4.12), we obtain π‘Ÿ=πœ…π‘Ÿξ“π‘–=1π‘”ξ€·πœ‰π‘–,πœ‰π‘–ξ€Έ+πœ…π‘šξ“π‘Ž=π‘Ÿ+1πœ–π‘Žπ‘”ξ€·π‘‹π‘Ž,π‘‹π‘Žξ€Έ=πœ…(π‘šβˆ’π‘Ÿ).(4.14) Thus, we have π‘ŸRic(𝑋,π‘Œ)=π‘šβˆ’π‘Ÿπ‘”(𝑋,π‘Œ),(4.15) which provides a geometric interpretation of half lightlike Einstein submanifold (the same as in Riemannian case) as we have shown that the constant πœ…=π‘Ÿ/(π‘šβˆ’π‘Ÿ).

5. Proof of Theorem 1.2

Assume that 𝜁 is tangent to 𝑀, 𝑆(𝑇𝑀) is totally umbilical, and 𝑆(π‘‡π‘€βŸ‚) is a conformal Killing vector field. Using (1.1), (2.26) reduces toξ€·βˆ‡π‘‹β„Žπ‘ π›Όξ€Έξ€·βˆ‡(π‘Œ,𝑍)βˆ’π‘Œβ„Žπ‘ π›Όξ€Έ(𝑋,𝑍)=π‘Ÿξ“π‘–=1ξ€½β„Žβ„“π‘–(𝑋,𝑍)πœŒπ‘–π›Ό(π‘Œ)βˆ’β„Žβ„“π‘–(π‘Œ,𝑍)πœŒπ‘–π›Όξ€Ύ+(𝑋)𝑛𝛽=π‘Ÿ+1ξ‚†β„Žπ‘ π›½(𝑋,𝑍)πœƒπ›½π›Ό(π‘Œ)βˆ’β„Žπ‘ π›½(π‘Œ,𝑍)πœƒπ›½π›Όξ‚‡,(𝑋)(5.1)

for all 𝑋,π‘Œ,π‘βˆˆΞ“(𝑇𝑀). Replacing π‘Š by 𝑁 to (1.1), we have̃𝑔𝑅(𝑋,π‘Œ)𝑍,𝑁𝑖=ξ€½π›Όπœ‚π‘–(𝑋)+π‘’π‘–ξ€Ύβˆ’ξ€½π›½πœƒ(𝑋)𝑔(π‘Œ,𝑍)π›Όπœ‚π‘–(π‘Œ)+π‘’π‘–ξ€Ύξ€½π›½πœƒ(π‘Œ)𝑔(𝑋,𝑍)+π›½πœƒ(π‘Œ)πœ‚π‘–(𝑋)βˆ’πœƒ(𝑋)πœ‚π‘–ξ€Ύ(π‘Œ)πœƒ(𝑍),(5.2)

for all 𝑋,π‘Œ,π‘βˆˆΞ“(𝑇𝑀) and where 𝑒𝑖=πœƒ(𝑁𝑖). Applying βˆ‡π‘‹ to (3.12) and using (2.13), we haveξ€·βˆ‡π‘‹β„Žβˆ—π‘–ξ€Έξ€·π‘‹ξ€Ίπ›Ύ(π‘Œ,𝑃𝑍)=𝑖𝑔(π‘Œ,𝑃𝑍)+π›Ύπ‘–π‘Ÿξ“π‘—=1β„Žβ„“π‘—(𝑋,𝑃𝑍)πœ‚π‘—(π‘Œ),(5.3)

for all 𝑋,π‘Œ,π‘βˆˆΞ“(𝑇𝑀). Substituting this equation into (2.30), we obtain̃𝑔𝑅(𝑋,π‘Œ)𝑃𝑍,𝑁𝑖=ξƒ―π‘‹ξ€Ίπ›Ύπ‘–ξ€»βˆ’π‘Ÿξ“π‘—=1π›Ύπ‘—πœπ‘–π‘—ξƒ°ξƒ―π‘Œξ€Ίπ›Ύ(𝑋)𝑔(π‘Œ,𝑃𝑍)βˆ’π‘–ξ€»βˆ’π‘Ÿξ“π‘—=1π›Ύπ‘—πœπ‘–π‘—ξƒ°(π‘Œ)𝑔(𝑋,𝑃𝑍)+π›Ύπ‘–π‘Ÿξ“π‘—=1β„Žβ„“π‘—(𝑋,𝑃𝑍)πœ‚π‘—(π‘Œ)βˆ’π›Ύπ‘–π‘Ÿξ“π‘—=1β„Žβ„“π‘—(π‘Œ,𝑃𝑍)πœ‚π‘—(𝑋),βˆ€π‘‹,π‘Œ,π‘βˆˆΞ“(𝑇𝑀).(5.4)

Substituting this equation and (5.2) into (2.27) and using πœƒ(πœ‰π‘–)=0, we obtainξƒ―π‘‹ξ€Ίπ›Ύπ‘–ξ€»βˆ’π‘Ÿξ“π‘—=1π›Ύπ‘—πœπ‘–π‘—(𝑋)βˆ’π›Όπœ‚π‘–(𝑋)βˆ’π‘’π‘–π›½πœƒ(𝑋)βˆ’π‘›ξ“π›Ό=π‘Ÿ+1π›Ώπ›ΌπœŒπ‘–π›Όξƒ°βˆ’ξƒ―π‘Œξ€Ίπ›Ύ(𝑋)𝑔(π‘Œ,𝑍)π‘–ξ€»βˆ’π‘Ÿξ“π‘—=1π›Ύπ‘—πœπ‘–π‘—(π‘Œ)βˆ’π›Όπœ‚π‘–(π‘Œ)βˆ’π‘’π‘–π›½πœƒ(π‘Œ)βˆ’π‘›ξ“π›Ό=π‘Ÿ+1π›Ώπ›ΌπœŒπ‘–π›Όξƒ°(π‘Œ)𝑔(𝑋,𝑍)=π›Ύπ‘–ξƒ―π‘Ÿξ“π‘—=1β„Žβ„“π‘—(π‘Œ,𝑃𝑍)πœ‚π‘—(𝑋)βˆ’π‘Ÿξ“π‘—=1β„Žβ„“π‘—(𝑋,𝑃𝑍)πœ‚π‘—ξƒ°ξ€½(π‘Œ)+π›½πœƒ(π‘Œ)πœ‚π‘–(𝑋)βˆ’πœƒ(𝑋)πœ‚π‘–ξ€Ύ(π‘Œ)πœƒ(𝑍),βˆ€π‘‹,π‘Œ,π‘βˆˆΞ“(𝑇𝑀).(5.5)

Replacing π‘Œ by πœ‰π‘– to this and using (2.20)1 and the fact πœƒ(πœ‰π‘–)=0, we haveπ›Ύπ‘–β„Žβ„“π‘–ξƒ―πœ‰(𝑋,π‘Œ)=π‘–ξ€Ίπ›Ύπ‘–ξ€»βˆ’π‘Ÿξ“π‘—=1π›Ύπ‘—πœπ‘–π‘—ξ€·πœ‰π‘–ξ€Έβˆ’π›Όβˆ’π‘›ξ“π›Ό=π‘Ÿ+1π›Ώπ›ΌπœŒπ‘–π›Όξ€·πœ‰π‘–ξ€Έξƒ°π‘”(𝑋,π‘Œ)βˆ’π›½πœƒ(𝑋)πœƒ(π‘Œ),(5.6)for all 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀). Differentiating (3.16) and using (5.1), we haveπ‘Ÿξ“π‘–=1ξ€½π›Ώπ›Όπœ‚π‘–(𝑋)βˆ’πœ–π›ΌπœŒπ‘–π›Όξ€Ύβ„Ž(𝑋)ℓ𝑖(π‘Œ,𝑍)βˆ’π‘Ÿξ“π‘–=1ξ€½π›Ώπ›Όπœ‚π‘–(π‘Œ)βˆ’πœ–π›ΌπœŒπ‘–π›Όξ€Ύβ„Ž(π‘Œ)ℓ𝑖=𝑋𝛿(𝑋,𝑍)𝛼+πœ–π›Όπ‘›ξ“π›½=π‘Ÿ+1πœ–π›½π›Ώπ›½πœƒπ›½π›Όξƒ°βˆ’ξƒ―π‘Œξ€Ίπ›Ώ(𝑋)𝑔(π‘Œ,𝑍)𝛼+πœ–π›Όπ‘›ξ“π›½=π‘Ÿ+1πœ–π›½π›Ώπ›½πœƒπ›½π›Όξƒ°(π‘Œ)𝑔(𝑋,𝑍).(5.7)

Replacing π‘Œ by πœ‰π‘– in the last equation and using (2.20)1, we obtainξ€½π›Ώπ›Όβˆ’πœ–π›ΌπœŒπ‘–π›Όξ€·πœ‰π‘–β„Žξ€Έξ€Ύβ„“π‘–ξƒ―πœ‰(𝑋,𝑍)=𝑖𝛿𝛼+πœ–π›Όπ‘›ξ“π›½=π‘Ÿ+1πœ–π›½π›Ώπ›½πœƒπ›½π›Όξ€·πœ‰π‘–ξ€Έξƒ°π‘”(𝑋,𝑍).(5.8)

As the conformal factor 𝛿𝛼 is nonconstant, we show that π›Ώπ›Όβˆ’πœ–π›ΌπœŒπ‘–π›Ό(πœ‰π‘–)β‰ 0. Thus, we haveβ„Žβ„“π‘–(𝑋,π‘Œ)=πœŽπ‘–π‘”(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀),(5.9) where πœŽπ‘–={πœ‰π‘–[𝛿𝛼]+πœ–π›Όβˆ‘π‘›π›½=π‘Ÿ+1πœ–π›½π›Ώπ›½πœƒπ›½π›Ό(πœ‰π‘–)}(π›Ώπ›Όβˆ’πœ–π›ΌπœŒπ‘–π›Ό(πœ‰π‘–))βˆ’1. From (3.17)1 and (5.9), we show that the second fundamental form tensor β„Ž, given by βˆ‘β„Ž(𝑋,π‘Œ)=π‘Ÿπ‘–=1β„Žβ„“π‘–(𝑋,π‘Œ)𝑁𝑖+βˆ‘π‘›π›Ό=π‘Ÿ+1β„Žπ‘ π›Ό(𝑋,π‘Œ)π‘Šπ›Ό, satisfiesβ„Ž(𝑋,π‘Œ)=ℋ𝑔(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀).(5.10)

Thus, 𝑀 is totally umbilical [5]. Substituting (5.9) into (5.6), we haveξƒ―πœ‰π‘–ξ€Ίπ›Ύπ‘–ξ€»βˆ’π‘Ÿξ“π‘—=1π›Ύπ‘—πœπ‘–π‘—ξ€·πœ‰π‘–ξ€Έβˆ’π›Ύπ‘–πœŽπ‘–βˆ’π›Όβˆ’π‘›ξ“π›Ό=π‘Ÿ+1π›Ώπ›ΌπœŒπ‘–π›Όξ€·πœ‰π‘–ξ€Έξƒ°π‘”(𝑋,π‘Œ)=π›½πœƒ(𝑋)πœƒ(π‘Œ),(5.11)

for all 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀). Taking 𝑋=π‘Œ=𝜁 to this equation, we have𝛽=πœ‰π‘–ξ€Ίπ›Ύπ‘–ξ€»βˆ’π‘Ÿξ“π‘—=1π›Ύπ‘—πœπ‘–π‘—ξ€·πœ‰π‘–ξ€Έβˆ’π›Ύπ‘–πœŽπ‘–βˆ’π›Όβˆ’π‘›ξ“π›Ό=π‘Ÿ+1π›Ώπ›ΌπœŒπ‘–π›Όξ€·πœ‰π‘–ξ€Έ.(5.12)

Assume that 𝛽≠0, then we have𝑔(𝑋,π‘Œ)=πœƒ(𝑋)πœƒ(π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀).(5.13) Substituting (5.13) into (1.1) and using (2.25), (3.12), (3.17)1, and (5.9), we have=𝑔(𝑅(𝑋,π‘Œ)𝑍,π‘Š)𝛼+2𝛽+π‘Ÿξ“π‘–=1πœŽπ‘–π›Ύπ‘–+𝑛𝛼=π‘Ÿ+1πœ–π›Όπ›Ώ2𝛼ξƒͺ{𝑔(π‘Œ,𝑍)𝑔(𝑋,π‘Š)βˆ’π‘”(𝑋,𝑍)𝑔(π‘Œ,π‘Š)},(5.14) for all 𝑋,π‘Œ,𝑍,π‘ŠβˆˆΞ“(𝑇𝑀). Substituting (5.9) and (5.13) into (3.18), we haveRic(𝑋,π‘Œ)=(π‘šβˆ’1)(𝛼+𝛽)+(π‘šβˆ’π‘Ÿβˆ’1)π‘Ÿξ“π‘–=1πœŽπ‘–π›Ύπ‘–+𝑛𝛼=π‘Ÿ+1πœ–π›Όπ›Ώ2𝛼ξƒͺ+π‘›ξ“π‘Ÿπ›Ό=π‘Ÿ+1𝑖=1π›Ώπ›ΌπœŒπ‘–π›Όξ€·πœ‰π‘–ξ€Έξƒ°π‘”(𝑋,π‘Œ).(5.15) On the other hand, substituting (5.14) and the fact thatξ€·π‘…ξ€·πœ‰Μƒπ‘”π‘–ξ€Έ,π‘Œπ‘‹,𝑁𝑖=𝛼+𝛽+𝑛𝛼=π‘Ÿ+1πœ–π›Όπ›Ώπ›ΌπœŒπ‘–π›Όξ€·πœ‰π‘–ξ€Έξƒ°π‘”(𝑋,π‘Œ)(5.16)

into (3.4), we haveRic(𝑋,π‘Œ)=(π‘šβˆ’1)𝛼+2(π‘šβˆ’1)𝛽+(π‘šβˆ’π‘Ÿβˆ’1)π‘Ÿξ“π‘–=1πœŽπ‘–π›Ύπ‘–+𝑛𝛼=π‘Ÿ+1πœ–π›Όπ›Ώ2𝛼ξƒͺ+π‘›ξ“π‘Ÿπ›Ό=π‘Ÿ+1𝑖=1π›Ώπ›ΌπœŒπ‘–π›Όξ€·πœ‰π‘–ξ€Έξƒ°π‘”(𝑋,π‘Œ).(5.17) Comparing (5.15) and (5.17), we obtain (π‘šβˆ’1)𝛽=0. As π‘š>1, we have 𝛽=0, which is a contradiction. Thus, we have 𝛽=0. Consequently, by (1.1), (2.29), and (5.14), we show that 𝑀 and π‘€βˆ— are spaces of constant curvatures 𝛼 and βˆ‘(𝛼+2π‘Ÿπ‘–=1πœŽπ‘–π›Ύπ‘–+βˆ‘π‘›π›Ό=π‘Ÿ+1πœ–π›Όπ›Ώ2𝛼), respectively. Letξƒ―ξƒ©πœ…=(π‘šβˆ’1)𝛼+(π‘šβˆ’π‘Ÿβˆ’1)π‘Ÿξ“π‘–=1πœŽπ‘–π›Ύπ‘–+𝑛𝛼=π‘Ÿ+1πœ–π›Όπ›Ώ2𝛼ξƒͺ+π‘›ξ“π‘Ÿπ›Ό=π‘Ÿ+1𝑖=1π›Ώπ›ΌπœŒπ‘–π›Όξ€·πœ‰π‘–ξ€Έξƒ°,(5.18)

then (5.15) and (5.17) reduce to𝑅(0,2)(𝑋,π‘Œ)=Ric(𝑋,π‘Œ)=πœ…π‘”(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀).(5.19)

Thus, 𝑀 is an Einstein manifold. The scalar quantity 𝑐 of 𝑀 is given by𝑐=π‘Ÿξ“π‘–=1𝑅(0,2)ξ€·πœ‰π‘–,πœ‰π‘–ξ€Έ+π‘šξ“π‘Ž=π‘Ÿ+1πœ–π‘Žπ‘…(0,2)ξ€·π‘‹π‘Ž,π‘‹π‘Žξ€Έ=π‘Ÿξ“π‘–=1ξ€·πœ‰πœ…π‘”π‘–,πœ‰π‘–ξ€Έ+πœ…π‘šξ“π‘Ž=π‘Ÿ+1πœ–π‘Žπ‘”ξ€·π‘‹π‘Ž,π‘‹π‘Žξ€Έ=πœ…(π‘šβˆ’π‘Ÿ).(5.20)

Thus, we have𝑐Ric(𝑋,π‘Œ)=π‘”π‘šβˆ’π‘Ÿ(𝑋,π‘Œ).(5.21)

Example 5.1. Let (𝑀,𝑔) be a lightlike hypersurface of an indefinite Kenmotsu manifold 𝑀 equipped with a screen distribution 𝑆(𝑇𝑀), then there exist an almost contact metric structure (𝐽,𝜁,πœ—,𝑔) on 𝑀, where 𝐽 is a (1,1)-type tensor field, 𝜁 is a vector field, πœ— is a 1-form, and 𝑔 is the semi-Riemannian metric on 𝑀 such that 𝐽2𝑋=βˆ’π‘‹+πœ—(𝑋)𝜁,𝐽𝜁=0,πœ—βˆ˜π½=0,πœ—(𝜁)=1,πœ—(𝑋)=𝑔(𝜁,𝑋),𝑔(𝐽𝑋,π½π‘Œ)=𝑔(𝑋,π‘Œ)βˆ’πœ—(𝑋)πœ—(π‘Œ),βˆ‡π‘‹ξ‚€πœ=βˆ’π‘‹+πœ—(𝑋)𝜁,βˆ‡π‘‹π½ξ‚π‘Œ=βˆ’π‘”(𝐽𝑋,π‘Œ)𝜁+πœ—(π‘Œ)𝐽𝑋,(5.22) for any vector fields 𝑋,π‘Œ on 𝑀, where βˆ‡ is the Levi-Civita connection of 𝑀. Using the local second fundamental forms 𝐡 and 𝐢 of 𝑀 and 𝑆(𝑇𝑀), respectively, and the projection morphism 𝑃 of 𝑀 on 𝑆(𝑇𝑀), the curvature tensors 𝑅,𝑅, and π‘…βˆ— of the connections βˆ‡, βˆ‡, and βˆ‡βˆ— on 𝑀,𝑀, and 𝑆(𝑇𝑀), respectively, are given by (see [9]) 𝑔𝑅𝑅(𝑋,π‘Œ)𝑍,π‘ƒπ‘Š=𝑔(𝑅(𝑋,π‘Œ)𝑍,π‘ƒπ‘Š)+𝐡(𝑋,𝑍)𝐢(π‘Œ,π‘ƒπ‘Š)βˆ’π΅(π‘Œ,𝑍)𝐢(𝑋,π‘ƒπ‘Š),𝑔(𝑅(𝑋,π‘Œ)𝑃𝑍,π‘ƒπ‘Š)=π‘”βˆ—ξ€Έ(𝑋,π‘Œ)𝑃𝑍,π‘ƒπ‘Š+𝐢(𝑋,𝑃𝑍)𝐡(π‘Œ,π‘ƒπ‘Š)βˆ’πΆ(π‘Œ,𝑃𝑍)𝐡(𝑋,π‘ƒπ‘Š),(5.23) for any 𝑋,π‘Œ,𝑍,π‘ŠβˆˆΞ“(𝑇𝑀). In case the ambient manifold 𝑀 is a space form 𝑀(𝑐) of constant 𝐽-holomorphic sectional curvature 𝑐, 𝑅 is given by (see [10]) 𝑅(𝑋,π‘Œ)𝑍=𝑔(𝑋,𝑍)π‘Œβˆ’π‘”(π‘Œ,𝑍)𝑋.(5.24) Assume that 𝑀 is almost screen conformal, that is, 𝐢(𝑋,π‘ƒπ‘Œ)=πœ‘π΅(𝑋,π‘ƒπ‘Œ)+πœ‚(𝑋)πœ—(π‘Œ),(5.25) where πœ‘ is a nonvanishing function on a neighborhood 𝒰 in 𝑀, and 𝜁 is tangent to 𝑀, then, by the method in Section 2 of [9], we have 𝐡(𝑋,π‘Œ)=𝜌{𝑔(𝑋,π‘Œ)βˆ’πœ—(𝑋)πœ—(π‘Œ)},(5.26) where 𝜌 is a nonvanishing function on a neighborhood 𝒰. Then the leaf π‘€βˆ— of 𝑆(𝑇𝑀) is a semi-Riemannian manifold of quasiconstant curvature such that 𝛼=βˆ’1+2πœ‘πœŒ2, 𝛽=βˆ’2πœ‘πœŒ2, and πœƒ=πœ— in (1.1).


The authors are thankful to the referee for making various constructive suggestions and corrections towards improving the final version of this paper.


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