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International Journal of Mathematics and Mathematical Sciences
Volumeย 2011, Article IDย 140259, 11 pages
http://dx.doi.org/10.1155/2011/140259
Research Article

Characteristic Lightlike Submanifolds of an Indefinite ๐’ฎ-Manifold

Department of Mathematics, Sogang University, Sinsu-Dong, Mapo-Gu, Seoul 121-742, Republic of Korea

Received 12 April 2011; Revised 12 August 2011; Accepted 23 August 2011

Academic Editor: Christianย Corda

Copyright ยฉ 2011 Jae Won 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.

Abstract

We study characteristic ๐‘Ÿ-lightlike submanifolds ๐‘€ tangent to the characteristic vector fields in an indefinite metric ๐’ฎ-manifold, and we also discuss the existence of characteristic lightlike submanifolds of an indefinite ๐’ฎ-space form under suitable hypotheses: (1) ๐‘€ is totally umbilical or (2) its screen distribution ๐‘†(๐‘‡๐‘€) is totally umbilical in ๐‘€.

1. Introduction

In the theory of submanifolds of semi-Riemannian manifolds, it is interesting to study the geometry of lightlike submanifolds due to the fact that the intersection of normal vector bundle and the tangent bundle is nontrivial, making it interesting and remarkably different from the study of nondegenerate submanifolds. In particular, many authors study lightlike submanifolds on indefinite Sasakian manifolds (e.g., [1โ€“4]).

Similar to Riemannian geometry, it is natural that indefinite ๐’ฎ-manifolds are generalizations of indefinite Sasakian manifolds. Brunetti and Pastore analyzed some properties of indefinite ๐’ฎ-manifolds and gave some characterizations in terms of the Levi-Civita connection and of the characteristic vector fields [5]. After then, they studied the geometry of lightlike hypersurfaces of indefinite ๐’ฎ-manifold [6]. As Jin's generalizations of lightlike submanifolds of the Sasakian manifolds with the general codimension [3, 4, 7], Lee and Jin recently extended lightlike hypersurfaces on indefinite ๐’ฎ-manifold to lightlike submanifolds with codimension 2 on an indefinite ๐’ฎ-manifold, called characteristic half lightlike submanifolds [8]. However, a general notion of characteristic lightlike submanifolds of an indefinite ๐’ฎ-manifold have not been introduced as yet.

The objective of this paper is to study characteristic ๐‘Ÿ-lightlike submanifolds ๐‘€ of an indefinite ๐’ฎ-manifold ๐‘€ subject to the conditions: (1) ๐‘€ is totally umbilcial, or (2) ๐‘†(๐‘‡๐‘€) is totally umbilcal in ๐‘€. In Section 2, we begin with some fundamental formulae in the theory of ๐‘Ÿ-lightlike submanifolds. In Section 3, for an indefinite metric ๐‘”.๐‘“.๐‘“-manifold we consider a lightlike submanifold ๐‘€ tangent to the characteristic vector fields, we recall some basic information about indefinite ๐’ฎ-manifolds and deal with the existence of irrotational characteristic submanifolds of an indefinite ๐’ฎ-space form. Afterwards, we study characteristic ๐‘Ÿ-lightlike submanifolds of ๐‘€ in Sections 4 and 5.

2. Preliminaries

Let (๐‘€,๐‘”) be an ๐‘š-dimensional lightlike submanifold of an (๐‘š+๐‘›)-dimensional semi-Riemannian manifold (๐‘€,๐‘”). Then the radical distribution Rad(๐‘‡๐‘€)=๐‘‡๐‘€โˆฉ๐‘‡๐‘€โŸ‚ is a vector subbundle of the tangent bundle ๐‘‡๐‘€ and the normal bundle ๐‘‡๐‘€โŸ‚, of rank ๐‘Ÿ(1โ‰ค๐‘Ÿโ‰คmin{๐‘š,๐‘›}). In general, there exist two complementary nondegenerate distributions ๐‘†(๐‘‡๐‘€) and ๐‘†(๐‘‡๐‘€โŸ‚) of Rad(๐‘‡๐‘€) in ๐‘‡๐‘€ and ๐‘‡๐‘€โŸ‚, respectively, called the screen and coscreen distributions on ๐‘€, such that๐‘‡๐‘€=Rad(๐‘‡๐‘€)โŠ•orth๐‘†(๐‘‡๐‘€),๐‘‡๐‘€โŸ‚=Rad(๐‘‡๐‘€)โŠ•orth๐‘†๎€ท๐‘‡๐‘€โŸ‚๎€ธ,(2.1) where the symbol โŠ•orth denotes the orthogonal direct sum. We denote such a lightlike submanifold by (๐‘€,๐‘”,๐‘†(๐‘‡๐‘€),๐‘†(๐‘‡๐‘€โŸ‚)). Denote by ๐น(๐‘€) the algebra of smooth functions on ๐‘€ and by ฮ“(๐ธ) the ๐น(๐‘€) module of smooth sections of a vector bundle ๐ธ over ๐‘€. We use the same notation for any other vector bundle. We use the following range of indices:๐‘–,๐‘—,๐‘˜,โ€ฆโˆˆ{1,โ€ฆ,๐‘Ÿ},๐›ผ,๐›ฝ,๐›พ,โ€ฆโˆˆ{๐‘Ÿ+1,โ€ฆ,๐‘›}.(2.2)

Let tr(๐‘‡๐‘€) and ltr(๐‘‡๐‘€) be complementary (but not orthogonal) vector bundles to ๐‘‡๐‘€ in ๐‘‡๐‘€|๐‘€ and ๐‘‡๐‘€โŸ‚ in ๐‘†(๐‘‡๐‘€)โŸ‚, respectively, and let {๐‘1,โ€ฆ,๐‘๐‘Ÿ} be a lightlike basis of ฮ“(ltr(๐‘‡๐‘€)|๐’ฐ) consisting of smooth sections of ๐‘†(๐‘‡๐‘€โŸ‚)โŸ‚|๐’ฐ, where ๐’ฐ is a coordinate neighborhood of ๐‘€, such that๐‘”๎€ท๐‘๐‘–,๐œ‰๐‘—๎€ธ=๐›ฟ๐‘–๐‘—,๐‘”๎€ท๐‘๐‘–,๐‘๐‘—๎€ธ=0,(2.3) where {๐œ‰1,โ€ฆ,๐œ‰๐‘Ÿ} is a lightlike basis of ฮ“(Rad(๐‘‡๐‘€)). Then we have๐‘‡๐‘€=๐‘‡๐‘€โŠ•tr(๐‘‡๐‘€)={Rad(๐‘‡๐‘€)โŠ•tr(๐‘‡๐‘€)}โŠ•orth๐‘†(๐‘‡๐‘€)={Rad(๐‘‡๐‘€)โŠ•ltr(๐‘‡๐‘€)}โŠ•orth๐‘†(๐‘‡๐‘€)โŠ•orth๐‘†๎€ท๐‘‡๐‘€โŸ‚๎€ธ.(2.4)

We say that a lightlike submanifolds (๐‘€,๐‘”,๐‘†(๐‘‡๐‘€),๐‘†(๐‘‡๐‘€โŸ‚)) of ๐‘€ are characterized as follows:(1)๐‘Ÿ-lightlike if 1โ‰ค๐‘Ÿ<min{๐‘š,๐‘›};(2)coisotropic if 1โ‰ค๐‘Ÿ=๐‘›<๐‘š;(3)isotropic if 1โ‰ค๐‘Ÿ=๐‘š<๐‘›;(4)totally lightlike if 1โ‰ค๐‘Ÿ=๐‘š=๐‘›.

The above three classes (2)โ€“(4) are particular cases of the class (1) as follows: ๐‘†(๐‘‡๐‘€โŸ‚)={0},๐‘†(๐‘‡๐‘€)={0}, and ๐‘†(๐‘‡๐‘€)=๐‘†(๐‘‡๐‘€โŸ‚)={0}, respectively. The geometry of ๐‘Ÿ-lightlike submanifolds is more general form than that of the other three type submanifolds. For this reason, in this paper we consider only ๐‘Ÿ-lightlike submanifolds ๐‘€โ‰ก(๐‘€,๐‘”,๐‘†(๐‘‡๐‘€),๐‘†(๐‘‡๐‘€โŸ‚)), with the following local quasiorthonormal field of frames on ๐‘€: ๎€ฝ๐œ‰1,โ€ฆ,๐œ‰๐‘Ÿ,๐‘1,โ€ฆ,๐‘๐‘Ÿ,๐น๐‘Ÿ+1,โ€ฆ,๐น๐‘š,๐‘Š๐‘Ÿ+1,โ€ฆ,๐‘Š๐‘›๎€พ,(2.5) where the sets {๐น๐‘Ÿ+1,โ€ฆ,๐น๐‘š} and {๐‘Š๐‘Ÿ+1,โ€ฆ,๐‘Š๐‘›} are orthonormal basis of ฮ“(๐‘†(๐‘‡๐‘€)) and ฮ“(๐‘†(๐‘‡๐‘€โŸ‚)), respectively.

Let โˆ‡ be the Levi-Civita connection of ๐‘€ and ๐‘ƒ the projection morphism of ฮ“(๐‘‡๐‘€) on ฮ“(๐‘†(๐‘‡๐‘€)) with respect to (2.1). For an ๐‘Ÿ-lightlike submanifold, the local Gauss-Weingartan formulas are given byโˆ‡๐‘‹๐‘Œ=โˆ‡๐‘‹๐‘Œ+๐‘Ÿ๎“๐‘–=1โ„Žโ„“๐‘–(๐‘‹,๐‘Œ)๐‘๐‘–+๐‘›๎“๐›ผ=๐‘Ÿ+1โ„Ž๐‘ ๐›ผ(๐‘‹,๐‘Œ)๐‘Š๐›ผ,(2.6)โˆ‡๐‘‹๐‘๐‘–=โˆ’๐ด๐‘๐‘–๐‘‹+๐‘Ÿ๎“๐‘—=1๐œ๐‘–๐‘—(๐‘‹)๐‘๐‘—+๐‘›๎“๐›ผ=๐‘Ÿ+1๐œŒ๐‘–๐›ผ(๐‘‹)๐‘Š๐›ผ,(2.7)โˆ‡๐‘‹๐‘Š๐›ผ=โˆ’๐ด๐‘Š๐›ผ๐‘‹+๐‘Ÿ๎“๐‘–=1๐œ™๐›ผ๐‘–(๐‘‹)๐‘๐‘–+๐‘›๎“๐›ฝ=๐‘Ÿ+1๐œŽ๐›ผ๐›ฝ(๐‘‹)๐‘Š๐›ฝ,(2.8)โˆ‡๐‘‹๐‘ƒ๐‘Œ=โˆ‡โˆ—๐‘‹๐‘ƒ๐‘Œ+๐‘Ÿ๎“๐‘–=1โ„Žโˆ—๐‘–(๐‘‹,๐‘ƒ๐‘Œ)๐œ‰๐‘–,(2.9)โˆ‡๐‘‹๐œ‰๐‘–=โˆ’๐ดโˆ—๐œ‰๐‘–๐‘‹โˆ’๐‘Ÿ๎“๐‘—=1๐œ๐‘—๐‘–(๐‘‹)๐œ‰๐‘—,(2.10) for any ๐‘‹,๐‘Œโˆˆฮ“(๐‘‡๐‘€), where โˆ‡ and โˆ‡โˆ— are induced linear connections on ๐‘‡๐‘€ and ๐‘†(๐‘‡๐‘€), respectively, the bilinear forms โ„Žโ„“๐‘– and โ„Ž๐‘ ๐›ผ on ๐‘€ are called the local lightlike and screen second fundamental forms on ๐‘‡๐‘€, respectively, โ„Žโˆ—๐‘– are called the local radical second fundamental forms on ๐‘†(๐‘‡๐‘€). ๐ด๐‘๐‘–,๐ดโˆ—๐œ‰๐‘–, and ๐ด๐‘Š๐›ผ are linear operators on ฮ“(๐‘‡๐‘€) and ๐œ๐‘–๐‘—,๐œŒ๐‘–๐›ผ,๐œ™๐›ผ๐‘–, and ๐œŽ๐›ผ๐›ฝ are 1-forms on ๐‘‡๐‘€. Since โˆ‡ is torsion-free, โˆ‡ is also torsion-free and both โ„Žโ„“๐‘– and โ„Ž๐‘ ๐›ผ are symmetric. From the fact โ„Žโ„“๐‘–(๐‘‹,๐‘Œ)=๐‘”(โˆ‡๐‘‹๐‘Œ,๐œ‰๐‘–), we know that โ„Žโ„“๐‘– are independent of the choice of a screen distribution. We say that โ„Ž(๐‘‹,๐‘Œ)=๐‘Ÿ๎“๐‘–=1โ„Žโ„“๐‘–(๐‘‹,๐‘Œ)๐‘๐‘–+๐‘›๎“๐›ผ=๐‘Ÿ+1โ„Ž๐‘ ๐›ผ(๐‘‹,๐‘Œ)๐‘Š๐›ผ(2.11) is the second fundamental tensor of ๐‘€.

The induced connection โˆ‡ on ๐‘‡๐‘€ is not metric and satisfies๎€ทโˆ‡๐‘‹๐‘”๎€ธ(๐‘Œ,๐‘)=๐‘Ÿ๎“๐‘–=1๎€ฝโ„Žโ„“๐‘–(๐‘‹,๐‘Œ)๐œ‚๐‘–(๐‘)+โ„Žโ„“๐‘–(๐‘‹,๐‘)๐œ‚๐‘–๎€พ,(๐‘Œ)(2.12) for all ๐‘‹,๐‘Œโˆˆฮ“(๐‘‡๐‘€), where ๐œ‚๐‘–s are the 1-forms such that๐œ‚๐‘–(๐‘‹)=๐‘”๎€ท๐‘‹,๐‘๐‘–๎€ธ,โˆ€๐‘‹โˆˆฮ“(๐‘‡๐‘€).(2.13) But the connection โˆ‡โˆ— on ๐‘†(๐‘‡๐‘€) is metric. The above three local second fundamental forms are related to their shape operators byโ„Žโ„“๐‘–๎‚€๐ด(๐‘‹,๐‘Œ)=๐‘”โˆ—๐œ‰๐‘–๎‚โˆ’๐‘‹,๐‘Œ๐‘Ÿ๎“๐‘˜=1โ„Žโ„“๐‘˜๎€ท๐‘‹,๐œ‰๐‘–๎€ธ๐œ‚๐‘˜(๐‘Œ),(2.14)โ„Žโ„“๐‘–๎‚€๐ด(๐‘‹,๐‘ƒ๐‘Œ)=๐‘”โˆ—๐œ‰๐‘–๎‚,๐‘‹,๐‘ƒ๐‘Œ๐‘”๎‚€๐ดโˆ—๐œ‰๐‘–๐‘‹,๐‘๐‘—๎‚=0,(2.15)๐œ–๐›ผโ„Ž๐‘ ๐›ผ๎€ท๐ด(๐‘‹,๐‘Œ)=๐‘”๐‘Š๐›ผ๎€ธโˆ’๐‘‹,๐‘Œ๐‘Ÿ๎“๐‘–=1๐œ™๐›ผ๐‘–(๐‘‹)๐œ‚๐‘–(๐‘Œ),(2.16)๐œ–๐›ผโ„Ž๐‘ ๐›ผ(๎€ท๐ด๐‘‹,๐‘ƒ๐‘Œ)=๐‘”๐‘Š๐›ผ๎€ธ,๐‘‹,๐‘ƒ๐‘Œ๐‘”๎€ท๐ด๐‘Š๐›ผ๐‘‹,๐‘๐‘–๎€ธ=๐œ–๐›ผ๐œŒ๐‘–๐›ผ(๐‘‹),(2.17)โ„Žโˆ—๐‘–๎€ท๐ด(๐‘‹,๐‘ƒ๐‘Œ)=๐‘”๐‘๐‘–๎€ธ๐‘‹,๐‘ƒ๐‘Œ,๐œ‚๐‘—๎€ท๐ด๐‘๐‘–๐‘‹๎€ธ+๐œ‚๐‘–๎‚€๐ด๐‘๐‘—๐‘‹๎‚=0,(2.18) where ๐‘‹,๐‘Œโˆˆฮ“(๐‘‡๐‘€) and ๐œ–๐›ผ is the sign of ๐‘Š๐›ผ but it is ยฑ1 related to the causal character of ๐‘Š๐›ผ. From (2.18), we know that each ๐ด๐‘๐‘– is shape operator related to the local second fundamental form โ„Žโˆ—๐‘– on ๐‘†(๐‘‡๐‘€). Replacing ๐‘Œ by ๐œ‰๐‘— in (2.14), we haveโ„Žโ„“๐‘–๎€ท๐‘‹,๐œ‰๐‘—๎€ธ+โ„Žโ„“๐‘—๎€ท๐‘‹,๐œ‰๐‘–๎€ธ=0,(2.19) for all ๐‘‹โˆˆฮ“(๐‘‡๐‘€). It followsโ„Žโ„“๐‘–๎€ท๐‘‹,๐œ‰๐‘–๎€ธ=0,โ„Žโ„“๐‘–๎€ท๐œ‰๐‘—,๐œ‰๐‘˜๎€ธ=0.(2.20) Also, replacing ๐‘‹ by ๐œ‰๐‘— in (2.14) and using (2.20), we haveโ„Žโ„“๐‘–๎€ท๐‘‹,๐œ‰๐‘—๎€ธ๎‚€=๐‘”๐‘‹,๐ดโˆ—๐œ‰๐‘–๐œ‰๐‘—๎‚,๐ดโˆ—๐œ‰๐‘–๐œ‰๐‘—+๐ดโˆ—๐œ‰๐‘—๐œ‰๐‘–=0,๐ดโˆ—๐œ‰๐‘–๐œ‰๐‘–=0.(2.21) For an ๐‘Ÿ-lightlike submanifold, replace ๐‘Œ by ๐œ‰๐‘– in (2.16), we haveโ„Ž๐‘ ๐›ผ๎€ท๐‘‹,๐œ‰๐‘–๎€ธ=โˆ’๐œ–๐›ผ๐œ™๐›ผ๐‘–(๐‘‹),โˆ€๐‘‹โˆˆฮ“(๐‘‡๐‘€).(2.22)

Note 1. Using (2.14) and the fact that โ„Žโ„“๐‘– are symmetric, we have ๐‘”๎‚€๐ดโˆ—๐œ‰๐‘–๎‚๎‚€๐‘‹,๐‘Œโˆ’๐‘”๐‘‹,๐ดโˆ—๐œ‰๐‘–๐‘Œ๎‚=๐‘Ÿ๎“๐‘˜=1๎€ฝโ„Žโ„“๐‘˜๎€ท๐‘‹,๐œ‰๐‘–๎€ธ๐œ‚๐‘˜(๐‘Œ)โˆ’โ„Žโ„“๐‘˜๎€ท๐‘Œ,๐œ‰๐‘–๎€ธ๐œ‚๐‘˜๎€พ.(๐‘‹)(2.23) From this, (2.20) and (2.21), we show that ๐ดโˆ—๐œ‰๐‘– are self-adjoint on ฮ“(๐‘‡๐‘€) with respect to ๐‘” if and only if โ„Žโ„“๐‘–(๐‘‹,๐œ‰๐‘—)=0 for all ๐‘‹โˆˆฮ“(๐‘‡๐‘€),๐‘– and ๐‘— if and only if ๐ดโˆ—๐œ‰๐‘–๐œ‰๐‘—=0 for all ๐‘–,๐‘—. We call self-adjoint ๐ดโˆ—๐œ‰๐‘– the lightlike shape operators of ๐‘€. It follows from the above equivalence and (2.10) that the radical distribution Rad(๐‘‡๐‘€) of a lightlike submanifold ๐‘€, with the lightlike shape operators ๐ดโˆ—๐œ‰๐‘–, is always an integrable distribution.

3. Characteristic Lightlike Submanifolds

A manifold ๐‘€ is called a globally framed f-manifold (or ๐‘”.๐‘“.๐‘“-manifold) if it is endowed with a nonnull (1,1)-tensor field ๐œ™ of constant rank, such that ker๐œ™ is parallelizable, that is, there exist global vector fields ๐œ‰๐›ผ, ๐›ผโˆˆ{1,โ€ฆ,๐‘˜}, with their dual 1-forms ๐œ‚๐›ผ, satisfying ๐œ™2=โˆ’๐ผ+๐œ‚๐›ผโŠ—๐œ‰๐›ผ and ๐œ‚๐›ผ(๐œ‰๐›ฝ)=๐›ฟ๐›ผ๐›ฝ.

The ๐‘”.๐‘“.๐‘“-manifold (๐‘€2๐‘›+๐‘Ÿ,๐œ™,๐œ‰๐›ผ,๐œ‚๐›ผ), ๐›ผโˆˆ{1,โ€ฆ,๐‘˜}, is said to be an indefinite metric ๐‘”.๐‘“.๐‘“-manifold if ๐‘” is a semi-Riemannian metric, with index ๐œˆ, 0<๐œˆ<2๐‘›+๐‘˜, satisfying the following compatibility condtion ๐‘”๎‚€๐œ™๐‘‹,๎‚=๐œ™๐‘Œ๐‘”(๐‘‹,๐‘Œ)โˆ’๐‘Ÿ๎“๐›ผ=1๐œ–๐›ผ๐œ‚๐›ผ(๐‘‹)๐œ‚๐›ผ,(๐‘Œ)(3.1) for any ๐‘‹,๐‘Œโˆˆฮ“(๐‘‡๐‘€), being ๐œ–๐›ผ=ยฑ1 according to whether ๐œ‰๐›ผ is spacelike or timelike. Then, for any ๐›ผโˆˆ{1,โ€ฆ,๐‘˜}, one has ๐œ‚๐›ผ(๐‘‹)=๐œ–๐›ผ๐‘”(๐‘‹,๐œ‰๐›ผ). An indefinite metric ๐‘”.๐‘“.๐‘“-manifold is called an indefinite ๐’ฎ-manifold if it is normal and ๐‘‘๐œ‚๐›ผ=ฮฆ, for any ๐›ผโˆˆ{1,โ€ฆ,๐‘˜}, where ฮฆ(๐‘‹,๐‘Œ)=๐‘”(๐‘‹,๐œ™๐‘Œ) for any ๐‘‹,๐‘Œโˆˆฮ“(๐‘‡๐‘€). The normality condition is expressed by the vanishing of the tensor field ๐‘=๐‘๐œ™+2๐‘‘๐œ‚๐›ผโŠ—๐œ‰๐›ผ, ๐‘๐œ™ being the Nijenhuis torsion of ๐œ™. Furthermore, as proved in [5], the Levi-Civita connection of an indefinite ๐’ฎ-manifold satisfies:๎‚€โˆ‡๐‘‹๐œ™๎‚๐‘Œ=๐‘”๎‚€๐œ™๐‘‹,๎‚๐œ™๐‘Œ๐œ‰+๐œ‚(๐‘Œ)๐œ™2(๐‘‹),(3.2) where โˆ‘๐œ‰=๐‘˜๐›ผ=1๐œ‰๐›ผ and โˆ‘๐œ‚=๐‘˜๐›ผ=1๐œ–๐›ผ๐œ‚๐›ผ. We recall that โˆ‡๐‘‹๐œ‰๐›ผ=โˆ’๐œ–๐›ผ๐œ™๐‘‹ and ker๐œ™ is an integrable flat distribution since โˆ‡๐œ‰๐›ผ๐œ‰๐›ฝ=0 (more details in [5]).

Following the notations in [9], we adopt the curvature tensor ๐‘…, and thus we have ๐‘…(๐‘‹,๐‘Œ,๐‘)=โˆ‡๐‘‹โˆ‡๐‘Œ๐‘โˆ’โˆ‡๐‘Œโˆ‡๐‘‹๐‘โˆ’โˆ‡[๐‘‹,๐‘Œ]๐‘, and ๐‘…(๐‘‹,๐‘Œ,๐‘,๐‘Š)=๐‘”(๐‘…(๐‘,๐‘Š,๐‘Œ),๐‘‹), for any ๐‘‹, ๐‘Œ, ๐‘, ๐‘Šโˆˆฮ“(๐‘‡๐‘€).

An indefinite ๐’ฎ-manifold (๐‘€,๐œ™,๐œ‰๐›ผ,๐œ‚๐›ผ) is called an indefinite ๐’ฎ-space form, denoted by ๐‘€(๐‘), if it has the constant ๐œ™-sectional curvature ๐‘ [5]. The curvature tensor ๐‘… of this space form ๐‘€(๐‘) is given by4๎‚†๐‘…(๐‘‹,๐‘Œ,๐‘,๐‘Š)=โˆ’(๐‘+3๐œ–)๐‘”๎‚€๐œ™๐‘Œ,๎‚๐œ™๐‘๐‘”๎‚€๐œ™๐‘‹,๎‚โˆ’๐œ™๐‘Š๐‘”๎‚€๐œ™๐‘‹,๎‚๐œ™๐‘๐‘”๎‚€๐œ™๐‘Œ,๎‚†๐œ™๐‘Š๎‚๎‚‡โˆ’(๐‘โˆ’๐œ–){ฮฆ(๐‘Š,๐‘‹)ฮฆ(๐‘,๐‘Œ)โˆ’ฮฆ(๐‘,๐‘‹)ฮฆ(๐‘Š,๐‘Œ)+2ฮฆ(๐‘‹,๐‘Œ)ฮฆ(๐‘Š,๐‘)}โˆ’4๐œ‚(๐‘Š)๐œ‚(๐‘‹)๐‘”๎‚€๐œ™๐‘,๎‚โˆ’๐œ™๐‘Œ๐œ‚(๐‘Š)๐œ‚(๐‘Œ)๐‘”๎‚€๐œ™๐‘,๎‚+๐œ™๐‘‹๐œ‚(๐‘Œ)๐œ‚(๐‘)๐‘”๎‚€๐œ™๐‘Š,๎‚โˆ’๐œ™๐‘‹๐œ‚(๐‘)๐œ‚(๐‘‹)๐‘”๎‚€๐œ™๐‘Š,,๐œ™๐‘Œ๎‚๎‚‡(3.3) for any vector fields ๐‘‹,๐‘Œ,๐‘,๐‘Šโˆˆฮ“(๐‘‡๐‘€).

Note 2. Although ๐‘†(๐‘‡๐‘€) is not unique, it is canonically isomorphic to the factor vector bundle ๐‘†(๐‘‡๐‘€)โˆ—=๐‘‡๐‘€/Rad(๐‘‡๐‘€) considered by Kupeli [10]. Thus all screen distributions ๐‘†(๐‘‡๐‘€) are mutually isomorphic. For this reason, we newly define generic lightlike submanifolds of ๐‘€ as follows.

Definition 3.1. Let ๐‘€ be a ๐‘Ÿ-lightlike submanifold of ๐‘€ such that all the characteristic vector fields ๐œ‰๐›ผ are tangent to ๐‘€. A screen distribution ๐‘†(๐‘‡๐‘€) is said to be characteristic if ker๐œ™โŠ‚๐‘†(๐‘‡๐‘€) and ๐œ™(๐‘†(๐‘‡๐‘€)โŸ‚)โŠ‚ฮ“(๐‘†(๐‘‡๐‘€)).

Definition 3.2. A ๐‘Ÿ-lightlike submanifold ๐‘€ of ๐‘€ is said to be characteristic if ker๐œ™โŠ‚๐‘‡๐‘€ and a characteristic screen distribution (๐‘†(๐‘‡๐‘€)) is chosen.

Proposition 3.3 (see [6]). Let (๐‘€,๐‘”,๐‘†(๐‘‡๐‘€)) be a lightlike hypersurface of an indefinite ๐’ฎ-manifold (๐‘€,๐œ™,๐œ‰๐›ผ,๐œ‚๐›ผ,๐‘”) such that the characteristic vector fields are tangent to ๐‘€. Then there exists a screen distribution such that ker๐œ™โŠ‚๐‘‡๐‘€ and ๐œ™(๐ธ)โŠ‚ฮ“(๐‘†(๐‘‡๐‘€)), where ๐ธ is a nonzero section of Rad(๐‘‡๐‘€).

Proposition 3.4 (see [8]). Let (๐‘€,๐‘”,๐‘†(๐‘‡๐‘€)) be a 1-lightlike submanifold of codimension 2 of an indefinite ๐’ฎ-manifold (๐‘€,๐œ™,๐œ‰๐›ผ,๐œ‚๐›ผ,๐‘”). Then ๐‘€ is a characteristic lightlike submanifold of ๐‘€.

Definition 3.5. A lightlike submanifold ๐‘€ is said to be irrotational [10] if โˆ‡๐‘‹๐œ‰๐‘–โˆˆฮ“(๐‘‡๐‘€) for any ๐‘‹โˆˆฮ“(๐‘‡๐‘€) and ๐œ‰๐‘–โˆˆฮ“(Rad(๐‘‡๐‘€)) for all ๐‘–.

Note 3. For an ๐‘Ÿ-lightlike ๐‘€, the above definition is equivalent to โ„Žโ„“๐‘—๎€ท๐‘‹,๐œ‰๐‘–๎€ธ=0,โ„Ž๐‘ ๐›ผ๎€ท๐‘‹,๐œ‰๐‘–๎€ธ=๐œ™๐›ผ๐‘–(๐‘‹)=0,โˆ€๐‘‹โˆˆฮ“(๐‘‡๐‘€).(3.4)

The extrinsic geometry of lightlike hypersurfaces depends on a choice of screen distribution, or equivalently, normalization. Since the screen distribution is not uniquely determined, a well-defined concept of ๐’ฎ-manifold is not possible for an arbitrary lightlike submanifold of a semi-Riemannian manifold, then one must look for a class of normalization for which the induced Riemannian curvature has the desired symmetries. Let (๐‘€,๐‘”) be a semi-Riemannian manifold, ๐‘โˆˆ๐‘€. ๐นโŠ—4๐‘‡โˆ—๐‘๐‘€ is said to be an algebraic curvature tensor [11] on ๐‘‡๐‘๐‘€ if it satisfies the following symmetries: ๐น๐น(๐‘‹,๐‘Œ,๐‘,๐‘Š)=โˆ’๐น(๐‘Œ,๐‘‹,๐‘,๐‘Š)=๐น(๐‘,๐‘Š,๐‘‹,๐‘Œ),(๐‘‹,๐‘Œ,๐‘,๐‘Š)+๐น(๐‘Œ,๐‘,๐‘‹,๐‘Š)+๐น(๐‘,๐‘‹,๐‘Œ,๐‘Š)=0.(3.5)

Definition 3.6. A screen distribution ๐‘†(๐‘‡๐‘€) is said to be admissible if the associated induced Riemannian curvature is an algebraic curvature tensor.

Theorem 3.7. Let (๐‘€,๐‘”,๐‘†(๐‘‡๐‘€)) be an irrotational generic characteristic lightlike submanifold of an indefinite ๐’ฎ-space form (๐‘€(๐‘),๐œ™,๐œ‰๐›ผ,๐œ‚๐›ผ,๐‘”) with an admissible screen distribution ๐‘†(๐‘‡๐‘€). Then one has ๐‘=๐œ–.

Proof. Denote by ๐‘… and ๐‘… the curvature tensors of โˆ‡ and โˆ‡, respectively. Using the local Gauss-Weingarten formulas for ๐‘€, we obtain ๐‘…(๐‘‹,๐‘Œ)๐‘=๐‘…(๐‘‹,๐‘Œ)๐‘+๐‘Ÿ๎“๐‘–=1๎€ฝโ„Žโ„“๐‘–(๐‘‹,๐‘)๐ด๐‘๐‘–๐‘Œโˆ’โ„Žโ„“๐‘–(๐‘Œ,๐‘)๐ด๐‘๐‘–๐‘‹๎€พ+๐‘›๎“๐›ผ=๐‘Ÿ1๎€ฝโ„Ž๐‘ ๐›ผ(๐‘‹,๐‘)๐ด๐‘Š๐›ผ๐‘Œโˆ’โ„Ž๐‘ ๐›ผ(๐‘Œ,๐‘)๐ด๐‘Š๐›ผ๐‘‹๎€พ+๐‘Ÿ๎“๐‘–=1๎ƒฏ๎€ทโˆ‡๐‘‹โ„Žโ„“๐‘–๎€ธ๎€ทโˆ‡(๐‘Œ,๐‘)โˆ’๐‘Œโ„Žโ„“๐‘–๎€ธ+(๐‘‹,๐‘)๐‘Ÿ๎“๐‘—=1๎€บ๐œ๐‘—๐‘–(๐‘‹)โ„Žโ„“๐‘—(๐‘Œ,๐‘)โˆ’๐œ๐‘—๐‘–(๐‘Œ)โ„Žโ„“๐‘—(๎€ป+๐‘‹,๐‘)๐‘›๎“๐›ผ=๐‘Ÿ+1๎€บ๐œ™๐›ผ๐‘–(๐‘‹)โ„Ž๐‘ ๐›ผ(๐‘Œ,๐‘)โˆ’๐œ™๐›ผ๐‘–(๐‘Œ)โ„Ž๐‘ ๐›ผ๎€ป๎ƒฐ๐‘(๐‘‹,๐‘)๐‘–+๐‘›๎“๐›ผ=๐‘Ÿ+1๎ƒฏ๎€ทโˆ‡๐‘‹โ„Ž๐‘ ๐›ผ๎€ธ(๎€ทโˆ‡๐‘Œ,๐‘)โˆ’๐‘Œโ„Ž๐‘ ๐›ผ๎€ธ(+๐‘‹,๐‘)๐‘Ÿ๎“๐‘–=1๎€บ๐œŒ๐‘–๐›ผ(๐‘‹)โ„Žโ„“๐‘–(๐‘Œ,๐‘)โˆ’๐œŒ๐‘–๐›ผ(๐‘Œ)โ„Ž๐‘ ๐›ผ๎€ป+(๐‘‹,๐‘)๐‘›๎“๐›ฝ=๐‘Ÿ+1๎‚ƒ๐œŽ๐›ฝ๐›ผ(๐‘‹)โ„Ž๐‘ ๐›ฝ(๐‘Œ,๐‘)โˆ’๐œŽ๐›ฝ๐›ผ(๐‘Œ)โ„Ž๐‘ ๐›ฝ(๎‚„๎ƒฐ๐‘Š๐‘‹,๐‘)๐›ผ,(3.6) for all ๐‘‹,๐‘Œ,๐‘โˆˆฮ“(๐‘‡๐‘€). Replace ๐‘ by ๐œ‰๐‘˜ in (3.6) and use (2.10), (2.15), (2.17), and (3.4), we have ๐‘…(๐‘‹,๐‘Œ)๐œ‰๐‘˜=๐‘…(๐‘‹,๐‘Œ)๐œ‰๐‘˜+๐‘Ÿ๎“๐‘–=1๎‚†๐‘”๎‚€๐ดโˆ—๐œ‰๐‘–๐‘Œ,๐ดโˆ—๐œ‰๐‘˜๐‘‹๎‚๎‚€๐ดโˆ’๐‘”โˆ—๐œ‰๐‘–๐‘‹,๐ดโˆ—๐œ‰๐‘˜๐‘Œ๐‘๎‚๎‚‡๐‘–+๐‘›๎“๐›ผ=๐‘Ÿ+1๐œ–๐›ผ๎‚†๐‘”๎‚€๐ด๐‘Š๐›ผ๐‘Œ,๐ดโˆ—๐œ‰๐‘˜๐‘‹๎‚๎‚€๐ดโˆ’๐‘”W๐›ผ๐‘‹,๐ดโˆ—๐œ‰๐‘˜๐‘Œ๐‘Š๎‚๎‚‡๐›ผ.(3.7) Using (3.7), the fact ๐‘…(๐‘‹,๐‘Œ)๐‘โˆˆฮ“(๐‘‡๐‘€) for ๐‘‹,๐‘Œ,๐‘โˆˆฮ“(๐‘‡๐‘€), and a screen distribution ๐‘†(๐‘‡๐‘€) is admissible, we get ๐‘”๎‚€๐‘…(๐‘‹,๐‘Œ)๐‘,๐œ‰๐‘˜๎‚=โˆ’๐‘”๎‚€๐‘…(๐‘‹,๐‘Œ)๐œ‰๐‘˜๎‚๎€ท,๐‘=โˆ’๐‘”๐‘…(๐‘‹,๐‘Œ)๐œ‰๐‘˜๎€ธ+,๐‘๐‘Ÿ๎“๐‘–=1๎‚†๐‘”๎‚€๐ดโˆ—๐œ‰๐‘–๐‘‹,๐ดโˆ—๐œ‰๐‘˜๐‘Œ๎‚๎‚€๐ดโˆ’๐‘”โˆ—๐œ‰๐‘–๐‘Œ,๐ดโˆ—๐œ‰๐‘˜๐‘‹๐œ‚๎‚๎‚‡๐‘–(๎€ท๐‘)=๐‘”๐‘…(๐‘‹,๐‘Œ)๐‘,๐œ‰๐‘˜๎€ธ+๐‘Ÿ๎“๐‘–=1๎‚†๐‘”๎‚€๐ดโˆ—๐œ‰๐‘–๐‘‹,๐ดโˆ—๐œ‰๐‘˜๐‘Œ๎‚๎‚€๐ดโˆ’๐‘”โˆ—๐œ‰๐‘–๐‘Œ,๐ดโˆ—๐œ‰๐‘˜๐‘‹๐œ‚๎‚๎‚‡๐‘–=(๐‘)๐‘Ÿ๎“๐‘–=1๎‚†๐‘”๎‚€๐ดโˆ—๐œ‰๐‘–๐‘‹,๐ดโˆ—๐œ‰๐‘˜๐‘Œ๎‚๎‚€๐ดโˆ’๐‘”โˆ—๐œ‰๐‘–๐‘Œ,๐ดโˆ—๐œ‰๐‘˜๐‘‹๐œ‚๎‚๎‚‡๐‘–(๐‘),โˆ€๐‘‹,๐‘Œ,๐‘โˆˆฮ“(๐‘‡๐‘€).(3.8) On the other hand, since ๐œ‚(๐œ‰๐›ผ)=0 and ๐‘”(๐œ™๐œ‰๐›ผ,๐œ™๐‘‹)=0 for any ๐‘‹โˆˆฮ“(๐‘‡๐‘€), ๐‘€(๐‘) is an indefinite ๐’ฎ-space form implies the Riemannian curvature ๐‘… in (3.3) is given by 4๐‘…๎€ท๐‘‹,๐‘Œ,๐‘,๐œ‰๐›ผ๎€ธ๎€ฝฮฆ๎€ท๐œ‰=(๐‘โˆ’๐œ–)๐›ผ๎€ธ๎€ท๐œ‰,๐‘‹ฮฆ(๐‘,๐‘Œ)โˆ’ฮฆ(๐‘,๐‘‹)ฮฆ๐›ผ๎€ธ๎€ท๐œ‰,๐‘Œ+2ฮฆ(๐‘‹,๐‘Œ)ฮฆ๐›ผ๎‚†,๐‘๎€ธ๎€พ=โˆ’(๐‘โˆ’๐œ–)๐‘”๎‚€๐œ™๐œ‰๐›ผ๎‚,๐‘‹ฮฆ(๐‘,๐‘Œ)โˆ’ฮฆ(๐‘,๐‘‹)๐‘”๎‚€๐œ™๐œ‰๐›ผ๎‚,๐‘Œ+2ฮฆ(๐‘‹,๐‘Œ)๐‘”๎‚€๐œ™๐œ‰๐›ผ,,๐‘๎‚๎‚‡(3.9) for any ๐‘‹,๐‘Œ,๐‘,โˆˆฮ“(๐‘‡๐‘€). So, replacing ๐‘‹, ๐‘Œ, ๐‘ by ๐‘ƒ๐‘‹, ๐œ‰, ๐‘ƒ๐‘ in (3.9), we find 4๐‘…๎€ท๐‘‹,๐‘Œ,๐‘,๐œ‰๐›ผ๎€ธ๎‚†โˆ’=โˆ’(๐‘โˆ’๐œ–)๐‘”๎‚€๐œ™๐œ‰๐›ผ๎‚,๐‘ƒ๐‘‹๐‘”๎‚€๐‘ƒ๐‘,๐œ™๐œ‰๐›ผ๎‚โˆ’2๐‘”๎‚€๐‘‹,๐œ™๐œ‰๐›ผ๎‚๐‘”๎‚€๐œ™๐œ‰๐›ผ,๐‘๎‚๎‚‡=3(๐‘โˆ’๐œ–)๐‘”๎‚€๐œ™๐œ‰๐›ผ๎‚,๐‘ƒ๐‘‹๐‘”๎‚€๐œ™๐œ‰๐›ผ๎‚.,๐‘ƒ๐‘(3.10) Then, using (3.3), (3.8), and (3.9), we get 4๐‘Ÿ๎“๐‘–=1๎‚†๐‘”๎‚€๐ดโˆ—๐œ‰๐‘–๐‘‹,๐ดโˆ—๐œ‰๐‘˜๐‘Œ๎‚๎‚€๐ดโˆ’๐‘”โˆ—๐œ‰๐‘–๐‘Œ,๐ดโˆ—๐œ‰๐‘˜๐‘‹๐œ‚๎‚๎‚‡๐‘–(๐‘)=โˆ’3(๐‘โˆ’๐œ–)๐‘”๎‚€๐œ™๐œ‰๐›ผ๎‚,๐‘ƒ๐‘‹๐‘”๎‚€๐œ™๐œ‰๐›ผ๎‚,๐‘ƒ๐‘,โˆ€๐‘‹,๐‘Œ,๐‘โˆˆฮ“(๐‘‡๐‘€).(3.11) Choosing ๐‘‹=๐‘=๐œ™๐‘๐›ผโˆˆฮ“(๐‘†(๐‘‡๐‘€)), we obtain ๐‘=๐œ–.

Corollary 3.8. There exist no irrotational characteristic ๐‘Ÿ-lightlike submanifolds (๐‘€,๐‘”,๐‘†(๐‘‡๐‘€)) of an indefinite ๐’ฎ-space form (๐‘€(๐‘),๐œ™,๐œ‰๐›ผ,๐œ‚๐›ผ,๐‘”) with ๐‘โ‰ ๐œ– such that the screen distribution ๐‘†(๐‘‡๐‘€) is admissible.

4. Totally Umbilical Characteristic Lightlike Submanifolds

Definition 4.1. An ๐‘Ÿ-lightlike submanifold ๐‘€ of ๐‘€ is said to be totally umbilical [1] if there is a smooth vector field โ„‹โˆˆฮ“(tr(๐‘‡๐‘€)) such that โ„Ž(๐‘‹,๐‘Œ)=โ„‹๐‘”(๐‘‹,๐‘ƒ๐‘Œ),(4.1) for all ๐‘‹,๐‘Œโˆˆฮ“(๐‘‡๐‘€). In case โ„‹=0, we say that ๐‘€ is totally geodesic.

It is easy to see that ๐‘€ is totally umbilical if and only if, on each coordinate neighborhood ๐’ฐ, there exist smooth functions ๐ด๐‘– and ๐ต๐›ผ such that โ„Žโ„“๐‘–(๐‘‹,๐‘Œ)=๐ด๐‘–๐‘”(๐‘‹,๐‘Œ),โ„Ž๐‘ ๐›ผ(๐‘‹,๐‘Œ)=๐ต๐›ผ๐‘”(X,๐‘Œ),(4.2) for any ๐‘‹,๐‘Œโˆˆฮ“(๐‘‡๐‘€). From (4.2) we show that any totally umbilical ๐‘Ÿ-lightlike submanifold of ๐‘€ is irrotational. Thus, by Theorem 3.7, we have the following.

Theorem 4.2. Let (๐‘€,๐‘”,๐‘†(๐‘‡๐‘€)) be a totally umbilical characteristic ๐‘Ÿ-lightlike submanifold of an indefinite ๐’ฎ-space form (๐‘€(๐‘),๐œ™,๐œ‰๐›ผ,๐œ‚๐›ผ,๐‘”). Then one has ๐‘=๐œ–.

Theorem 4.3. Let (๐‘€,๐‘”,๐‘†(๐‘‡๐‘€)) be a totally umbilical characteristic ๐‘Ÿ-lightlike submanifold of an indefinite ๐’ฎ-manifold (๐‘€,๐œ™,๐œ‰๐›ผ,๐œ‚๐›ผ,๐‘”). Then ๐‘€ is totally geodesic.

Proof. Apply โˆ‡๐‘‹ to ๐‘”(๐œ™๐œ‰๐‘–,๐‘Š๐›ผ)=0 with ๐‘‹โˆˆฮ“(๐‘‡๐‘€), for all ๐‘– and ๐›ผ, and use (2.8), (2.10), (2.15), (2.17), (2.22), and (3.2), we have โ„Žโ„“๐‘–๎‚€๐‘‹,๐œ™๐‘Š๐›ผ๎‚=๐œ–๐›ผโ„Ž๐‘ ๐›ผ๎‚€๐‘‹,๐œ™๐œ‰๐‘–๎‚,โˆ€๐‘‹โˆˆฮ“(๐‘‡๐‘€).(4.3) Assume that ๐‘€ is totally umbilical. Then we have ๐ด๐‘–๐‘”๎‚€๐‘‹,๐œ™๐‘Š๐›ผ๎‚=๐œ–๐›ผ๐ต๐›ผ๐‘”๎‚€๐‘‹,๐œ™๐œ‰๐‘–๎‚,โˆ€๐‘‹โˆˆฮ“(๐‘‡๐‘€).(4.4) Replace ๐‘‹ by ๐œ™๐‘๐‘– and ๐‘‹ by ๐œ™๐‘Š๐›ผ by turns, we get ๐ด๐‘–=0 for all ๐‘– and ๐ต๐›ผ=0 for all ๐›ผ. Thus we show that โˆ‘โ„‹=๐‘Ÿ๐‘–=1๐ด๐‘–๐‘๐‘–+โˆ‘๐‘›๐›ผ=๐‘Ÿ+1๐ต๐›ผ๐‘Š๐›ผ=0 and ๐‘€ is totally geodesic.

Corollary 4.4 (see [1]). Let (๐‘€,๐‘”,๐‘†(๐‘‡๐‘€)) be a totally umbilical characteristic ๐‘Ÿ-lightlike submanifold of an indefinite ๐’ฎ-manifold (๐‘€,๐œ™,๐œ‰๐›ผ,๐œ‚๐›ผ,๐‘”). Then there exists a unique torsion-free metric connection โˆ‡ on ๐‘€ induced by the connection โˆ‡ on ๐‘€.

Proof. From (4.2) and Theorem 4.3, we have โ„Žโ„“๐‘–(๐‘‹,๐‘Œ)=0 for all ๐‘‹,๐‘Œโˆˆฮ“(๐‘‡๐‘€) and ๐‘–. Thus, using (2.12), we obtain our assertion.

5. Totally Umbilical Screen Distributions

Definition 5.1. A screen distribution ๐‘†(๐‘‡๐‘€) of ๐‘€ is said to be totally umbilical [1] in ๐‘€ if, for each locally second fundamental form โ„Žโˆ—๐‘–, there exist smooth functions ๐ถ๐‘– on any coordinate neighborhood ๐’ฐ in ๐‘€ such that โ„Žโˆ—๐‘–(๐‘‹,๐‘ƒ๐‘Œ)=๐ถ๐‘–๐‘”(๐‘‹,๐‘Œ),โˆ€๐‘‹,๐‘Œโˆˆฮ“(๐‘‡๐‘€).(5.1) In case ๐ถ๐‘–=0 for all ๐‘–, we say that ๐‘†(๐‘‡๐‘€) is totally geodesic in ๐‘€.

Due to (2.18) and (5.1), we know that ๐‘†(๐‘‡๐‘€) is totally umbilical in ๐‘€ if and only if each shape operators ๐ด๐‘๐‘– of ๐‘†(๐‘‡๐‘€) satisfies๐‘”๎€ท๐ด๐‘๐‘–๎€ธ๐‘‹,๐‘ƒ๐‘Œ=๐ถ๐‘–๐‘”(๐‘‹,๐‘ƒ๐‘Œ),โˆ€๐‘‹,๐‘Œโˆˆฮ“(๐‘‡M),(5.2) for some smooth functions ๐ถ๐‘– on ๐’ฐโŠ†๐‘€.

In general, ๐‘†(๐‘‡๐‘€) is not necessarily integrable. The following result gives equivalent conditions for the integrability of a screen ๐‘†(๐‘‡๐‘€).

Theorem 5.2 (see [1]). Let ๐‘€ be an ๐‘Ÿ-lightlike submanifold of a semi-Riemannian manifold (๐‘€,๐‘”). Then the following assertions are equivalent: (1)๐‘†(๐‘‡๐‘€) is integrable,(2)โ„Žโˆ—๐‘– is symmetric on ฮ“(๐‘†(๐‘‡๐‘€)), for each ๐‘–,(3)๐ด๐‘๐‘– is self-adjoint on ฮ“(๐‘†(๐‘‡๐‘€)) with respect to ๐‘”, for each ๐‘–.

We know that, from (5.2), each shape operator ๐ด๐‘๐‘– is self-adjoint on ฮ“(๐‘†(๐‘‡๐‘€)) with respect to ๐‘”, which further follows from that above theorem that any totally umbilical screen distribution ๐‘†(๐‘‡๐‘€) of ๐‘€ is integrable.

Theorem 5.3. Let (๐‘€,๐‘”,๐‘†(๐‘‡๐‘€)) be a characteristic ๐‘Ÿ-lightlike submanifold of an indefinite ๐’ฎ-manifold (๐‘€,๐œ™,๐œ‰๐›ผ,๐œ‚๐›ผ,๐‘”). If ๐‘†(๐‘‡๐‘€) is totally umbilical in ๐‘€, then ๐‘†(๐‘‡๐‘€) is totally geodesic in ๐‘€.

Proof. Apply the operator โˆ‡๐‘‹ to ๐‘”(๐œ™๐‘๐‘˜,๐‘๐‘—)=0 for some ๐‘˜,๐‘— such that ๐‘˜โ‰ ๐‘—, and use (2.7) and (2.18) โ„Žโˆ—๐‘˜๎‚€๐‘‹,๐œ™๐‘๐‘—๎‚=โ„Žโˆ—๐‘—๎‚€๐‘‹,๐œ™๐‘๐‘˜๎‚,โˆ€๐‘‹โˆˆฮ“(๐‘‡๐‘€).(5.3) Assume that ๐‘†(๐‘‡๐‘€) is totally umbilical in ๐‘€. Then we have ๐ถ๐‘˜๐‘”๎‚€๐‘‹,๐œ™๐‘๐‘—๎‚=๐ถ๐‘—๐‘”๎‚€๐‘‹,๐œ™๐‘๐‘˜๎‚,โˆ€๐‘‹โˆˆฮ“(๐‘‡๐‘€).(5.4) Replacing ๐‘‹ by ๐œ™๐œ‰๐‘— in (5.4) and taking (๐‘˜,๐‘—)=(1,2),(2,3),โ€ฆ,(๐‘Ÿโˆ’1,๐‘Ÿ) and (๐‘Ÿ,1) by turns and use the above method, we have ๐ถ๐‘–=0 for all ๐‘–โˆˆ{1,โ€ฆ,๐‘Ÿ}. Thus we have our assertion.

Theorem 5.4. Let (๐‘€,๐‘”,๐‘†(๐‘‡๐‘€)) be a characteristic ๐‘Ÿ-lightlike submanifold of an indefinite ๐’ฎ-manifold (๐‘€,๐œ™,๐œ‰๐›ผ,๐œ‚๐›ผ,๐‘”) such that ๐‘†(๐‘‡๐‘€) is totally umbilical in ๐‘€. Then ๐‘€ is not irrotational.

Proof. Apply the operator โˆ‡๐‘‹ to ๐‘”(๐œ™๐œ‰๐‘–,๐‘๐‘—)=0 for all ๐‘– and ๐‘—, and use (2.6), (2.10), (2.15), (2.18), and Theorem 5.3, we have โ„Žโ„“๐‘–๎‚€๐‘‹,๐œ™๐‘๐‘—๎‚=โ„Žโˆ—๐‘—๎‚€๐‘‹,๐œ™๐œ‰๐‘–๎‚,โˆ€๐‘‹โˆˆฮ“(๐‘‡๐‘€).(5.5) Since ๐‘†(๐‘‡๐‘€) is totally umbilical, by Theorem 5.3, we have that ๐‘†(๐‘‡๐‘€) is totally geodesic. Then, by (5.5), we have โ„Žโ„“๐‘–๎‚€๐‘‹,๐œ™๐‘๐‘—๎‚=0,โˆ€๐‘‹โˆˆฮ“(๐‘‡๐‘€).(5.6) Apply the operator โˆ‡๐‘‹ to ๐‘”(๐œ‰๐›ผ,๐œ‰๐‘–)=0 and use (2.6), (2.10), (2.15), we have โ„Žโ„“๐‘–๎‚€๐‘‹,๐œ‰๐›ผ๎‚=โˆ’๐œ–๐›ผ๐‘”๎‚€๐‘‹,๐œ™๐œ‰๐‘–๎‚,โˆ€๐‘‹โˆˆฮ“(๐‘‡๐‘€).(5.7) Replace ๐‘‹ by ๐œ™๐‘๐‘– in this equation and (5.6), we have 0=โ„Žโ„“๐‘–๎‚€๐œ™๐‘๐‘–,๐œ‰๐›ผ๎‚๎‚€=โˆ’๐‘”๐œ™๐‘๐‘–,๐œ™๐œ‰๐‘–๎‚=โˆ’1.(5.8) It is a contradiction. Thus ๐‘€ is not irrotational.

Since any totally umbilical ๐‘Ÿ-lightlike submanifold of ๐‘€ is irrotational, by Theorem 5.4, we have the following result.

Corollary 5.5. There exist no totally umbilical characteristic ๐‘Ÿ-lightlike submanifolds (๐‘€,๐‘”,๐‘†(๐‘‡๐‘€)) of an indefinite ๐’ฎ-manifold (๐‘€,๐œ™,๐œ‰๐›ผ,๐œ‚๐›ผ,๐‘”) equipped with a totally umbilical screen distribution ๐‘†(๐‘‡๐‘€) in ๐‘€.

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