INTERNATIONAL JOURNAL OF GEOMETRYVol. 7 (2018), No. 1, 37 - 53

HALF-LIGHTLIKE SUBMANIFOLDSWITH DEGENERATE AND NON-DEGENERATE

PLANAR NORMAL SECTIONSIN PSEUDO-EUCLIDEAN SPACES

FEYZA ESRA ERDO¼GAN, SELCEN YÜKSEL PERKTAŞAND RIFAT GÜNEŞ

Abstract. In this paper we consider half-lightlike submanifolds withdegenerate and non-degenerate planar normal sections in pseudo-Euclideanspaces with index 2. We obtain necessary and su¢ cient conditions for a half-lightlike submanifold to have degenerate and non-degenerate planar normalsections.

1. Introduction

The theory of submanifolds of Riemannian (or semi-Riemannian) mani-folds is one of the most important topics in di¤erential geometry. In case theinduced metric on the submanifold of semi-Riemannian manifold is degener-ate, the study becomes more di¢ cult. Since normal vector bundle intersectswith the tangent bundle in lightlike submanifolds, the geometry of lightlikesubmanifolds is di¤erent from the study of non-degenerate submanifolds.Lightlike submanifolds of semi-Riemannian manifolds was introduced by K.L. Duggal and A. Bejancu in [5]. Several authors have studied lightlike sub-manifolds of semi-Riemannian manifolds (see [9] and many more referencestherein). For a 2 codimensional lightlike submanifold, the dimension of rad-ical distribution is either one or two. A codimension 2 lightlike submanifoldis called half-lightlike [6] if the dimension of radical distribution is 1.Submanifolds with planar normal sections in Euclidean spaces were rst

studied by B.-Y. Chen ([2], [3], [4]). In [12], Y. H. Kim initiated the study ofsemi-Riemannian setting of such submanifolds (see also [10]). Both authorsobtained similar results in these spaces. The authors [8] initiated the studyof lightlike submanifolds with planar normal sections in pseudo-Euclideanspaces.Let M be a half-lightlike submanifold in Rn2 . For a point p in M and a

lightlike vector � tangent to M at p which spans radical distribution, thevector � and transversal space tr(TM) toM at p determine a 2- dimensional Keywords and phrases: Half-lightlike submanifolds, planar normal sections(2010)Mathematics Subject Classication: 53C42, 53C50Received: 15.10.2017. In revised form: 04.01.2018. Accepted: 10.10.1018.

38 Feyza Esra Erdo¼gan, Selcen Yüksel Perktaş and R¬fat Güneş

subspace E(p; �) in Rn2 through p. The intersection ofM and E(p; �) gives alightlike curve in a neighborhood of p; which will be called the degeneratenormal section ofM at the point p in the direction of �. Let v be a spacelikevector tangent to M at p. The vector v and transversal space tr(TM) toM at p then determine a 2-dimensional subspace E(p; v) in Rn2 through p.In this case the intersection of M and E(p; v) gives a spacelike curve ina neighborhood of p; which is called the non-degenerate normal section ofM at the point p in the direction of v. Then, M is said to have degener-ate pointwise and nondegenerate pointwise planar normal sections, if eachdegenerate normal section at p satises 0 ^ 00 ^ 000 = 0 [3, 10, 12, 13].In this paper we investigate necessary and su¢ cient conditions for a half-

lightlike submanifold of Rn2 to have degenerate planar normal sections. Weprove that a half-lightlike submanifold with degenerate planar normal sec-tions in Rn2 is irrotational, if the normal section at p is a geodesic arc on asu¢ ciently small neighborhood of p. Moreover we study half-lightlike sub-manifolds with degenerate planar normal sections in Rn2 in case the normalsection at p is not a geodesic arc on a su¢ ciently small neighborhood of p.It is shown that if a screen conformal half-lightlike submanifold with degen-erate planar normal sections in semi-Euclidean space form Rn2 (c) is minimalthen its null sectional curvature vanishes identically. Finally we considerhalf-lightlike submanifolds of Rn2 (n = 5; 6) with non-degenerate planar nor-mal sections and obtain a generalization for pseudo-Euclidean spaces.

2. Preliminaries

The codimension 2 lightlike submanifold (M; g) is called a half-lightlikesubmanifold if rank(radTM) = 1: In this case, there exists two complemen-tary non-degenerate distributions S(TM) and S(TM?) of RadTM in TMand TM?, respectively, called the screen and co-screen distribution on M .Then we have the following two orthogonal decompositions

TM = RadTM �orth S(TM); TM? = RadTM �orth S(TM?);

where the symbol �orthdenotes the orthogonal direct sum.It is known that [5], for any smooth null section � of RadTM on a co-

ordinate neighborhood U � M , there exists a uniquely dened null vectoreld N 2 �(ltrTM) satisfying

�g (N; �) = 1; �g (N;N) = �g (N;X) = 0;8X 2 �(S(TM)):

We call N , ltr(TM) and tr(TM) = S(TM?) �orth ltr(TM) the lightliketransversal vector eld, lightlike transversal bundle and transversal vectorbundle ofM with respect to the screen S(TM) respectively. Since RadTM isa 1-dimensional vector sub-bundle of TM? we may consider a supplementarydistribution D to RadTM such that it is locally represented by u.We call D a screen transversal bundle of M . Thus we say that the vector

bundle tr(TM) dened over M by

tr(TM) = D �orth Itr(TM);

Half-lightlike submanifolds with degenerate and non-degenerate... 39

Therefore we have [9]

T �M = S(TM) ? (RadTM � tr (TM))= S(TM) ? D ? (RadTM � Itr(TM)) :(1)

Denote by P the projection of TM on S(TM) with respect to the decom-position (1). Then we write

X = PX + � (X) �; 8X 2 � (TM) ;

where � is a local di¤erential 1-form on M dened by � (X) = g (X;N) :Suppose �r is the metric connection on �M: Since f�;Ng is locally a pair oflightlike sections on U � M , we dene symmetric F (M)-bilinear forms D1and D2 and 1-forms �1; �2; "1 and "2 on U: Using (1), we put

�rXY = rXY +D1 (X;Y )N +D2 (X;Y )u(2)�rXN = �ANX + �1 (X)N + �2 (X)u(3)�rXu = �AuX + "1 (X)N + "2 (X)u(4)

for any X;Y 2 � (TM) ; where rXY; ANX and AuX belong to � (TM). Wecalled D1 and D2 the lightlike second fundamental form and screen secondfundamental form of M with respect to tr(TM), respectively. Both ANand Au are linear operators on � (TM). The rst one is � (S (TM))-valued,called the shape operator of M . Since u is a unit vector eld, (4) implies"2 (X) = 0: In a similar way, since � and N are lightlike vector elds, from(2)-(4) we obtain

D1 (X; �) = 0; �g (ANX;N) = 0;(5)

�g (AuX;Y ) = �D2 (X;Y ) + "1 (X) � (Y )(6)

"1 (X) = ��D2 (X; �) ; 8X 2 � (TM)(7)

Next, consider the decomposition (1) then we have

rXPY = r�XPY + E1 (X;PY ) �;(8)rX� = �A��X + u1 (X) �;(9)

where r�XPY and A�� belong to � (S(TM)) : A�� is a linear operator on� (TM) and r� is a metric connection on S(TM). We call E1 the localsecond fundamental form of S(TM) with respect to Rad(TM) and A�� theshape operator of the screen distribution. The geometric object from Gaussand Weingarten equations (2)-(4) on one side and (8 )-(9) on the other sideare related by

E1 (X;PY ) = g (ANX;PY ) ;(10)

D1 (X;PY ) = g�A��X;PY

�;(11)

u1 (X) = ��1 (X) ;

for any X;Y 2 � (TM). From (5) and (11) we derive

(12) A��� = 0:

A half-lightlike submanifold (M; g) of a semi-Riemannian manifold ( �M; �g)is said to be totally umbilical in �M if there is a normal vector eld �Z 2

40 Feyza Esra Erdo¼gan, Selcen Yüksel Perktaş and R¬fat Güneş

� (tr (TM)) on M , called an a¢ ne normal curvature vector eld of M , suchthat

h(X;Y ) = D1 (X;Y )N +D2 (X;Y )u = �Z�g (X;Y ) ; 8X;Y 2 � (TM) :

In particular, (M; g) is said to be totally geodesic if its second fundamentalform h(X;Y ) = 0, for any X;Y 2 � (TM). By direct calculation it is easyto see that M is totally geodesic if and only if both the lightlike and thescreen second fundamental tensors D1 and D2 respectively vanish on M:Moreover, from (3), (6), (7) and (11) we obtain

A� = Au = "1 = �2 = 0:

The notion of screen locally conformal half-lightlike submanifolds has beenintroduced by K.L. Duggal and B.Sahin [9] as follows.A half-lightlike submanifold M of a semi-Riemannian manifold is called

screen locally conformal if on any coordinate neighborhood U there exists anon-zero smooth function ' such that for any null vector eld � 2 �(TM?)the relation

(13) ANX = 'A��X; 8X 2 �(TM jU );

holds between the shape operators AN and A�� of M and S(TM), respec-tively [9].On the other hand the notion of minimal lightlike submanifolds has been

dened by K. L. Duggal and A. Bejancu [5] as follows.

Denition 2.1. LetM be a half-lightlike submanifold of a semi-Riemannianmanifold �M: Then, we say that M is a minimal half-lightlike submanifold if�tr jS(TM) h = 0

�and "1 (X) = 0 [5].

Denition 2.2. A half-lightlike submanifold M is said to be irrotational if�rX� 2 � (TM), for any X 2 � (TM), where � 2 � (RadTM)[9].

For a half-lightlike M , since D1 (X; �) = 0, the above denition is equiv-alent to D2 (X; �) = 0 = "1 (X) ; 8X 2 � (TM).

Corollary 2.1. LetM be an irrotational screen conformal half-lightlike sub-manifold of a semi-Riemannian manifold �M . Then

(i) M is totally geodesic,(ii) M is totally umbilical,(iii) M is minimal,

if and only if a leaf M 0 of any S(TM) is so immersed as a submanifold of�M [9].

3. Non-degenerate planar normal sections of half-lightlikesubmanifolds in Rn2 (n = 5; 6)

In this section, rstly we consider half-lightlike submanifolds which havenon-degenerate planar normal sections and generalize the obtained resultsto pseudo-Euclidean spaces.Let (M; g; S(TM)) be a screen conformal half-lightlike submanifold of�R52; g

�. In this case S(TM) is integrable. We denote integral submanifold

Half-lightlike submanifolds with degenerate and non-degenerate... 41

of S(TM) by M 0: For a spacelike curve on M: we have

0 (s) = v ,(14)

00 (s) = �rvv = r�vv + E1 (v; v) � +D1 (v; v)N +D2 (v; v)u,(15)

000 (s) = r�vr�vv + E1 (v;r�vv) � +D1 (v;r�vv)N +D2 (v;r�vv)u

+v (E1 (v; v)) � + v (D1 (v; v))N + v (D2 (v; v))u

�E1 (v; v)A��v + E1 (v; v)u1 (v) � + E1 (v; v)D2 (v; �)u(16)�D1 (v; v)ANv +D1 (v; v) �1 (v)N +D1 (v; v) �2 (v)u�D2 (v; v)Auv +D2 (v; v) "1 (v)N ,

where r� is the induced connection of M 0. Since S(TM) = Sp fv1; v2g, wewrite

v = av1 + bv2;(17)

r�vv = a2r�v1v1 + abr�v1v2

+abr�v2v1 + b2r�v2v2;(18)

r�vr�vv = a3r�v1r�v1v1 + a

2br�v1r�v1v2 + a

2br�v1r�v2v1

+ab2r�v1r�v2v2 + a

2br�v2r�v1v1 + ab

2r�v2r�v1v2(19)

+ab2r�v2r�v2v1 + b

3r�v2r�v2v2:

Also by using g (v1; v1) = 1, g (v2; v2) = 1 and g (v1; v2) = 0 we have

r�v1v1 = �1v2;r�v2v1 = �2v2(20)

r�v1v2 = �3v1;r�v2v2 = �4v1(21)

�1 + �3 = 0; �2 + �4 = 0(22)

where a; b; �1; �2; �3; �4 2 R. From (17)-(22) we obtainr�vv = �b (a�1 + b�2) v1 + a (a�1 + b�2) v2;(23)

r�vr�vv = �a (a�1 + b�2)2 v1 � b (a�1 + b�2)2 v2;(24)

g (v;r�vv) = 0:(25)Since v is a non-null vector, it is clear that a2 + b2 6= 0. Therefore we nd(26) v ^r�vv = 0:Then from (25) and (26) we have r�vv = 0: Moreover we obtain(27) v ^r�vr�vv = 0:So we have

Theorem 3.1. Let M be a screen conformal half-lightlike submanifold inR52. Then M has spacelike planar normal sections if and only if

(28) T (v; v) ^ �rvT (v; v) = 0;where v 2 �(S(TM)) and T (v; v) = E1 (v; v) � +D1 (v; v)N +D2 (v; v)u:

Proof. Let be a spacelike curve on a screen conformal half-lightlike sub-manifold M . Then by using (66)-(16) and taking into account the fact thatS(TM) = Sp fvg, we have(29) v ^r�vv = 0 and v ^r�vr�vv = 0:

42 Feyza Esra Erdo¼gan, Selcen Yüksel Perktaş and R¬fat Güneş

Assume that M has planar non-degenerate normal sections. Then we have

000 (s) ^ 00 (s) ^ 0 (s) = 0:Thus from (29) one can see that

E1 (v; v) � +D1 (v; v)N +D2 (v; v)u

and

E1 (v;r�vv) � +D1 (v;r�vv)N +D2 (v;r�vv)u+v (E1 (v; v)) � + v (D1 (v; v))N + v (D2 (v; v))u

�E1 (v; v)A��v + E1 (v; v)u1 (v) � + E1 (v; v)D2 (v; �)u�D1 (v; v)ANv +D1 (v; v) �1 (v)N +D1 (v; v) �2 (v)u�D2 (v; v)Auv +D2 (v; v) "1 (v)N

are linearly dependent. We put

�rvT (v; v) = E1 (v;r�vv) � +D1 (v;r�vv)N +D2 (v;r�vv)u+v (E1 (v; v)) � + v (D1 (v; v))N + v (D2 (v; v))u

�E1 (v; v)A��v + E1 (v; v)u1 (v) � + E1 (v; v)D2 (v; �)u�D1 (v; v)ANv +D1 (v; v) �1 (v)N +D1 (v; v) �2 (v)u�D2 (v; v)Auv +D2 (v; v) "1 (v)N;

where is assumed to be parameterized by arc-length. Thus, we obtain

T (v; v) ^ �rvT (v; v) = 0:Conversely, let T (v; v) ^ �rvT (v; v) = 0; for a spacelike tangent vector v ofM at p: Then either T (v; v) = 0 or �rvT (v; v) = 0: If T (v; v) = 0; thenfrom (66)-(16) and (29), M has non-degenerate planar normal sections. If�rvT (v; v) = 0; from (29), we obtain

000 (s) ^ 00 (s) ^ 0 (s) = v ^ T (v; v) ^ �rvT (v; v) = 0:

Example 3.1. Let M be a submanifold in R52 given by

x4 =�x21 + x

22

� 12 ; x3 =

�1� x25

� 12 ; x5; x1; x2 > 0:

Then we have

TM = Span

�� = x1@x1 + x2@x2 + x4@x4; U = x4@x1 + x1@x4

; V = �x5@x3 + x3@x5

�;

TM? = Span = f�; u = x3@x3 + x5@x5g :Thus RadTM = Spanf�g is a distribution on M and S(TM?) = Spanfug.Hence M is a half-lightlike submanifold of R52 with S(TM) = SpanfU; V g.Also, the lightlike transversal bundle ltr(TM) is spanned by

N =1

2x22fx1@x1 � x2@x2 + x4@x4g :

By direct calculations, we obtain�rU� = U; �rV � = 0; �r�� = �;�rUN =

1

2x22U; �rVN = 0; �r�N = �N:

Half-lightlike submanifolds with degenerate and non-degenerate... 43

Then, from (3) and (9) we obtain

A��U = �U;A��V = 0;

ANU = �1

2x22U; �1 (U) = 0; �2 (U) = 0

ANV = 0; �1 (V ) = 0; �2 (V ) = 0;

AN� = 0; �1 (�) = �1; �2 (�) = 0:Hence we derive ANX = 12x22

A��X; 8X 2 � (TM), which shows that M is ascreen conformal lightlike submanifold with ' = 1

2x22: Also we calculate

�rUV = �rV U = �r�V = �rV � = 0; �rU� = U;�rUU =

1

2� + x22N;

�rV V = �u;�r�� = �;�rUu = 0;�rV u = V;�r�u = 0:

Thus from (2)-(4),(9) and (10) we derive

rUU =1

2�;

E1(U;U) =1

2;

AuU = 0;

AuV = �V;Au� = 0;

D1(U;U) = x22; D2(U;U) = 0;

D1(V; V ) = 0; D2(V; V ) = �1;D2(X; �) = 0; "1(X) = 0;

for all X 2 � (TM) [9]. Hence M is irrotational with a symmetric Riccitensor and vanishing null sectional curvature. Therefore, the intersection ofM and E(p; v) gives a non-degenerate curve in a neighborhood of p, whichis called the normal section of M at the point p in the direction of v, namelyv = U + V 2 S(TM) and p 2M . We denote subspace

E (p; v) = fvg [ tr (TM)and we have

E (p; v) \M = ;where is the normal section of M at p in the direction of v. Then we have

0 (s) = v = x4@x1 � x5@x3 + x1@x4 + x3@x5

00 (s) = �rvv = �rUU + �rV U + �rUV + �rV V

= x1@x1 � x3@x3 + x4@x4 � x5@x5

000 (s) = �rv �rvv = x4@x1 + x5@x3 + x1@x4 � x3@x5:

44 Feyza Esra Erdo¼gan, Selcen Yüksel Perktaş and R¬fat Güneş

Hence we get

000 (s) ^ 00 (s) ^ 0 (s) = 0;

which implies that M has non-degenerate planar normal sections.

Proposition 3.1. Let M be a half-lightlike submanifold in R52: If M hasplanar normal sections, then

(30) r�vv = 0

where is normal section in the direction v = 0 (s) for v 2 � (S(TM)) :

Now we dene a function L by

L(p; v) = Lp (v) = hT (v; v); T (v; v)i

onSpM; where

SpM =

nv 2 � (TM) j hv; vi

12 = 1

o: If L 6= 0; thenM has

non-degenerate pointwise normal sections. By a vertex of curve we mean apoint p on such that its curvature � satises d�

2(0)ds = 0: Let M has planar

normal sections. From Proposition 3.1 we obtain

��2 (s) = 2E1 (v; v)D1 (v; v) +D22 (v; v) �;

1

2

d�2 (0)

ds= v(E1 (v; v)D1 (v; v)) + v(D2 (v; v))D2 (v; v) �:

If M is totally geodesic, then D1 = D2 = 0: Thus has a vertex.Consequently, we have

Theorem 3.2. Let M be a half-lightlike submanifold of R52: Then M hasnon-degenerate planar normal sections and it is totally geodesic at p 2M ifand only if normal section curve has a vertex at p 2M .

Theorem 3.3. Let M be a screen conformal half-lightlike submanifold ofR52 with planar normal sections. Then normal section curve has a vertexif and only if M is minimal.

Proof. IfM is totally geodesic, then from (tr jS(TM) h = 0) and "1 (�) = 0,so we conclude.From Theorem 3.2 and Theorem 3.3, we give

Theorem 3.4. Let M be a half-lightlike submanifold in R52 (c) with planarnormal sections. Then K� (H) = 0 if and only if normal section curve hasa vertex at p 2M , where � 2 � (RadTM) :

Theorem 3.5. Let M be a half-lightlike submanifold of R52 and the normalsection at any p be a geodesic arc on a su¢ ciently small neighborhood ofp. Then M has non-degenerate planar normal sections if and only if

h (v; v) ^ ( �rvh) (v; v) = 0

where is h (v; v) = D1 (v; v)N +D2 (v; v)u:

Half-lightlike submanifolds with degenerate and non-degenerate... 45

Proof. If normal section at any p is a geodesic arc on a su¢ ciently smallneighborhood of p, we have

0 (s) = v;

00 (s) = D1 (v; v)N +D2 (v; v)u;

000 (s) = v(D1 (v; v))N + v(D2 (v; v))u

�D1 (v; v)ANv +D1 (v; v) �1 (v)N+D1 (v; v) �2 (v)u�D2 (v; v)Auv+D2 (v; v) "1 (v)N:

Since is a planar curve then we get

v^(D1 (v; v)N +D2 (v; v)u)^

0BB@v(D1 (v; v))N + v(D2 (v; v))u

�D1 (v; v)ANv +D1 (v; v) �1 (v)N+D1 (v; v) �2 (v)u�D2 (v; v)Auv

+D2 (v; v) "1 (v)N

1CCA = 0:Therefore, by taking the covariant derivative of

h (v; v) = D1 (v; v)N +D2 (v; v)u;

we obtain( �rvh) (v; v) = �rvh (v; v) = 000 (s) ;

which implies

000 (s) ^ 00 (s) ^ 0 (s) = v ^ h (v; v) ^ ( �rvh) (v; v) = 0:From the last equation above, we have

h (v; v) ^ ( �rvh) (v; v) = 0:Conversely, we assume that h (v; v)^ ( �rvh) (v; v) = 0. In this case, we haveeither h (v; v) = 0 or ( �rvh) (v; v) = 0. If h (v; v) = 0, we have D1 (v; v) = 0and D2 (v; v) = 0. In this way, we get

�r�v = �Av� + "1 (�)Nand so we write

�g��r�v; �

�= ��g (Av�; �) + "1 (�) ;

0 = "1 (�) ;

which shows that M is minimal and has planar normal sections. On theother hand, if ( �rvh) (v; v) = 0; from ( �rvh) (v; v) = �rvh (v; v) = 000 (s) = 0;we obtain

000 (s) ^ 00 (s) ^ 0 (s) = 0;that is M has non-degenerate planar normal sections.

We also have the following result:

Theorem 3.6. Let M be a half-lightlike submanifold in R52 and the normalsection at for any p be a geodesic arc on a su¢ ciently small neighborhoodof p. Then the following statements are equivalent;

(i) ( �rvh) (v; v) = 0;(ii) �rh = 0;

46 Feyza Esra Erdo¼gan, Selcen Yüksel Perktaş and R¬fat Güneş

(iii) M has non degenerate planar normal sections of p 2 M and hasvertex point at p 2M;

(iv) D2 = 0 in S(TM):

Proof. For curvature � at p point of , we have

��2 (s) =

00 (s) ; 00 (s)

�= D22 (v; v) �(31)

1

2�d�2 (s)

ds= v (D2 (v; v))D2 (v; v) �:

Since ��2 (s) = h00 (s) ; 00 (s)i then we have1

2�d�2 (s)

ds=

000 (s) ; 00 (s)

�(32)

=

( �rvh) (v; v) ; h (v; v)

�= 0:

Hence, from (31) and (32), we obtain D2 (v; v) = 0. This completes theproof.Now let M be a half-lightlike submanifold in R62: As a second step we

investigate the conditions for a half-lightlike submanifold of R62 to have non-degenerate planar normal sections.Consider (M; g; S(TM)) is a screen conformal half-lightlike submanifold

of�R62; g

�. Then S(TM) is integrable. We denote integral submanifold of

S(TM) by M 0; where r� is the induced connection of M 0 and 0 (s) = v;

0 (0) = v: From denition of planar normal sections and since S(TM) =Sp fv1; v2; v3g, we have

v = av1 + bv2 + cv3;(33)

r�vv = a2r�v1v1 + abr�v1v2 + acr

�v1v3 + abr

�v2v1(34)

+b2r�v2v2 + bcr�v2v3 + acr

�v3v1 + cbr

�v3v2 + c

2r�v3v3;and

(35) r�v1v1 = �1v2 + �2v3;r�v2v2 = �3v1 + �4v3;r

�v3v3 = �5v1 + �6v2:

From

(36) g (v1; v2) = 0; g (v1; v3) = 0; g (v2; v3) = 0;

we obtain

r�v1v1 = �1v2 + �2v3;r�v2v2 = �3v1 + �4v3;(37)

r�v3v3 = �5v1 + �6v2;r�v2v1 = �7v2 + �8v3;(38)

r�v3v1 = �9v2 + �10v3;r�v1v2 = �11v1 + �12v3;(39)

r�v3v2 = �13v1 + �14v3;r�v1v3 = �15v1 + �16v2;(40)

r�v2v3 = �17v1 + �18v2;(41)where a; b; c; �i; i = 1; :::18 are real constants. From (33)-(41) we obtain

(42) g (v;r�vv) = 0:Since a2 + b2 + c2 6= 0; we nd(43) v ^r�vv = 0:

Half-lightlike submanifolds with degenerate and non-degenerate... 47

Then from (42) and (43) we have r�vv = 0: Moreover we obtain

(44) v ^r�vr�vv = 0:

So we have

Theorem 3.7. Let M be a screen conformal half-lightlike submanifold inR62. Then M has spacelike planar normal sections if and only if

T (v; v) ^ �rvT (v; v) = 0;

where v 2 �(S(TM)) and T (v; v) = E1 (v; v) � +D1 (v; v)N +D2 (v; v)u:

Proposition 3.2. Let M be a half-lightlike submanifold in R62. If M hasplanar normal sections, then

(45) r�vv = 0;

where is normal section in the direction v = 0 (s) for v 2 � (S(TM)) :

Theorem 3.8. Let M be a half-lightlike submanifold of R62: Then M hasnon-degenerate planar normal sections and submanifold is totally geodesicat p 2M if and only if normal section curve has a vertex at p 2M .

Theorem 3.9. Let M be a screen conformal half-lightlike submanifold ofR62 with planar normal sections. Then normal section curve has a vertexif and only if M is minimal.

Theorem 3.10. Let M be a half-lightlike submanifold in R62 with planarnormal sections. Then K� (H) = 0 if and only if normal section curve hasa vertex at p 2M where � 2 � (RadTM)

Theorem 3.11. Let M be a half-lightlike submanifold of R62 and the normalsection at for any p be a geodesic arc on a su¢ ciently small neighborhoodof p. Then M has non-degenerate planar normal sections if and only if

h (v; v) ^ ( �rvh) (v; v) = 0;

where is h (v; v) = D1 (v; v)N +D2 (v; v)u:

We also have the following result

Theorem 3.12. Let M be a half-lightlike submanifold in R62 and the normalsection at for any p be a geodesic arc on a su¢ ciently small neighborhoodof p. Then the following statements are equivalent;

(i) ( �rvh) (v; v) = 0;(ii) �rh = 0;(iii) M has non degenerate planar normal sections of p 2 M and has

vertex point at p 2M;� D2 = 0 in S(TM):

Remark 3.1. The characterizations given above can be generalized for an n-dimensional pseudo-Euclidean space with index 2 by following similar ways.

48 Feyza Esra Erdo¼gan, Selcen Yüksel Perktaş and R¬fat Güneş

4. Degenerate planar normal sections of half-lightlikesubmanifolds in Rn2

Now we shall investigate the conditions for a half-lightlike submanifold inRn2 to have degenerate planar normal sections. In this case since rank(RadTM) =1, for a half-lightlike submanifold of n-dimensional pseudo-Euclidean spaceswith index 2, normal section curves are always in the direction of �. So inthis section we can investigate normal section curves for an arbitrary n.

Theorem 4.1. A half-lightlike submanifold M in Rn2 has degenerate planarnormal sections if and only if

(46) D2 (�; �)u ^ �r�D2 (�; �)u = 0;where D2 is the screen second fundamental form of M:

Proof. For a null curve in M; we have

0 (s) = �;(47)

00 (s) = r�� +D2 (�; �)u;(48)

000 (s) = r�r�� +D2 (r��; �)u(49)

+� (D2 (�; �))u+D2 (�; �) (�Au� + "1 (�)N) :From the denition of planar normal section and using Rad(TM) = Sp f�g ;we get

(50) r�� ^ � = 0; r�r�� ^ � = 0:Now assume that M has degenerate planar normal sections. Then

(51) 000 (s) ^ 00 (s) ^ 0 (s) = 0:Thus, by using (47)-(50) in (51), one can see that

D2 (�; �)u

and

D2 (r��; �)u+ � (D2 (�; �))u�D2 (�; �)Au� +D2 (�; �) "1 (�)Nare linearly dependent. Taking covariant derivative of D2 (�; �)u we obtain

�r�(D2 (�; �)u) = � (D2 (�; �))u�D2 (�; �)Au� +D2 (�; �) "1 (�)N;where is assumed to be parameterized by distinguished parameter. Hencewe get (46).Conversely, assume that D2 (�; �)u ^ �r�D2 (�; �)u = 0 for the degen-

erate tangent vector � of M at p: In this case, either D2 (�; �)u = 0 or�r�D2 (�; �)u = 0: If D2 (�; �)u = 0; then M is totally geodesic in �M andM is totally umbilical. Thus, we obtain

0 (s) = � ;

00 (s) = u1 (�) �;(52)

000 (s) = �(u1 (�))� + u21 (�) �;(53)

which imply that M has degenerate planar normal sections. On the otherhand, if �r�D2 (�; �)u = 0; then M is screen conformal. Hence we have

000 (s) ^ 00 (s) ^ 0 (s) = � ^D2 (�; �)u ^ �r�D2 (�; �)u = 0:

Half-lightlike submanifolds with degenerate and non-degenerate... 49

The proof is completed.

Now we dene a function

Lp : RadTpM ! R;� ! Lp (�) = D22 (�; �) �;

where p 2 M and (0) = p: If Lp (�) = D22 (�; �) � = 0, then we obtainD2 (�; �) = 0 and "1 (�) = 0: From (52) we see that 000 (s)^00 (s)^0 (s) = 0;which implies that M has degenerate planar normal sections.We say that the curve has a vertex at the point p if the curvature � of

satises d�2(p)ds = 0 and �

2 = h00 (s) ; 00 (s)i : Now let M has degenerateplanar normal sections. Then Lp = 0 and so D2 (�; �) = 0: Hence, we get

h (�; �) = D2(�; �)u = 0;��r�h

�(�; �) = 0;

which imply �rh = 0: Moreover, we have��2 (s) =

00 (s) ; 00 (s)

�= 0;

for any p 2M:

Consequently, we have the following result.

Theorem 4.2. Let M be a half-lightlike submanifold in Rn2 with degenerateplanar normal sections such that

Lp : RadTpM ! R;� ! Lp (�) = D22 (�; �) �;

where p 2M: Then the following statements are equivalent(i) D2 (�; �) = 0;(ii)

��r�h

�(�; �) = 0;

(iii) �rh = 0;(iv) For any p 2M; � = 0:

Now, let us assume that a half-lightlike submanifold M of Rn2 has degen-erate planar normal sections. Then for null vector � 2 RadTM; we have

r�� 6= 0;where � = 0 (s), namely, the normal section is not a geodesic arc on asu¢ ciently small neighborhood of p. Thus from (47)-(49) we write

(54) 000 (s) = a (s) 00 (s) + b (s) 0 (s) ;

where a and b are di¤erentiable functions for all p 2M: Hence, by equatingthe components of reciprocal bundles in ( 54), we get D2 (�; �) = "1 (�) = 0.

Consequently, we have

Theorem 4.3. Let M be a half-lightlike submanifold with degenerate planarnormal sections in Rn2 . If the degenerate normal section at any p 2 M isnot a geodesic arc on a su¢ ciently small neighborhood of p; then D2 = 0 atRadTM:

50 Feyza Esra Erdo¼gan, Selcen Yüksel Perktaş and R¬fat Güneş

Next, assume that is parameterized by distinguished parameter, namely,

is a geodesic arc on a small neighborhood of p = (0), i.e., r�� = 0. Sinceu1 (�) = �1 (�) = 0; we obtain

0 (0) = �;

00 (0) = D2(�; �)u ;(55)

000 (0) = �r�D2 (�; �)u= � (D2 (�; �))u�D2 (�; �)Au� � �D22 (�; �)N;(56)

Now, let us suppose thatM has degenerate planar normal sections at (0) =p: Therefore from 000 (s)^00 (s)^0 (s) = 0; we have �^h (�; �)^ �r�h (�; �) =0: From (55)-(56), one can see that �; h (�; �) and �r�h (�; �) can not belinearly dependent. In this case, either h (�; �) = 0 or �r�h (�; �) = 0: If�r�h (�; �) = 0; then , we calculate

hh (�; �) ; h (�; w)i =

h (�; �) ; �r�w

�=

�r�h (�; �) ; w

�= 0:(57)

and

hh (�; �) ; h (�; w)i =

h (�; �) ; �rw�

�� hh (�; �) ;rw�i

= �D2 (�; �)D2 (w; �) :(58)

which imply D2 = 0 at � (TM). Furthermore, from �rw� 2 � (TM) ; (� 2RadTM and w 2 � (TM)); we see that M is irrotational.So we have the following result.

Theorem 4.4. Let M be a half-lightlike submanifold with degenerate planarnormal sections in Rn2 . If the normal section at any p 2M is a geodesicarc on a su¢ ciently small neighborhood of p; then M is irrotational.

Corollary 4.1. Let M be a half-lightlike submanifold with degenerate pla-nar normal sections in Rn2 . Then Au� is RadTM -valued.

Proof. LetM be a half-lightlike submanifold with degenerate planar normalsections in Rn2 . Since is a planar curve we write

(59) 000 (s) = a (s) 00 (s) + b (s) 0 (s) ;

where a and b are di¤erentiable functions on M:Then from (52) and (53) weget

a (s) = u1 (�) + � (ln (D2 (�; �))) ;(60)

b (s) = � (u1 (�))�D2 (�; �) �2 (�) �� u1 (�) � (ln (D2 (�; �))) :(61)Moreover, we have ��2 (s) = h00 (s) ; 00 (s)i = 0 for any p 2M , which givesD2 (�; �) = "1 (�) = 0. Thus, we obtain

000 (s) = u21 (�) � + u1 (�)D2(�; �)u

+� (ln (D2 (�; �)))D2 (�; �)u(62)

+� (u1 (�)) � � �D2(�; �)�2 (�) �:Then, by using (60),(61) in (59) and equating the components of reciprocalbundles, we have

(63) Au� = ��2 (�) �:

Half-lightlike submanifolds with degenerate and non-degenerate... 51

This completes the proof.

Theorem 4.5. Let M be half-lightlike submanifold of Rn2 such that thenormal section at for any p is not a geodesic arc on a su¢ ciently smallneighborhood of p. Then half-lightlike submanifold M has degenerate planarnormal sections if and only if

(64) D2(�; �)u ^��r�h

�(�; �) = 0

is satised.

Proof. From (62) and (63), we get��r�h

�(�; �) = � (ln (D2 (�; �)))D2 (�; �)u

��D2(�; �)�2 (�) � � 2u1 (�)D2(�; �)u:(65)

Also, from at any p 2 M is not a geodesic arc on a su¢ ciently smallneighborhood of p and we use r�� = u1 (�) �

0 (0) = �;

00 (0) = u1 (�) � +D2(�; �)u;(66)

000 (0) = �(u1 (�))� + u21 (�) � + u1 (�)D2(�; �)u� � (D2(�; �))u(67)

�D2 (�; �)Au� +D2 (�; �) "1 (�)N

Consider that M is a half-lightlike submanifold with degenerate planar nor-mal sections in Rn2 such that the normal section at for any p is not ageodesic arc on a su¢ ciently small neighborhood of p: If we use (66),(67) to(65), we obtain (64).Conversely, assume that (64) is satised for any degenerate tangent vector

eld � of M; then either D2(�; �)u = 0 or��r�h

�(�; �) = 0. If D2(�; �)u = 0;

then from Theorem 4.1, one can see that M has degenerate planar normalsections. On the other hand, if

��r�h

�(�; �) = 0; then, by taking into account

(66),(67), we obtain

000 (s) ^ 00 (s) ^ 0 (s) = � ^D2(�; �)u ^��r�h

�(�; �) = 0;

which completes the proof.

Theorem 4.6. LetM be a half-lightlike submanifold with degenerate planarnormal sections in Rn2 : If the normal section at any p is a geodesic arc ona su¢ ciently small neighborhood of p; then D2(�; �) = 0 or "1 (�) = 0:

Proof. Assume that normal section is a geodesic arc on a su¢ cientlysmall neighborhood of p; namely, r�� = 0 = u1 (�) : SinceM has degenerateplanar normal sections, we obtain

0 = 000 (s) ^ 00 (s) ^ 0 (s)= (� ^D2(�; �)u ^D2 (�; �)Au�)(68)

+(� ^D2(�; �)u ^D2 (�; �) "1 (�)N):

From Corollary 4.1, we conclude.

Let M be a screen conformal half-lightlike submanifold of R42(c) withdegenerate planar normal sections. We denote Riemann curvature tensor of

52 Feyza Esra Erdo¼gan, Selcen Yüksel Perktaş and R¬fat Güneş

�M and M by �R and R, hence we have

�g��R (X;Y )Z;PW

�= ' [D1(X;Z)D1(Y; PW )�D1(Y; Z)D1(X;PW )]

+� [D2(X;Z)D2(Y; PW )�D2(Y; Z)D2(X;PW )] :Let p 2M and � be a null vector of TpM . A plane H of TpM is called a nullplane directed by �; if it contains �, �g(�;W ) = 0 for any W 2 H and thereexits W0 2 H such that �g(W0;W0) 6= 0. Then the null sectional curvatureof H with respect to � and �r is dened by [1]

(69) K� (H) =Rp (W; �; �;W )

gp (W;W ):

Theorem 4.7. Let M be a screen conformal half-lightlike submanifoldwith degenerate planar normal sections in Rn2 (c) . If M is minimal, thenK� (H) = 0:

Proof. Since v 2 � (S (TM)) and � 2 � (RadTM) ; we haveK� (H) = ' [D1(v; �)D1(�; v)�D1(�; �)D1(v; v)]

+� [D2(v; �)D2(�; v)�D2(�; �)D2(v; v)] ;By using D1(v; �) = 0 in the last equation we obtain

(70) K� (H) = � [D2(v; �)D2(�; v)�D2(�; �)D2(v; v)] ;which completes the proof.

Example 4.1. Let M be a submanifold given in Example 3.1. If we takeinto account the intersection of M and E(p; �), we have a lightlike curve in a neighborhood of p, which is called the degenerate normal section of Mat the point p in the direction of �, namely

0(s) = �;

00(s) = u1(�)� = �;

000(s) = �:

Hence we obtain 000(s) ^ 00(s) ^ 0(s) = 0:

We would like to thank the referee for his/her valuable suggestions andcomments for improving the paper.

References

[1] Beem, J. K., Ehrlich, P. E., Global Lorentzian Geometry, Marcel Dekker, Inc. NewYork, First Edition, 1981.

[2] Chen, B. -Y., Submanifolds with planar nornal sections, Soochow J. Math., 7(1981),19-24.

[3] Chen, B. -Y., Di¤erential Geometry of submanifolds with planar normal sections,Ann. Mat. Pura Appl., 130(1982), 59-66.

[4] Chen B. -Y., Classication of surfaces with planar normal sections, J. Geom.,20(2)(1983), 122-127.

[5] Duggal, K. L., Bejancu, A., Lightlike Submanifolds of Semi-Riemannian Manifoldsand Applications, Kluwer Academic Publishers, 1996, 334.

[6] Duggal, K. L., Bejancu, A., Lightlike submanifolds of codimension two, Math. J.Toyama Univ., 15(1992), 59-82.

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[7] Duggal, K. L., Jin, D. H., Half lightlike submanifolds of codimension 2, Math. J.Toyama Univ., 22(1999), 121-161.

[8] Erdo¼gan, F. E., Güneş, R., Şahin, B., Half-Lightlike Submanifolds with planar normalsections in R42, Turk J Math, 38(2014), 764-777.

[9] Duggal, K. L., Şahin, B., Di¤erential Geometry of Lightlike Submanifolds, Springer,Birkhauser, 2010.

[10] Kim, Y. H., Surfaces in a pseudo-Euclidean Space with planar normal sections, J.Geom., 35(1-2)(1989), 120-131.

[11] Kim, Y. H., Minimal surfaces of pseudo-Euclidean spaces with geodesic normal sec-tions, Di¤erential Geom. Appl., 5(4)(1995), 321-329.

[12] Kim, Y. H., Pseudo-Riemannian submanifolds with pointwise planar normal sections,Math. J. Okayama Univ., 34(1992), 249-257.

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DEPARTMENT OF MATHEMATICS,FACULTY OF EDUCATION,ADIYAMAN UNIVERSITY,02040, ADIYAMAN, TURKEYE-mail Adress: ferdogan@adiyaman.edu.tr

DEPARTMENT OF MATHEMATICS,FACULTY OF ARTS AND SCIENCES,ADIYAMAN UNIVERSITY,02040, ADIYAMAN, TURKEYE-mail Adress: sperktas@adiyaman.edu.tr

DEPARTMENT OF MATHEMATICS,FACULTY OF ARTS AND SCIENCES,·INÖNÜ UNIVERSITY,44280, MALATYA, TURKEYE-mail Adress: rifat.gunes@inonu.edu.tr