C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat. Volum e 69, N umb er 1, Pages 354–368 (2020) D O I: 10.31801/cfsuasm as.469747
ISSN 1303–5991 E-ISSN 2618-6470
http://com munications.science.ankara.edu.tr
INTEGRABILITY OF THE DISTRIBUTIONS OF GCR-LIGHTLIKE SUBMANIFOLDS OF (")-SASAKIAN
MANIFOLDS
AYSE FUNDA SAGLAMER AND NESRIN CALISKAN
Abstract. We study GCR-lightlike submanifolds of (")-Sasakian manifolds and derived some important structural characteristics equations for further uses. We also obtain some necessary and su¢ cient conditions for the integra-bility of various distributions of GCR-lightlike submanifolds of (")-Sasakian manifolds.
1. Introduction
As a generalization of complex and totally real submanifolds, Cauchy-Riemann (CR)-submanifolds of Kaehler manifolds were introduced by Bejancu [1] in 1978 and further studied by many authors on using positive de…nite metric. In [3], Duggal in-troduced the geometry of CR-submanifolds with Lorentz metric and showed mutual interplay between the Cauchy-Riemann structure and physical spacetime geometry. In [4], Duggal showed the interaction of Lorentz CR-submanifolds with relativity and also studied a new class of CR-submanifolds. Later on, Duggal and Bejancu [5] introduced the concept of CR-lightlike submanifolds of inde…nite Kaehler manifolds but which excluded the complex and totally real subcases, therefore Duggal and Sahin [6] introduced Screen Cauchy-Riemann (SCR)-lightlike submanifolds of in-de…nite Kaehler manifolds which included complex and screen real subcases but there was no inclusion relation between SCR and CR classes. Thus as an umbrella of complex, real hypersurfaces, screen real and CR-lightlike submanifolds, Duggal and Sahin [7] introduced Generalized Cauchy-Riemann (GCR)-lightlike submani-folds of inde…nite Kaehler manisubmani-folds and further studied by [10–15]. Since there are signi…cant applications of contact geometry in thermodynamics, optics, mechan-ics and many more. Therefore, Duggal and Sahin [8] introduced the geometry of (GCR)-lightlike submanifolds of inde…nite Sasakian manifolds and further studied by [16–18]. Recent developments in the geometry of GCR-lightlike submanifolds motivated us to extend this work. Kumar et al. [9] contributed in the study of
Received by the editors: October 12, 2018; Accepted: November 11, 2019. 2020 Mathematics Subject Classi…cation. 53C15; 53C40; 53C50.
Key words and phrases. (")-Sasakian manifold, lightlike submanifold, GCR-lightlike subman-ifold, degenerate metric.
c 2 0 2 0 A n ka ra U n ive rsity C o m m u n ic a tio n s Fa c u lty o f S c ie n c e s U n ive rs ity o f A n ka ra -S e rie s A 1 M a t h e m a t ic s a n d S ta t is t ic s
(")-Sasakian manifolds and our aim of this paper is to study GCR-lightlike sub-manifolds of (")-Sasakian sub-manifolds.
2. Preliminaries
2.1. (")-Sasakian Manifolds. Assume that M is a (2n + 1)-dimensional di¤er-entiable manifold endowed with an almost contact structure ( ; ; V ), where is a (1; 1)-type tensor …eld, is a 1-form and V is a vector …eld on M , called the characteristic vector …eld, satisfying
2
X = X + (X) V; (V ) = 1; (2.1)
( X) = 0; (V ) = 0; rank = 2n; (2.2)
then M , with the triple ( ; ; V ) is called an almost contact manifold. If there exists a semi-Riemannian metric g such that
g ( X; Y ) = g (X; Y ) " (X) (Y ) ; 8X; Y 2 T M; (2.3) (X) = "g (X; V ) ; g (V; V ) = "; 8X 2 T M; (2.4) for any vector …elds X; Y on M , where " = 1, then ( ; ; V; g) is called an (")-almost contact metric structure on M . If d (X; Y ) = g ( X; Y ), then (")-(")-almost contact metric structure is called an (")-contact metric structure and M endowed with this structure is called an (")- contact metric manifold. Furthermore, if the (")-contact metric structure is normal, that is, if satisfying
[ X; Y ] + 2[X; Y ] [X; Y ] [ X; Y ] = 2d (X; Y ) V; (2.5) then (")-contact metric structure is called an (")-Sasakian structure and M endowed with this structure is called as an (")-Sasakian manifold [2].
Remark 1. From the relations g (V; V ) = " and " = 1, it is clear that the vector …eld V can never be null. If " = 1 and the index of g is odd; then M is called a time-like Sasakian manifold. If " = 1 and the index of g is even; then M is called a space-like Sasakian manifold. In particular, if " = 1 and the index of g is either zero or one; then M is said to be a usual Sasakian manifold or a Lorentz-Sasakian manifold, respectively.
Theorem 1 ( [2, Theorem 3]). The necessary and su¢ cient conditions for an (")-almost contact metric structure ( ; ; V; g) to be an (")-Sasakian structure is
rX Y = g (X; Y ) V " (Y ) X; 8X; Y 2 T M; (2.6)
for any vector …elds X; Y on M , where r denotes the Levi-Civita connection with respect to g. Moreover, we also have
rXV = " X; (2.7)
2.2. Lightlike Submanifolds. Suppose that Mm+n; g is a semi-Riemann
man-ifold and Mmis its immersed submanifold. Then, Mmis called a lightlike
subman-ifold; if the metric g on M induced from g has a radical distribution Rad (T M ) of rank r, for 1 r m, for details see [5]. Then, its semi-Riemannian complemen-tary distribution in T M , denoted by S(T M ), is known as the screen distribution and it follows that T M = Rad (T M ) ?S (T M). The orthogonal complementary of Rad (T M ) in T M?, denoted by S T M? , is also a semi-Riemannian bundle and
known as a screen transversal bundle of M . Since S(T M ) is a non-degenerate vector subbundle of T M jM; then, we have T M jM = S(T M )?S(T M)? where S(T M )? is
the complementary orthogonal vector bundle of S(T M ) in T M jM. Then, clearly
we have S(T M )? = S(T M?)?S(T M?)?. If (M; g) is an r-lightlike submanifold
of (M ; g); then, for the local basis f igri=1 of Rad (T M ) on a coordinate
neigh-bourhood U of M, there exist smooth sections fNigri=1 of S(T M?)?jU such that
g ( i; Nj) = ij and g(Ni; Nj) = 0, for any i; j 2 f1; : : : ; rg. Then, there
ex-ists a vector subbundle of S(T M?)? spanned by fN
igri=1, known as the lightlike
transversal vector bundle of M and denoted by ltr(T M ). Consider a vector bun-dle tr(T M ) = ltr(T M )?S(T M?), which is a complementary (but not orthogonal)
vector bundle to T M in T M jM and known as the transversal vector bundle of M .
Thus, we have the following decomposition
T M jM = T M tr(T M ) = S(T M )?fRad(T M) ltr(T M )g?S(T M?):
Let r be the Levi-Civita connection on M; then using above decomposition, the Gauss and Weingarten formulae are given by
rXY = rXY + h (X; Y ) ; 8X; Y 2 (T M) ; (2.8)
rXU = AUX + rtxU; 8X 2 (T M) ; U 2 (tr (T M)) ; (2.9)
where frXY; AUXg and h (X; Y ) ; rtXU are the elements of (T M ) and
(tr (T M )), respectively. Here r and rt are the linear connections on T M and tr (T M ), respectively and the linear operator AU on M is called the shape operator
and the symmetric bilinear form h on T M is called the second fundamental form. Consider projection morphisms L and S of tr(T M) on ltr(T M) and S(T M?),
respectively, then particularly Gauss and Weingarten formulas are given by rXY = rXY + h`(X; Y ) + hs(X; Y ) ; (2.10)
rXN = ANX + r`X(N ) + Ds(X; N ) ; (2.11)
rXW = AWX + rsX(W ) + D`(X; W ) ; (2.12)
for any X; Y 2 (T M), N 2 (`tr (T M)) and W 2 S T M? , where hl(X; Y ) = L(h(X; Y )) and hs(X; Y ) = S(h(X; Y )) are the lightlike second fundamental form and the screen second fundamental form of M , respectively. It should be noted that Dl: (T M ) (S(T M?)) ! (ltr(T M)) and Ds: (T M ) (ltr(T M )) !
(S(T M?)) are F(M)-bilinear mappings. r`
and rs are the lightlike and the screen transversal connection on M , respectively. In the consequence of (2.8), (2.10), (2.11) and (2.12), we have
g ANX; P Y = g N; rXP Y ; (2.14)
where P is the projection of T M on S (T M ). Furthermore, we also have
rXP Y = rXP Y + h X; P Y ; (2.15)
rX = A X + rXt ; (2.16)
for any X; Y 2 (T M ) and 2 (Rad(T M )), where r and r t are the linear connections on S (T M ) and Rad (T M ), respectively. h and A are (Rad(T M ))-valued and (S(T M ))-))-valued bilinear forms and called as second fundamental forms of distributions S(T M ) and Rad(T M ), respectively. By the virtue of (2.16) and (2.17), we have
g h X; P Y ; N = g ANX; P Y (2.17)
3. GCR-lightlike submanifolds of (")-Sasakian manifolds
De…nition 1. Suppose M ; g is an (")-Sasakian manifold and (M; g; S (T M )) is its real lightlike submanifold, where V is tangent to M . Then, M is called a GCR-lightlike submanifold of M if the following conditions are satis…ed:
(A) There exist two subbundles D1and D2of Rad(T M ); such that Rad(T M ) =
D1 D2, where (D1) = D1 and (D2) S(T M ).
(B) There exist two subbundles D0 and D of S(T M ); such that S(T M ) =
f D2 Dg?D0?V and (D) = L?S,
where D0is invariant non-degenerate distribution on M , fV g is one dimensional
distribution spanned by V , L and S are vector subbundles of ltr(T M ) and S(T M?),
respectively. Then, the tangent bundle T M of M is decomposed as T M = fD D fV gg, where D = Rad(T M) D0 (D2).
Suppose (M; g; S (T M )) is a GCR-lightlike submanifold of an (")-Sasakian man-ifold M . Then, any X 2 T M can be written as
X = P1X + P2X + P0X + P2X + QX + (X) V; (3.1)
where P1X, P2X, P0X, P2X and QX belong to the distributions D1, D2, D0,
D2 and D, respectively. Assume that L1 represents the orthogonal complement
of the vector subbundle L in `tr (T M ); then using the de…nition of GCR-lightlike submanifold, for any N 2 (`tr (T M)), we have
N = T N + CN; (3.2)
where T N 2 ( L) is the tangential part of N and CN 2 L? is the transversal
part of N . Similarly, suppose that S? represents the orthogonal complement of
the vector subbundle S in S T M? ; then for any W 2 S T M? , we have
W = T W + CW; (3.3)
where T W 2 ( S) is the tangential part of W and CW 2 S? is the screen
transversal part of W . Using (3.1), we obtain
where QX 2 (L?S) and we can write
QX = LX + SY; (3.5)
where LX 2 (L) and SY 2 (S). So, we have
U (X; Y ) = rX( P1Y ) + rX( P2Y ) rX(P2Y ) (3.6)
+rX( P0Y ) ALYX ASYX;
for any X; Y 2 T M.
Lemma 1. Let (M; g; S (T M )) be a GCR-lightlike submanifold of an (")-Sasakian manifold M . Then, for any X; Y 2 T M the following equalities hold
P1U (X; Y ) P1rXY = " (Y ) P1X; (3.7) P2U (X; Y ) + P2rXY = " (Y ) P2X; (3.8) P0U (X; Y ) P0rXY = " (Y ) P0X; (3.9) P2U (X; Y ) P2rXY = " (Y ) P2X; (3.10) QU (X; Y ) QT h`(X; Y ) QT hs(X; Y ) = " (Y ) QX; (3.11) f (U (X; Y )) g (X; Y )g V = " (Y ) (X) V; (3.12) r`X(LY ) + D`(X; SY ) LrXY + h`(X; P1Y ) +h`(X; P2Y ) + h`(X; P0Y ) h`(X; P2Y ) Ch`(X; Y ) = 0; (3.13) rsX(SY ) SrXY + Ds(X; LY ) + hS(X; P1Y ) +hS(X; P2Y ) hs(X; P2Y ) + hs(X; P0Y ) Chs(X; Y ) = 0: (3.14)
Proof. Let Y 2 (T M ); then using (3.4) and (3.5), it follows that (P1Y )
P2Y , (P2Y ) + P0Y , LY and SY belong to Rad (T M ), S (T M ), `tr (T M ) and
S T M? , respectively. Also for any X; Y 2 (T M), it is known that
rX Y = rX( Y ) rXY : (3.15)
Using (2.10), (2.11), (2.12) and (3.4) in (3.15) and afterwards applying (3.6), we obtain rX Y = (P1U (X; Y ) P1rXY ) + (P2U (X; Y ) + P2rXY ) + (P0U (X; Y ) P0rXY ) + (P2U (X; Y ) P2rXY ) + QU (X; Y ) T h`(X; Y ) T hs(X; Y ) + (U (X; Y )) V + r`X(LY ) + D`(X; SY ) LrXY + h`(X; P1Y ) + h`(X; P2Y ) +h`(X; P0Y ) h`(X; P2Y ) Ch`(X; Y ) + (rsX(SY ) SrXY + hs(X; P1Y ) ; (3.16)
for any X; Y 2 (T M). Also from (2.6) and (3.1), it follows that
rX Y = " (Y ) (P1X) " (Y ) (P2X) " (Y ) (P0X) " (Y ) ( P2X)
By using (3.16) and (3.17), it is easy to obtain ~ NXf Y = P1U (X; Y ) f P1N~XY + P2 U (X; Y ) + ~NXY + P0U (X; Y ) f P0N~XY + f P2 U (X; Y ) N~XY +QU (X; Y ) T hl(X; Y ) T hs(X; Y ) + (U (X; Y )) V + N~Xl (LY ) + Dl(X; SY ) L ~NXY + hl(X; f P1Y ) + hl(X; f P2Y ) +hl(X; f P0Y ) hl(X; P2Y ) Chl(X; Y ) + N~Xs (SY ) S ~NXY + hs(X; f P1Y + hs(X; f P2Y ) hs(X; P2Y ) + hs(X; f P0Y ) + Ds(X; LY ) Chs(X; Y )) :
Then, (3.7) to (3.14) follow on comparing the components of the vector bundles D1, D2, D0, D2, D, fV g, `tr (T M) and S T M? , respectively.
Lemma 2. Let (M; g; S (T M )) be a GCR-lightlike submanifold of an (")-Sasakian manifold M . Then, for any X 2 (T M) and N 2 `tr (T M), the following relations hold P1rX(T N ) P1ACNX + P1(ANX) = 0; (3.18) P2rX(T N ) P2ACNX P2(ANX) = 0; (3.19) P0rX(T N ) P0ACNX + P0(ANX) = 0; (3.20) P2(rX(T N )) P2(ACNX) + (P2ANX) = 0; (3.21) QrX(T N ) QACNX T r`XN T Ds(X; N ) = 0; (3.22) (rXT N ACNX) = g (P1X; N ) + g (P2X; N ) ; (3.23) h`(X; T N ) + r`X(CN ) Cr`XN + LANX = 0; (3.24) hs(X; T N ) + Ds(X; CN ) CDs(X; N ) + SANX = 0: (3.25)
Proof. Let X 2 (T M) and N 2 (`tr (T M)), then we have rX N = rX(T N ) + rX(CN )
+ (ANX) r`X(N ) (Ds(X; N )) : (3.26)
Further on using the equations (2.10) and (2.11) in (3.26), we get rX N = rX(T N ) + h`(X; T N ) + hS(X; T N ) ACNX
+r`X(CN ) + Ds(X; CN ) + (ANX)
r`X(N ) Ds(X; N ) (3.27)
Using (3.1) to (3.3), we also have
rX(T N ) = P1rX(T N ) + P2rX(T N ) + P0rX(T N )
+ P2rX(T N ) + QrX(T N ) + (rX(T N )) V; (3.28)
ACNX = P1ACNX + P2ACNX + P0ACNX
(ANX) = (P1ANX) + (P2ANX)
+ (P0ANX) P2(ANX) + Q (ANX) ; (3.30)
r`XN = T r`XN + C r`XN ; (3.31)
(Ds(X; N )) = T Ds(X; N ) + CDs(X; N ) : (3.32) On using the equations from (3.28) to (3.32) in equation (3.27), we get
rX N = fP1rXT N P1ACNX + P1(ANX)g + fP2(rXT N ) P2(ACNX) P2(ANX)g + fP0rXT N P0ACNX + P0ANXg + f P2(rXT N ) P2(ACNX) + P2(ANX)g + n QrXT N QACNX T r`XN T Ds(X; N ) o + f (rXT N ) (ACNX)g V + n h`(X; T N ) + r`X(CN ) Cr`XN + LANX o + fhs(X; T N ) + Ds(X; CN ) CDs(X; N ) + SANXg ;(3.33)
which implies Q (ANX) = L (ANX) + S (ANX), where L (ANX) 2 (L) and
S (ANX) 2 (S). Also using (2.6), we have
rX N = g (P1X; N ) V + g (P2X; N ) V: (3.34) On using (3.33) in (3.34), we obtain g (P1X; N ) V + g (P2X; N ) V = fP1rXT N P1ACNX + P1(ANX)g + fP2(rXT N ) P2(ACNX) P2(ANX)g + fP0rXT N P0ACNX + P0ANXg + f P2(rXT N ) P2(ACNX) + P2(ANX)g + n QrXT N QACNX T r`XN T Ds(X; N ) o + f (rXT N ) (ACNX)g V +nh`(X; T N ) + r`X(CN ) Cr`XN + LANX o + fhs(X; T N ) + Ds(X; CN ) CDs(X; N ) + SANXg :
Then, the relations from (3.18) to (3.25) are obtained on comparing the compo-nents of the vector bundles D1, D2, D0, D2, D, fV g, `tr (T M) and S T M? ,
respectively.
Lemma 3. Let (M; g; S (T M )) be a GCR-lightlike submanifold of an (")-Sasakian manifold M . Then, for any X 2 (T M ) and W 2 (S T M? ), the following
relations hold
P1frXT W ACWX + (ANX)g = 0; (3.35)
P2frXT W ACWX AWXg = 0; (3.36)
P2frXT W ACWX + AWXg = 0; (3.38)
QrXT W QACWX T rsXW T D`(X; W ) = 0; (3.39)
(rXT W ACWX) = 0; (3.40)
h`(X; T W ) CD`(X; W ) + D`(X; CW ) + LAWX = 0; (3.41)
hs(X; T W ) + rsX(CW ) C (rsXW ) + SAWX = 0: (3.42)
Proof. Let X 2 (T M ) and W 2 (S T M? ); then using (2.12) and (3.3), it
follows that
rX W = rX(T W ) + rX(CW ) + (AWX) (rsXW ) D`(X; W ) :
(3.43) Then, further using (2.10), (2.12), (3.1), (3.3) and (3.4) in equation (3.43), we obtain rX W = P1frX(T W ) (ACWX) + (AWX)g +P2frX(T W ) ACWX AWXg +P0frX(T W ) ACWX + (AWX)g + P2frXT W ACWX + AWXg + QrXT W QACWX T rsXW T D`(X; W ) + f (rXT W ) (ACWX)g V + h`(X; T W ) + D`(X; CW ) CD`(X; W ) + LANX + fhs(X; T W ) CrsXW + rsX(CW ) + S (ANX)g : (3.44)
In consequence of (2.6), we know that rX W = 0; then the relations from (3.35)
to (3.42) follow immediately on comparing the components of the vector bundles D1, D2, D0, D2, D, fV g, `tr (T M) and S T M? , respectively.
Lemma 4. Let (M; g; S (T M )) be a GCR-lightlike submanifold of an (")-Sasakian manifold M . Then, for any X 2 D and Y 2 D, we have the following relations
rXV = " X; h`(X; V ) = 0; hs(X; V ) = 0; (3.45)
rYV = 0; h`(Y; V ) = "LY; hs(Y; V ) = "SY; (3.46)
rVV = 0; h`(V; V ) = 0; hs(V; V ) = 0: (3.47)
Proof. The proof follows immediately by using (2.10), (3.1) and (3.4) in (2.6). 4. Integrability of the distributions
Theorem 2. Let (M; g; S (T M )) be a GCR-lightlike submanifold of an (")-Sasakian manifold M . Then, necessary and su¢ cient conditions for the radical distribution Rad (T M ) to be integrable are the following
(1) h`(X; Y ) = h`(Y; X) ; hs(X; Y ) = hs(Y; X), 8X; Y 2 Rad (T M).
(2) g h (X; Y ) ; Z~ = g h (Y; X) ; Z , 8X; Y 2 D~ 2; Z 2 `tr (T M).~
(3) g rXt Y; Z~ = g r t
(4) g h (X; Y ) ; Z~ = g rYt( X) ; Z , 8X 2 D~ 1; Y 2 D2; Z 2 `tr (T M).~
g h (Y; X) ; Z~ = g rXt( Y ) ; Z , 8X 2 D~ 2; Y 2 D1; Z 2 `tr (T M).~
(5) g A YX; Z0 = g A XY; Z0 , 8X; Y 2 D1; Z02 D0.
(6) g (rX Y; Z0) = g (rY ( X) ; Z0), 8X; Y 2 D2; Z02 D0.
(7) g (rX( Y ) ; Z0) = g A XY; Z0 , 8X 2 D1; Y 2 D2; Z02 D0.
Proof. (1) Assume that the radical distribution RadT M is integrable; then, this implies that [X; Y ] 2 RadT M for any X; Y 2 RadT M. Using (3.13), we have h`(X; Y ) = Lr
XY + Ch`(X; Y ) and h`(Y; X) = LrYX + Ch`(Y; X) this
further implies that h`(X; Y ) h`(Y; X) = L (r
XY rYX) = L [X; Y ] : If
[X; Y ] 2 RadT M, then L [X; Y ] = 0, therefore we get
h`(X; Y ) = h`(Y; X) : (4.1)
Similarly as a consequence of (3.14), we also have
hs(X; Y ) = hs(Y; X) ; (4.2)
and g ([X; Y ] ; V ) = 2"g (Y; X) = 0: Now using (2.6) for any X; Y 2 RadT M and ~
Z 2 D, we get
g rX( Y ) +h`(X; Y )+hs(X; Y ) rY ( X) h`(Y; X) hs(Y; X) Z~ =
0,
and then by using (4.1) and (4.2), we obtain
0 = g rX( Y ) ; Z~ g rY ( X) ; Z :~ (4.3)
For rX( Y ), rY ( X) 2 (T M) and Z 2 S T M~ ? , (4.3) is satis…ed.
(2) Let X; Y 2 S (T M) and Z 2 `tr (T M), then applying (2.16) in (4.3), we~ get
0 = g rX( Y ) ; Z~ g rY ( X) ; Z~
+g h (X; Y ) ; Z~ g h (Y; X) ; Z~ (4.4)
where rX( Y ) ; rY ( X) 2 S (T M) and h (X; Y ), h (Y; X) 2 Rad (T M).
Hence, (4.4) becomes
g h (X; Y ) ; Z~ = g h (Y; X) ; Z :~ (4.5)
(3) Similarly, for X; Y 2 S (T M) and Z 2 `tr (T M), using (2.17) in (4.3),~ we get
0 = g A XY; Z~ g A YX; Z~ g AYt X; Z + g A~ Xt Y; Z ;~ (4.6) where A XY; A YX 2 S (T M) and rYt X; rXt Y 2 Rad (T M). Hence, (4.6) becomes
g rXt Y; Z~ = g r t
(4) Let X 2 Rad (T M), Y 2 S (T M) and Z 2 `tr (T M); then using (2.16)~ and (2.17) in (4.3), we get
g h (X; Y ) ; Z~ = g rYt( X) ; Z :~ (4.8) Similarly, by using X 2 S (T M), Y 2 Rad (T M) and Z 2 `tr (T M), with the~ help of (2.16) and (2.17) in (4.3), it follows that
g h (Y; X) ; Z~ = g rXt( Y ) ; Z :~ (4.9) (5) For X; Y 2 D1 and Z0 2 D0, using (2.6) and (2.17), we have the following
relation immediately
g A XY; Z0 = g A YX; Z0 : (4.10)
(6) For X; Y 2 D2 and Z02 D0, by the use of (2.6) and (2.16), we have
g (rX( Y ) ; Z0) = g (rY ( X) ; Z0) : (4.11)
(7) Finally, let X 2 D1, Y 2 D2 and Z02 D0; then, using (2.6), (2.16) and (2.17),
we obtain
g (rX( Y ) ; Z0) = g A XY; Z0 : (4.12)
If we take X; Y 2 D2 and Z 2 D2, then on applying(2.17) and (2.16), we get
g (h (X; Y ) ; Z) g (h (Y; X) ; Z) = 0; (4.13) where g (h (X; Y ) ; Z) = 0 = g (h (Y; X) ; Z), for all Z 2 D2, this implies
(4.13) holds. For X; Y 2 D1 and Z 2 D2 with the help of (2.6) and (2.17) it is
possible to get g rXt Y; Z = g rYt X; Z = 0. For X 2 D1, Y 2 D2 and
Z 2 D2 by using (2.6) and (2.16) and (2.17), we get g (h (X; Y ) ; Z) = 0 =
g (h (Y; X) ; Z), for all Z 2 D2.
Theorem 3. Let (M; g; S (T M )) be a GCR-lightlike submanifold of an (")-Sasakian manifold M . Then, the distribution D0 is never integrable.
Proof. Assume that the distribution D0 is integrable, then g ([X; Y ] ; V ) = 0 for
any X; Y 2 D0. By using (2.6), we obtain g ([X; Y ] ; V ) = 2"g (Y; X) for X; Y 2
D0, then by using the above relation, we have g (Y; X) = 0. Since D0 is
non-degenerate then g (Y; X) 6= 0. This leads to a contradiction and hence the assertion follows.
Lemma 5. Let (M; g; S (T M )) be a GCR-lightlike submanifold of an (")-Sasakian manifold M . Then, [X; V ] 2 D fV g, for any X 2 D.
Proof. Let X 2 D and Y 2 D; then using (3.47), we get g ([X; V ] ; Y ) = "g (X; Y ) g (rVX; Y ). Particularly, on taking Y 2 S T M? , we have g ([X; V ] ; Y ) =
g (rVX; Y ) and further on putting X = X, then using (2.6), (2.12), (2.13)
and (3.47), we obtain g ([ X; V ] ; Y ) = g (hs(V; X) ; Y ) = 0. In particular, if we
take Y 2 `tr (T M) then by using (2.6), (2.15) and (2.11), we get g ([ X; V ] ; Y ) = g ( Y; rVX) = 0. We know that if X 2 D then this implies that X 2 D therefore
Theorem 4. Let (M; g; S (T M )) be a GCR-lightlike submanifold of an (")-Sasakian manifold M . Then, necessary and su¢ cient conditions for D fV g to be integrable are h`(X; Y ) = h`( X; Y ) and hs(X; Y ) = hs( X; Y ), for any X; Y 2 D fV g.
Proof. Assume that D fV g is integrable then [X; Y ] 2 D fV g, for any X; Y 2 D fV g. We know that for any X; Y 2 D fV g, we can write X = P X + (X) V and Y = P Y + (Y ) V , where P X; P Y 2 D. Using these relations, we obtain [X; Y ] (Y ) [P X; V ] (X) [V; P Y ] = [P X; P Y ] : Since [X; Y ], [P X; V ], [V; P Y ] 2 D fV g then [P X; P Y ] 2 D fV g. Thus Q [P X; P Y ] = 0 implies L [P X; P Y ] = 0 = S [P X; P Y ]. On using these equalities in (3.13), our assertion follows.
Theorem 5. Let (M; g; S (T M )) be a GCR-lightlike submanifold of an (")-Sasakian manifold M . Then, necessary and su¢ cient condition for the distribution D to be integrable is that A XY = A YX for X; Y 2 D.
Proof. Let X 2 S T M? , Y 2 D and Z 2 S (T M), then using (2.6), (2.10) and
(2.13), it follows that
g (A XY; Z) = g rZ( Y ) ; X : (4.14)
For Y 2 S T M? , by the use of (2.13) in (4.14), we have
g (A XY; Z) = g (A YX; Z) : (4.15)
Also for Y 2 `tr (T M), by using (2.17) in (4.14), we have
g (A XY; Z) = g (A YX; Z) : (4.16)
Hence by the use of (4.15) and (4.16), for any Z 2 S (T M), we obtain A XY =
A YX. If X 2 `tr (T M), Y 2 D and Z 2 S (T M) then from (2.6), (2.16) and
(2.17), it follows that g (A XY; Z) = g (rZX; Y ). Furthermore, particularly on
taking Y 2 `tr (T M) and using (2.16), we obtain g (A XY; Z) = g (h (X; Z) ; Y ) =
g (A YX; Z) : If particularly we take X 2 S T M? , Y 2 D and Z 2 Rad (T M),
then using (2.6), (2.10), (2.12) and (2.13), we get
g (A XY; Z) = g (hs(Y; Z) ; X) + g Z; D`(Y; X)
= g (hs(X; Z) ; Y ) + g Z; D`(Y; X) : (4.17) Now, for any Y 2 S T M? , using (3.13), we get D`(X; Y ) = D`(Y; X) : Then, using (4.17) for any Z 2 S (T M), we obtain g (A XY; Z) = g (A YX; Z).
If X 2 `tr (T M), Y 2 D and Z 2 S (T M); then using (2.6), (2.16) and (2.17), it yields
g (A XY; Z) = g ( X; rYZ) + g r`Y ( X) ; Z
= g (X; rY ( Z)) + g r`Y( X) ; Z : (4.18)
On applying (2.16) and (2.17) in (4.18) we get g (A XY; Z) = g r`Y ( X) ; Z
g r`X(Y ) ; Z = g (A YX; Z) : This implies that on considering Z 2 Rad (T M),
we have A XY = A YX:
Conversely, let X; Y 2 D; then using P rXY = (P1rXY ) + (P2rXY ) +
(P0rXY ) P2rXY and applying (2.6) and (2.10),by the use of the equations
from (3.2) to (3.5), it is possible to have
rX Y = g (X; Y ) V + P rXY + LrXY + SrXY
+T h`(X; Y ) + Ch`(X; Y ) + T hs(X; Y ) + Chs(X; Y ) : (4.19) For X; Y 2 `tr (T M), by using (2.11) in (4.19), we also have
A YX + r`X( Y ) + Ds(X; Y ) = g (X; Y ) V + P rXY + LrXY
+SrXY + T h`(X; Y ) + Ch`(X; Y )
+T hs(X; Y ) + Chs(X; Y ) : (4.20) On separating the tangential and transversal components of (4.20), we obtain
A YX = g (X; Y ) V P rXY T h`(X; Y ) T hs(X; Y ) ; (4.21)
r`X( Y ) + Ds(X; Y ) = LrXY + SrXY + Ch`(X; Y ) + Chs(X; Y ) : (4.22)
From (4.21), we get A YX A XY = P [X; Y ]. Since A YX = A XY ; then we
get P [X; Y ] = 0 and this implies that [X; Y ] 2 D fV g. Therefore, g ([X; Y ] ; V ) = g (Y; rXV )+g (X; rYV ) = 0, for any X; Y 2 D, X; Y 2 `tr (T M) and [X; Y ] 2
D. Similarly, for any X; Y 2 S T M? , by using (2.12) in (4.19), we obtain
A YX + rsX( Y ) + D`(X; Y ) = g (X; Y ) V + P rXY + LrXY
+SrXY + T h`(X; Y ) + T hs(X; Y )
+Ch`(X; Y ) + Chs(X; Y ) : (4.23) On separating the tangential and transversal components of (4.23), we get
A YX = g (X; Y ) V P rXY T h`(X; Y ) T hs(X; Y ) ; (4.24)
rsX( Y ) + D`(X; Y ) = LrXY + SrXY + Ch`(X; Y ) + Chs(X; Y ) : (4.25)
From (4.24), we get A YX A XY = P [X; Y ]. Since A YX = A XY ; then
P [X; Y ] = 0. We know that g ([X; Y ] ; V ) = 0; therefore, for any X; Y 2 D, X; Y 2 S T M? , it follows that [X; Y ] 2 D. Hence, the proof is complete.
Theorem 6. Let (M; g; S (T M )) be a GCR-lightlike submanifold of an (")-Sasakian manifold M . Then, the distribution D de…nes a totally geodesic foliation in M if T h (X; Y ) = 0 for any X; Y 2 (D).
Proof. From the De…nition 1, for any X; Y 2 (D), Z 2 (D), W 2 (S), we have g (rXY; Z) = g (rXY; W ) = 0. Particularly, from (2.6) and (2.10), for any
X; Y 2 (D) and Z 2 (D1) Rad (T M ), it follows that
g (rXY; Z) = g h`(X; Y ) ; Z = 0: (4.26)
Similarly, using (2.10) and (2.3), for any X; Y 2 (D), W 2 (S), we have g (rXY; W ) = g (hs(X; Y ) ; W ) = 0: (4.27)
Thus, from (4.26) and (4.27), it is clear that if the distribution D de…nes a totally geodesic foliation in M then hs(X; Y ) and h`(X; Y ) have no components in S
and L, respectively. Thus, using these results with (3.2) and (3.3), the proof is complete.
Theorem 7. Let (M; g; S (T M )) be a GCR-lightlike submanifold of an (")-Sasakian manifold M . Then, the distribution D de…nes a totally geodesic foliation in M if and only if A YX 2 D for any X; Y 2 D .
Proof. For the elements X; Y 2 D , by using (3.4), we obtain rXY =
P rXY + QrXY . If we set P rXY = T rXY and QrXY =
CrXY ; then, by the use of (2.6), (2.8) and (2.9), we further have A YX =
T rXY T h (X; Y ) for any X; Y 2 D . Assume that the distribution D is a
totally geodesic foliation in M ; then, it follows that A YX = T h (X; Y ).
There-fore, A YX 2 D for any X; Y 2 D . Conversely, let A YX 2 D , for any
X; Y 2 D then this implies that T rXY = 0 and hence rXY 2 D .
De…nition 2. A GCR-lightlike submanifold M is called D-geodesic if h (X; Y ) = 0, for any X; Y 2 (D). Using the decomposition of the transversal vector bun-dle, GCR-lightlike submanifold M is said to be a D-geodesic if h`(X; Y ) = 0 and hs(X; Y ) = 0 for any X; Y 2 (D) : Also, M is said to be a mixed geodesic if h`(X; Y ) = 0 and hs(X; Y ) = 0 for any X 2 (D) and Y 2 D .
Theorem 8. Let (M; g; S (T M )) be a GCR-lightlike submanifold of an (")-Sasakian manifold M . Then, the following assertions are equivalent
(1) M is mixed totally geodesic.
(2) rsD D D and A DD D.
Proof. Choose Y 2 D such that Y 2 S T M? ; then, there exists a W 2
S T M? such that W = T W = Y . Let X 2 D and W 2 S T M? ; then,
we have hs(X; Y ) = C (rs
XW ) S (AWX) : Using the hypothesis that M is a
mixed totally geodesic; then, for X 2 D and Y 2 D, hs(X; Y ) = 0 holds and we
further obtain C (rsXW ) = S (AWX) where S (AWX) 2 (S) S T M? and
C (rsXW ) 2 S? S T M? . For any rsXW 2 S T M? , on using (3.3), we
have
(CrsXW ) = rsXW (T rsXW ) ; (4.28)
where rsXW 2 S T M? and (T rsXW ) 2 D S T M? . Using (4.28), it
follows that CrsXW 2 S T M? D . Since S (AWX) 2 D; then, using
(4.27), we get C (rsXW ) = 0 and S (AWX) = 0. Thus, from (4.27) and (4.28),
we have rsXW = (T r s XW ) ; r
s
XW 2 D and AWX 2 D, for any X 2 D and
W 2 D. Consequently, we obtain that rsD D D and A DD D.
Next, choose Y 2 D such that there exists a N 2 `tr (T M) such that N = T N = Y , CN = 0. Using (3.26), for any X 2 D and N 2 `tr (T M), it follows that h`(X; Y ) = Cr`
XN LANX: Assume that M is the mixed totally geodesic; then,
h`(X; Y ) = 0 where X 2 D, Y 2 D, therefore we further obtain
where C r`XN 2 L? `tr (T M ) and LANX 2 (L) `tr (T M ). For
r`XN 2 `tr (T M), from (3.2), we also have
Cr`XN = r`XN T r`XN ; (4.30)
where r`XN 2 `tr (T M) and T r`XN 2 D `tr (T M ). From (4.30), it is
obvious that Cr`XN 2 D, this implies that Cr= `XN =2 D, that is, Cr`XN 2 `tr (T M ) D . Since L (ANX) 2 D therefore from (4.29), we get Cr`XN = 0
and LANX = 0. As a conclusion, we obtain r`XN 2 D and ANX 2 D, for
any X 2 D and N 2 D `tr (T M ). Consequently, we have r`D D D and
A DD D. Hence the proof is complete.
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Current address : Ayse Funda SAGLAMER: Department of Mathematics, Faculty of Art and Science, Dumlup¬nar University, 43100, Kütahya, Turkey
E-mail address : afundasaglamer@gmail.com, aysefunda.saglamer@dpu.edu.tr ORCID Address: http://orcid.org/0000-0001-5162-6378
Current address : Nesrin CALISKAN: Usak University, Faculty of Education, Department of Mathematics and Science Education, 64200, Usak-TURKEY.
E-mail address : caliskan.nesrin35@gmail.com, nesrin.caliskan@usak.edu.tr ORCID Address: http://orcid.org/0000-0002-3189-177X
