CC BY
78Probl. Anal. Issues Anal. Vol. 7 (25), No. 2, 2018, pp. 47-68 47
DOI: 10.15393/j3.art.2018.4650
UDC 514.76, 514.77
Dae Ho Jin
GENERIC LIGHTLIKE SUBMANIFOLDS OF AN INDEFINITE KAEHLER MANIFOLD WITH A SEMI-SYMMETRIC NON-METRIC CONNECTION
Abstract. Recently, this author studied lightlike hypersurfaces of
an indefinite Kaehler manifold endowed with a semi-symmetric
non-metric connection in [7]. Further we study this subject. The
object of study in this paper is generic lightlike submanifolds of an
indefinite Kaehler manifold endowed with a semi-symmetric non-
metric connection such that the induced structure tensor field on
the submanifolds is recurrent or Lie recurrent.
Key words: generic lightlike submanifold, semi-symmetric non-
metric connection, indefinite Kaehler manifold
2010 Mathematical Subject Classification: 53C25, 53C40,
53C50
1. Introduction. A lightlike submanifold M of an indefinite almost complex manifold M, with an indefinite almost complex structure J, is called generic lightlike submanifold if there exists a screen distribution S(TM) of M such that
J(S(TM)±) c S(TM), (1)
where S(TMis the orthogonal complement of S(TM) in the tangent bundle TM of M, i.e., TM = S(TM) ®orih S(TM)±. The notion of generic lightlike submanifold was introduced by Jin-Lee [8] at 2011 and later, studied by several authors (see [3-5]). The geometry of generic lightlike submanifold is an extension of that of lightlike hypersurface and half lightlike submanifold of codimension 2. Much of its theory will be immediately generalized in a formal way to general lightlike submanifolds.
A linear connection V on a semi-Riemannian manifold (M, g) is called a semi-symmetric non-metric connection if it and its torsion T satisfy
(Vxg)(Y,Z) = -d(Y)g(X,Z) - d(Z)g(X,Y), (2)
© Petrozavodsk State University, 2018
T(X,Y) = 9(Y)X - 9(X)Y, (3)
where 9 is a 1-form on M associated with a smooth unit vector field ( by 9(X) = g(X,(). In the followings, we denote by X,Y and Z the smooth vector fields on M. The notion of semi-symmetric non-metric connection on a Riemannian manifold was introduced by Ageshe-Chafle [1].
Remark. Denote by V a Levi-Civita connection of a semi-Riemannian manifold (M,g). It is known [7] that a linear connection V on M is a semi-symmetric non-metric connection if and only if V satisfies
Vx Y = V x Y + 9(Y )X. (4)
The object of present study is generic lightlike submanifolds of an indefinite Kaehler manifold with a semi-symmetric non-metric connection. First, we study the geometry of two types of generic lightlike submanifolds, named by recurrent and Lie recurrent, of such an indefinite Kaehler manifold. Next, we characterize generic lightlike submanifolds of an indefinite complex space form with a semi-symmetric non-metric connection.
2. Semi-symmetric non-metric connections. Let M = (M,g,J) be an indedinite Kaeler manifold, where g is a semi-Riemannian metric and J is an indefinite almost complex structure:
J2X = -X, gJJXjJY ) = g(X,Y), (V X J)Y = 0. (5)
Replacing the Levi-Civita connection V by the semi-symmetric non-metric connection V given by (4), the third equation of (5) is reduced to
(VXJ)Y = 9(JY)X - 9(Y)JX. (6)
Let (M, g) be an m-dimensional lightlike submanifold of an indefinite Kaehler manifold (M,g) of dimension (m + n). Then the radical distribution Rad(TM), defined by Rad(TM) = TM n TM±, of M is a subbundle of the tangent bundle TM and the normal bundle TM±, of rank r (1 ^ r ^ ~mm{m,n}). In general, there exist two complementary non-degenerate distributions S(TM) and S(TM±) of Rad(TM) in TM and TM± respectively, which are called the screen distribution and the co-screen distribution of M [2], such that
TM = Rad(TM) ®arth S(TM), TM± = Rad(TM) ®arth S(TM±),
where 0orth denotes the orthogonal direct sum. Denote by F(M) the algebra of smooth functions on M and by Г(Е) the F(M) module of smooth sections of a vector bundle E over M. Also denote by (5)j the i-th equation of (5). We use the same notations for any others. Let X,Y,Z and W be the vector fields on M, unless otherwise specified. We use the following range of indices:
i,j,k,... E {1,...,r}, a,b,c,... E {r + 1,...,n}.
Let tr(TM) and ltr(TM) be complementary vector bundles to TM in TM|m and TMx in S(TMrespectively and let {N1} ..., Nr} be a lightlike basis of ltr(TM)\u, where U is a neighborhood of M, such that
g(N&) = 6ц, g(Ni,N) = 0,
where {£1, ... ,£r} is a lightlike basis of Rad(TM)|u. Then we have
TM = TM 0 tr(TM) = {Rad(TM) 0 tr(TM)} ®orth S(TM) = = {Rad(TM) 0 ltr(TM)} 0orth S(TM) 0orth S(TM±).
A lightlike submanifold M = (M,g, S(TM),S(TM±)) of M is called
(1) r-lightlike submanifold if 1 ^ r < min{m,n};
(2) co-isotropic submanifold if 1 ^ r = n < m;
(3) isotropic submanifold if 1 ^ r = m < n;
(4) totally lightlike submanifold if 1 ^ r = m = n.
The above three classes (2) — (4) are particular cases of (1) as follows:
S (TM±) = {0}, S (TM) = {0}, S(TM) = S (TM±) = {0}
respectively. The geometry of r-lightlike submanifolds is more general than that of the other three types. Thus we consider only r-lightlike submanifolds M, with following quasi-orthonormal field of frames of M:
{£1, . . . ,£r , N1, . . . ,Nr , Fr+1, . . . ,Fm , Er+1, . . . ,En},
where {FA} and {Ea} are orthonormal bases of S(TM) and S(TM respectively. Denote ta = g(Ea, Ea). Then taSab = g(Ea, Eb).
Let P be the projection morphism of TM on S(TM). Then the local Gauss-Weingarten formulae of M and S(TM) are given respectively by
r n
VxY = VxY + ^ hf(X,Y)Ni + ^ K(X,Y)Ea, (7)
i= 1 a=r+1
rn
VxNi = -ANtX + ^ Tij(X)Nj + ^ Pia(X)Ea, (8)
j=1 a=r+1
rn
VxEa = -AEa X + ^ Aai (X)Ni + ^ Vab (X)Eb] (9)
i=1 b=r+1
r
Vx PY = VX PY + ^ h*(X,PY )£i, (10)
i=1 r
Vx& = —A*.X - £ Oji(X)j, (11)
j=1
where V and V* are induced linear connections on M and S(TM) respectively, hf and ha are called the local second fundamental forms on M, h* are called the local second fundamental forms on S(TM). AN ,AEa and A*, are called the shape operators, and Tij, pia, \ai, ^ab and j are 1-forms on M. Using (2), (3) and (7), we see that
r
(Vx g)(Y,Z) = Y,ihf(X,Y )m(Z) + hf (X,Z )m(Y)} —
i=1
— e(Y )g(X,Z) — B(Z )g(X,Y), (12)
T(X, Y) = d(Y)X — 6(X)Y, (13)
and both hf and hsa are symmetric, where ni's are 1-forms such that
Vi(X ) = g(X,Ni).
In the sequel, denote by ai,^i and Ya the functions given by
®i = d(£i), ^i = d(Ni), Ya = 0(Ea).
As hf(X,Y) = g(Vx Y,bi) and taha(X,Y) = g(V x Y, Ea), we know that hf and ha are independent of the choice of S(TM). The above three types local second fundamental forms are related to their shape operators by
r
hf(X,Y) = g(A*.X, Y) — ^ hek(X,ti)Vk(Y) + a,g(X,Y), (14)
k=1
eahsa(X,Y) = g(AEa X, Y) - £ Xak (X )Vk (Y)+ lag(X,Y), (15)
k=i
h*(X, PY) = g(AN,X, PY) + Vi(X)9(PY) + fcg(X, PY). (16)
Applying the operator Vx to g(&i, ) = 0, g(&i, Ea) = 0, g(Nu Nj) = 0, g(Ni, Ea) = 0, g(Ea, Eb) = t8ab and g(Ni, ) = 5j by turns, we have
:17)
hl(X,Zi) + hj (X,& = 0, ha (X,& = -ea\ai(X),
Vj(A^x) + vi(an x) = -fa(X) - fc^(X),
g(AEa X, Ni) = taPia(X) - YaVi(X), ibl^ab + CaVba = 0, Tj (X) = Oj (X) + aj ^(X).
Furthermore, using (17)i, we see that
hj(X,il) = 0, hj(j ,ik ) = 0, Al & = 0. (18)
Definition 1. We say that a lightlike submanifold of a semi-Riemannian manifold is irrotational [9] if VXE r(TM) for all i E {1, ■ ■ ■ ,r}.
Remark. From (7) and (17) the above definition is equivalent to hj (X,bi) = 0, hSa(X,Ci) = Xai (X )=0.
3. Structure equations. Let M be a generic lightlike submanifold of M. From (1) we show that J(Rad(TM)), J(ltr(TM)) and J(S(TM±)) are subbundles of S(TM). Thus there exist two non-degenerate almost complex distributions Ho and H with respect to J, i.e., J(Ho) = Ho and J(H) = H, such that
S(TM) ={J(Rad(TM)) © J(ltr(TM))} ©orth J(S(TM±)) ©orth Ho, H =Rad(TM) ©orth J(Rad(TM)) ©orth Ho.
In this case, the tangent bundle TM of M is decomposed as follow:
TM = H © J(ltr(TM)) ©orth J(S(TM±)). (19)
Consider r-th local null vector fields Ui and Vi, (n - r)-th local non-null unit vector fields Wa, and their 1-forms ui,vi and wa defined by
Ui = -JNi, Vi = -J&i, Wa = -JEa, (20)
Ui(X ) = g(X, Vi), Vi(X ) = g(X,Ui), wa(X ) = eag(X,Wa). (21)
Denote by S the projection morphism of TM on H and by F the tensor field of type (1,1) globally defined on M by F = J o S. Then X is expressed as X = SX + ^=1 Ui(X)U + YTa=r+i Wa(X)Wa. Therefore,
r a
JX = FX + Y Ui(X)Ni + Y Wa(X)Ea. (22)
i=1 a=r+1
Applying J to (22) and using (5)1, (20) and (22), we have
ra
F2X = -X + Y Ui(X)Ui + Y Wa(X)Wa. (23)
i=1 a=r+1
In the sequel, we say that F is the structure tensor field of M.
Applying the operator VX to (20) 1,2,3 and (22) by turns and using (6), (7)-(11), (14)-(16) and (20)-(22), we have
hj (X,Ui) =u3 (AN, X)+ ßiUj (X) = h* (X, Vj) - 0(V )Vi(X), h8a(X,Ui) = Wa(ANi X)+ ß%Wa(X) = 6a{h*(X,Wa)-0(Wa)m (X)}, ( )
hj(X, Vi) = hii(X, Vj), ha(X, Vi) = iahl(X, Wa), tbhSb (X,Wa)= ^(XW),
ra
VxUi = F(An.X) + YTij(X)Uj + Y Pia(X)Wa + (25)
j=1 a=r+1
+ ßi FX + d(Ui)X,
rr
Vx Vi = F (AI X) - Y «AX )V + Y hj (X,&Uj - (26)
j=1 j=1
- Y eaKi(X)Wa + aiFX + 9(V)X,
a=r+1
ra
Vx Wa = F (AEa X) + Y Aai(X )Ui + Y Vab(X )Wb + (27)
i=1 b=r+1
+ YaFX + 0(Wa)X,
r n
(VxF)(Y) = £Ul(Y)AN,X + £ wa(Y)ABaX - (28)
i=1 a=r+1
rn
- £ hf(X,Y )Ui - £ K(X,Y )Wa +
i=1 a=r+1
+ 9(JY)X - 9(Y)FX.
4. Recurrent and Lie recurrent structure tensors.
Definition 2. The structure tensor field F of M is said to be recurrent [6] if there exists a 1-form w on M such that
(Vx F )Y = w(X )FY.
A generic lightlike submanifold M of an indefinite Kaehler manifold M is called recurrent if it admits a recurrent structure tensor field F.
Theorem 1. Let M be a recurrent lightlike submanifold of an indefinite Kaehler manifold M with a semi-symmetric non-metric connection. Then
(1) F is parallel with respect to the induced connection V on M,
(2) M is irrotational and the 1-forms pia satisfy pia = 0,
(3) the 1-form 9 vanishes on TM,
(4) H, J(ltr(TM)) and J(S(TMx)) are parallel distributions on M,
(5) M is locally a product manifold Mr x Mn-r x M^ where Mr, Mn-r and MB are leaves of J(ltr(TM)), J(S(TM±)) and H, respectively.
Proof.
(1) From the above definition and (27), we obtain
w(X)FY =Y, Ui(Y)An. X + £ Wa(Y)ABaX-
i=1 a=r+1
rn
- £ h*(X,Y)Ui - £ h8a(X,Y)Wa + 9(JY)X - 9(Y)FX. (29)
i=1 a=r+1
Replacing Y by to this and using the fact that F£j = -Vj, we get
rn
w(X)Vj = £ hi(X,j)Uk + £ hsb(X,tj)Wb + 9(Vj)X + ajFX. (30)
k=1 b=r+1
Taking the scalar product with Ni to (30), we obtain
d{Vj )VZ{X) + a Vi(x) = 0.
Taking X = Vi and X = £i to this equation by turns, we have
a = o, e(v) = o, (31)
for all i. Taking the scalar product with Uj to (30), we get w = 0. Thus F is parallel with respect to the induced connection V on M.
(2) Taking the scalar product with Vi and Wa to (30) such that w = aj = 6(Vj) = 0, we obtain hii(X,£j) = 0 and h{(X,£j) = 0. Thus M is irrotational by Remark in Section 2.
Replacing Y by Wa to (29) such that w = 0, we have
r n
X = Y hi(X,Wa)Ui + Y hsb(X,Wa)Wb - YaX + d(Wa)FX. (32)
i=1 b=r+1
Taking the scalar product with Ni and Ui to this equation by turns and using (15), (17)4, we obtain
taPia(X) = 9(Wa)Vi(X), tahaaiX, Ui) = -d(Wa)Vi(X). (33)
Replacing X by £i to (33)2 and using the fact that ha(£i, Ui) = 0, we get Q(Wa) = 0. From this result and (33) 1, we see that pia = 0. Thus
0(Wa) = 0, Pia = 0, haa(X,Ui) = 0. (34)
(3) Replacing Y by Ui to (29) such that w = 0, we have
rn
An,X = Y hi(X, Ui)Uk + Y haa(X, Ui)Wa - PiX + 0(Ui)FX. (35)
fc=1 a=r+1
Taking the scalar product with Nj and Uj to this by turns, we get Vj(AN X) = -PiVj(X) - e(Ui)Vj(X),
(36)
g(ANtX, Uj) = -frvj(X) - d(Ui)Vj(X).
Taking i = j to (36)1 and using (17)3, we get Q(Ui)vi(X) = 0. It follows that 0(Ui) = 0. Using (16), (36)2 reduces h*(X, Uj) = 0. Thus
0(Ui) = 0, h*(X, Uj) = 0. (37)
Replacing X by Zj to (29) and using M is irrotational, we get
rn
£ Ui(Y)An. j + £ Wa(Y)ABa j + 9(JY)j + 9(Y)Vj = 0.
a=r+1
i=1
Taking the scalar product with Uj to this equation, we have
rn
£ Ui(Y )§(An. j, Uj) + £ Wa(Y )g(AEa j ,U3) + 9(Y) = 0. (38
a=r+1
i=1
Taking Y = Ui and Y = Wa by turns and using (34) 1 and (37) 1, we get
9(ANz j ,Uj ) = 0, g(ABa j ,Uj ) = 0.
Consequently, (38) is reduced to 9(X) = 0. Thus 9 vanishes on TM. (4) Using (2), (11), (14), (15), (22), (26) and (27), we get
f g(VxZi, Vj) = -hj(X, Vj) + a,Uj(X), g(VxWa) = -hj(X, Wa) + taaiWaW(X), g(Vx Vi,Vj) = hj (X, Zi) + 9(Vi)Uj (X),
g(VxVi, Wa) = -Ki(X) + 6a9(V,)Wa(X),
g(VxZ,Vj) = hj(X, FZ) + 9(Z)uj(X),
[g(VxZ, Wa) = ta[hSa(X, FZ) + 9(Z)Wa(X)},
for any X e r(TM) and Z e r(Ho). Taking Y = Vj and Y -Z e r(Ho) to (29) by turns and using (31) and the facts that 9 TM, uii(FZ) = Wa(FZ) = 0 and JFZ = F2Z = -Z, we have
h1(X,Vj ) = 0,
hSa(X,Vj ) = hj (X,Wa) = 0,
(39)
FZ, 0 on
h^X, FZ) = 0, hsa(X, FZ) = 0.
Using (31), (40), (41) and Xai = 0, (39) are equivalent to
VXY e r(H), VX e r(TM), VY e r(H).
It follows that H is a parallel distribution on M.
Applying F to (32) and (35) and using (34) 1 and (37) 1, we get
(40)
(41)
F (An. X) = -&FX,
F (Ae„ X) = -Ya FX.
Using these results together with (34), (37) and Xai = 0, (25) and (27) reduce to
r
VxUi = YTij(X)Uj, VxUi G r(J(ltr(TM))), (42) j=i
n
Vx ^Ki = Y VabWb, Vx Wa G r(J (S (TM±))). (43)
b=r+1
Thus J(ltr(TM)) and J(S(TMx)) are also parallel distributions on M.
(5) As H, J(ltr(TM)) and J(S(TM±)) are parallel distributions and satisfy (19), by the decomposition theorem of de Rham [10], M is locally a product manifold Mr x Mn-r x MN, where Mr, Mn-r and M^ are leaves of the distributions J(ltr(TM)), J(S(TMx)) and H respectively. □
Definition 3. The structure tensor Geld F of M is said to be Lie recurrent [6] if there exists a 1-form § on M such that
(Cx F )Y = §(X )FY,
where Lx denotes the Lie derivative on M with respect to X, that is,
(Lx F)Y = [X, FY] - F[X, Y]. (44)
In the case § = 0, i. e., Lx F = 0, we say that F is Lie parallel. A generic lightlike submanifold M of an indefinite Kaehler manifold M is called Lie recurrent if it admits a Lie recurrent structure tensor field F.
Theorem 2. Let M be a Lie recurrent generic lightlike submanifold of an indefinite Kaehler manifold M with a semi-symmetric non-metric connection. Then
(1) F is Lie parallel,
(2) Tij and Pia satisfy Tij(FX) = 0 and Pia(FX) = Moreover,
r
Tij (X ) = Y Uk (X )g(ANk Vj ,Ni). k=1
Proof.
(1) Using (13), (22), (28) and (44), we obtain
&(X )FY = —V fy X + FVy X+
r n
+ £ ut(Y)AN,X + £ wa(Y)AEaX—
i=1 a=r+1
rn
— £ hi(X,Y)Ui — £ ha(X,Y)
i=1 a=r+1
r n
+ {£ /3iUi(Y)+ £ YaWa(Y)}X. (45)
i=1 a=r+1
Replacing Y by Zj and Y by Vj to (45) respectively, we have — i)(X )Vj = Vv3 X + FVj X—
rn
— £ hi(X,Zj)Ui — £ hSa(X,Cj)Wa, (46)
i=1 a=r+1
V(X )Zj = —Vj X + F Vvj X—
rn
— £ h£i(X, Vj )Ui — £ hSa(X,Vj )Wa. (47)
i=1 a=r+1
Taking the scalar product with Ui to (46) and Ni to (47), we get
—5ij#(X) = g(VvjX, Ui) — g(VjX, Ni),
j j (48)
6i3$(X) = g(VvjX, Ui) — g(VjX, Ni),
respectively. It follows that = 0. Thus F is Lie parallel.
(2) Taking the scalar product with Ni to (46) such that X = Wa and using (15), (17)4 and (27), we get hsa{Ui,Vj) = pia(Zj). Also, taking the scalar product with Wa to (47) such that X = Ui and using (25), we have haa(Ui, Vj) = —pia(Zj). Thus Pia(Zj) = 0 and haa(Ui, Vj) = 0.
Taking the scalar product with Ui to (46) with X = Wa and using (15), (17)2,4 and (27), we get tapia(Vj) = Xaj(Ui). Also, taking the scalar product with Wa to (46) such that X = Ui and using (17)2 and (25), we get tapia(Vj) = —Xaj (Ui). Thus pia(Vj) = 0 and Xaj (Ui) = 0.
Taking the scalar product with Vi to (46) such that X = Wa and using (17)2, (24)4 and (27), we obtain Xai(Vj) = —Xaj(Vi). Also, taking the scalar product with Wa to (46) such that X = Vi and using (17)2 and (26), we have Xai(Vj) = Xaj(Vj). Thus we obtain Xai(Vj) = 0.
Taking the scalar product with Wa to (46) such that X = £i and using (11), (14) and (17)2, we get h£i(Vj,Wa) = Xai(£j). Also, taking the scalar product with Vi to (47) such that X = Wa and using (27), we have hi(Vj, Wa) = -Xai(j). Thus Xai(^j) = 0 and hl(Vj, Wa) = 0. Summarizing the above results, we obtain Pia(ij ) = 0, Pia(Vj )=0, Xai(Uj )=0, Xm(Vj ) = 0, Xai(j ) = 0,
hSa(Ui, Vj) = hi (Ui, Wa) = 0, hi(Vj , Wa) = ha(Vj, V) = 0. (49)
Taking the scalar product with Ni to (45) and using (17)4, we have
n
- g(VFyX, Ni) + g(VyX, Ui) + Y CaWa(Y)pia(X) +
a=r+1
r
+ Y Uk(Y){g(ANkX, Ni) + ¡3kVi(X)} = 0. (50) k=1
Taking X = and Y = Uk to (50) and using (11) and (14), we have
hj(Uk, Ui) = g(ANk j, Ni) + 3kSj. (51)
As hi is symmetric, applying (24)1 {take X = Ui} to (51), we obtain
hk(Ui, Vj) = hj(Ui, Uk) = g(ANk j, Ni) + 3kSij. (52)
On the other hand, applying (24)1 {take X = Uk} to (51), we obtain
h* (Uk ,Vj ) = g(ANk j ,Ni)+ 3k Sij. Exchanging i by k and k by i to this equation and using (17)3, we have
hk (Ui, Vj) = -g(ANr j, Nk) + 3iSkj = -g(ANk j ,N) - 3k Sij. (53) Comparing (52) with (53), we obtain
g(Ank j ,Ni)+ 3k Sij = 0. (54)
Replacing X by £j to (50) and using (11), (14), (17))a, (49)1 and (54), we get
hj (X,Ui) = Tij (FX). (55)
Taking X = Vj to (50) and using (17)6, (26) and (49)2, we have
r
hi (FX, Ui) + Tij (X) = Y Uk (X )g(ANk Vj, Ni). (56)
k=1
Taking X = Ui to (45) and using (16), (23), (24) 1,2 and (25), we get
rn
£ uk(Y)ANk Ui + £ Wa(Y)ABa Ui — An.Y—
n
' Wa(Y )AEa
k=1 a=r+1
rn
— F(A^ FY) — £ Tij (FY)Uj — £ pm(FY)Wa+
j=1 a=r+1
rn
+ {£ fjUj(Y)+ £ YaWa(Y)}Ui — f3i{F2Y + Y} = 0. (57)
j=1 a=r+1
Taking scalar product with Vj to (57) and using (54), we get
hj (X,Ui) = —Tij (FX). Comparing this equation with (55), we obtain
Ti3 (FX ) = 0, hj (X,Ui) = 0. (58)
Using (58)2, the equation (56) reduced to
r
Tij(X) = £ Uk(X)g(ANk Vj, Ni). (59)
k=1
Taking the scalar product with Uj to (57) and then, taking Y = Wa and using (15), (16) and (24)2, we have
h*(Wa, U3 ) = ta ha (Ui, Uj ) = tahaa(Uj , Ui) = h*(Uj , Wa). (60)
Taking the scalar product with Wa to (57) and using (23), we have
rn ' a) + / . uk (Y )hk W a) + / tbWb(Y )hb (ui, W a)
tapia(FY) = —h*(Y, Wa) + J] Uk (Y )h*k (Ui, Wa) tbWb(Y )hb (Ui, Wa)
k=1 b=r+1
by (15) and (16). Taking the scalar product with Ui to (45) such that X = Wa and using (17)4, (23), (24)2 and (60), we get
rn
tapia(FY) = h* (Y, Wa) — £ Uk (Y )h*k (Ui, Wa) — £ tbWb(Y )h§ (Ui, Wa).
k=1 b=r+1
Comparing the last two equations, we obtain pia(FY) = 0. □
5. Indefinite complex space forms.
Definition 4. An indefinite complex space form M(c) is an indefinite Kaehler manifold of constant holomorphic sectional curvature c such that
R(X,Y)Z = c{g(Y,Z)X - g(X,Z)Y + g(JY,Z)jX-
- g(JX,Z)JY + 2-(X,jY)JZ}, (61)
where R is the curvature tensor of the Levi-Civita connection V on M.
Let R be the curvature tensor of the semi-symmetric non-metric connection V on M. By directed calculations from (3) and (4), we get
R(X,Y)Z = R(X,Y)Z + (VXe)(Z)Y - (VYd)(Z)X. (62)
Denote by R and R* the curvature tensors of the induced linear connections V and V * on M and S(TM) respectively. Using the Gauss-Weingarten formulae, we obtain Gauss equations for M and S(TM) respectively:
R(X, Y)Z = R(X, Y)Z + J2{hi(X, Z)ANtY - hi(Y, Z)ANzX}+
i=1
n r
+ Y{hSa(X,Z)AEaY - hSa(Y,Z)AEaX} + Y {(Vxhi)(Y,Z)-
a=r+1 i=1
r
- (Vyhi)(X,Z) + Y[Tji(X)hj(Y, Z) - Tji(Y)hi(X, Z)] + j=1
n
+ Y [Xai(X)hSa(Y,Z) - Xai(Y)h'a(X,Z)] - e(X)hi(Y,Z) +
a=r+1
n
+ e(Y)hi(X,Z)}Ni + Y {(Vxha)(Y,Z) - (Vyha)(X,Z)+
a=r+1
rn
+ Y[Pia(X)hi(Y, Z) - Pia(Y)hi(xX, Z)] + Y [^ba(X)%(¥, Z)-
i=1 b=r+1
4 (x, Z)] - e(x )ha
Mba(Y )hb (X, Z)] - e(X )hSa(Y, Z) + e(Y )hSa(X, Z)} Ea, (63)
R(X, Y)PZ = R*(X, Y)PZ+
r
+ Y{h*(X, PZ)A*UY - h*(Y, PZ)A^X}+
i=1
+ £ {(Vxh*)(Y,PZ) - (Vyh*)(X,PZ) +
i=1 r
+ £[uik(Y)hk(X, PZ) - un-(X)hk(Y, PZ)]-
k=i
- d(X)h*(Y, PZ) + 9(Y)h*(X, PZ)}Ci. (64)
Comparing the tangential, lightlike transversal and radical components of the two equations (62) and (63) and using (22), we get
R(X,Y)Z = £{hi(Y,Z)ANíX - híi(X,Z)ANiY}+
i=1 n
+ £ {K(Y,Z)AEaX - K(X,Z)AEaY}+
a=r+1
+ (Vx0)(Z)Y - (Vyd)(Z)X+
c
+ 4{g(Y, Z)X - g(X, Z)Y + g(JY, Z)FX-
- g(JX, Z)FY + 2g (X, JY)FZ}, (65)
r
(Vxhi)(Y, Z) - (Vyhf)(X, Z) + Y,{hk(Y, Z)rkl(X)-
k=1
n
- hk(X, Z)Tki(Y)} + £ {hsa(Y, Z)\ai(X) - hsa(X, Z)\ai(Y)}-
a=r+1
- hi(Y, Z)d(X) + 9(Y)hi(X, Z)9(Y) = c
= 4{Ui(X)g(JY, Z) - Ui(Y)g(JX, Z) + 2m(Z)g(X, JY)}. (66)
(Vxh*)(Y, PZ) - (Vyh*)(X, PZ)-
r
- £ {hk(Y, PZ)uik(X) - hk(X, PZ)uik(Y)}-
k=i
r
- £ {hk(Y, PZ)Vi(ANkX) - hk(X, PZ)Vi(ANkY)}-k=i
n
- £ {hSa(Y, PZ)Vi(AEaX) - hS(X, PZ)n%(AEaY)} - h*(Y, PZ)Ô(X) +
a=r+1
+ h*(.X, PZ)9(Y) - (Vx9)(PZMY) + (Vyd)(PZMX) =
c
= 4{g(Y, PZ)m(X) - g(X, PZ)Vi(Y) + vt(X)g(JY, PZ)-
- Vi(Y)g(JX, PZ) + 2Vi(PZ)g(X, JY)}. (67)
Theorem 3. Let M be a generic lightlike submanifold of an indefinite complex space form M(c) with a semi-symmetric non-metric connection. If one of the following four statements
(1) M is recurrent,
(2) M is Lie recurrent,
(3) Ui is parallel with respect to the connection V, or
(4) Vi is parallel with respect to the connection V is satisfied, then M(c) is flat, i. e., c = 0.
Proof. (1) By Theorem 1, we get pia = 0 and 9 = 0 on TM, and we have (34)3, (36) and (37)1,2. From (36)i and (37)i: 9(Ui) = 0, we obtain
Vi(ANj X) = -j Vi(X). (68)
Applying VX to 9(Ui) = 0 and using (7), (34)3 and 9|TM = 0, we have
r
(Vx9)(Ui) = - Yhhek(X,Ui). (69)
k=1
Applying VX to (37)2: h*(Y, Uj) = 0 and using (42)1, we obtain
(Vxh*)(Y, Uj) = 0. Taking PZ = Uj to (67) and using (34)3, (37)2 (68) and (69), we have
-{Vj(Y)Vi(X) - Vj(X)m(Y) + Vi(Y)Vj(X) - Vi(X)Vj(Y)} = 0.
Taking X = £i and Y = Vj, we have c = 0 and M(c) is flat.
(2) Taking X = to (14) and using (18)2 and h\ is symmetric, we get h£i(X,£j) = g(A\.,X). From this result and (17)1, we obtain g(A^+ + A\ £i,X) = 0.' As S(TM) is non-degenerate, we get A^ j = -A| £i. Thus A*,£j is skew-symmetric with respect to i and j.
In the case M is Lie recurrent, taking Y = Uj to (57), we have
ANj Ui + Pj Ui = AN; Uj + Wj.
Applying F to this equation, we have F(AN.Ui) = F(AN.Uj). Thus F(An. Uj) is symmetric with respect to i and j. Therefore, we obtain
hKZj, F (Aj Ui))= g(Al j ,F (ANj U)) = 0. (70)
From (17)2, (24)4, (49)4 and the fact that ha is symmetric, we get
hf(Zj ,Wa) = ta ha (Zj ,Vi) = ^(Vi^) = —Xaj (V) = 0. (71) Applying VX to (58)2: hf (Y, Uj) = 0 and using (25), we have
(Vx hf)(Y,Uj ) = —hf(Y,F (ANj X)) —
a
— £ pja(X)hl(Y,Wa) — fjhl(Y,FX) — d(Uj)hl(X,Y).
a=r+1
Substituting this into (66) with Z = Uj and using (58)2, we get h(i(X)F (ANj Y)) — hf(Y,F (ANj X))+
n
+ £ {pja (Y )hi(X, Wa) — p3a(X )hf(Y, Wa)} +
a=r+1
+ £ {Xai(X )h?a(Y, Uj ) — Xai(Y )haa(X, Uj )} +
a=r+1
+ fj{hf(X,FY) — hf(Y, FX)} = c
= 4
-{Ui(Y)Vj(X) — Ui(X)Vj(Y) + 25ijg(X, JY)}.
Taking Y = Ui and X = Zj to this equation and using (49)3,5, (58)2, (70) and (71), we have c = 0. Consequently, M(c) is flat.
(3) As VXUi = 0, taking the scalar product with Uj to (25), we get Vj (An.X) = —ffiVj(X)+ d(Ui)Vj (X). Substituting this equation into the left term of (17)3, we have
d(Ui)Vj (X) + d(Uj )Vi(X ) = 0. Taking X = Vj to this equation, we obtain
9(Ui ) = 0, Vj (An. X ) = —fiVj (X). (72)
Applying VX to 9(Ui) = 0 and using (7) and VXUi = 0, we get
r n
(Vx9)(Ui) = - Y Pkhk(X, Ui) - Y Yah'a(X,Ui). (73)
k (X,Ui) - lahSa(
k=1 a=r+1
Taking the scalar product with Wa and Nj to (25) by turns and using (16) and (72) 1, we have
Pia = 0, h*(X, Uj) = 0. (74)
From (17)4 and (74) 1, we see that
Vi(AEa X) = -YaVi(X). (75)
Applying Vy to (74)2 and using the fact that VXUj = 0, we obtain
(Vxh*)(Y, Uj) = 0.
Replacing PZ by Uj to (66) and using (72)2, (73), (74)2, (75) and the last
equation, we have c
4{Vj(Y)Vi(X) - Vj(X)Vi(Y) + Vi(Y)Vj(X) - Vi(X)Vj(Y)} = 0.
Taking X = and Y = Vj to this equation, we have c = 0.
(4) As VXV = 0, taking the scalar product with Vj, Wa and Nj to (26) by turns and using (14) and (17)2, we obtain
h) (X,&) = -9(Vi)uj (X), ha (X,&) = -9(Vi)wa(X), (76)
hi(X,Uj ) = -9(V)Vj (X).
By using (24)4, (76)3 and the fact that h\ is symmetric, we see that
ha ( Uj ,Vk ) = tahi (Uj ,Wa) = 0. (77)
From (24)1 and (76)3, we obtain h*(Y,Vj) = 0. Applying VX to this equation and using the fact that VXVj = 0, we get
(VXh*)(Y, Vj) = 0.
Taking PZ = Vj to (66) and using the last two equations, we obtain
E {hi(X, VjMANkY) - hek(Y, Vj)n%(ANkX)}+
j=1 n
+ Y {K(X,Vj)vi(AEaY) - K(Y,Vj)vi(AEaX)}-
a=r+1
- (Vx0)(Vj)Vi(Y) + (Vy0)(Vj)Vi(X) =
c
= 4{Uj(Y)ni(X) - u-(X)ni(Y) + 2Sijg(X, JY)}.
Taking X = £i and Y = Uj and using (76) and (77), we get c = 0. □
Theorem 4. Let M be a generic lightlike submanifold of an indefinite complex space form M(c) with a semi-symmetric non-metric connection. If Wa is parallel with respect to V and Y1 i=1 hsa(Wa, Vk) = 0, then r = 1 and c = 0.
Proof. As VXWa = 0, taking the scalar product with Wb to (27), we get
Vab(X) = -9(Wa)wb(X). Substituting this equation into the left term of (17)5, we have
ebB(Wa)wb(X) + CaO(Wb)wa(X) = 0. Replacing X by Wb to the last equation, we obtain
d(Wa) = 0, Vab = 0. (78)
Applying VX to 9(Wa) = 0 and using (7) and VXWa = 0, we get
rn
(Vx 0)(Wa) = - Y Pih<i(X,Wa) - Y Ybhb (X, Wa). (79)
i=1 a=r+1
Taking the scalar product with Ui, Vi and Ni to (27) by turns and using (15), (17)4 and (78)1, we have
Vi(AEaX) = -YaVi(X), i. e., pa = 0, Aai = 0, hBa(X, Ui) = 0. (80)
As Aai = 0, from (17)2, we obtain
haa(X,£i ) = 0. (81)
From (24)2, (78)1 and (80)3, we obtain h*(X, Wa) = 0. Applying VY to this equation and using the fact that VXWa = 0, we get
(Vx h* )(Y,Wa) = 0.
Replacing PZ by Wa to (67) and using (24)4, (79), (80) 1 and the last two equations, we have
£ haa(X,Vk ){TH(ANk Y )+fk Vi(Y )} — £ haa(Y,Vk ){^(ANk X )+fk m(X )} =
^a(X,Vk ){Vi(ANk Y )+fk Vi(Y )} — £ ^
k=1 k=1
c
= 4 {Wa(Y )Vi(X ) — Wa (X )Vi(Y )}. Taking X = Zi and Y = Wa to this and using (81), we have
£ haa(Wa, VkH^A^ Zi) + fk} = — C. (82
k=1
Comparing the co-screen components of (62) and (63), we obtain (Vx haa)(Y,Z ) — (Vy haa)(X,Z ) +
r
+ £{pia(X)hi(Y, Z) — pia(Y)hi(X, Z)} +
i=1
+ £ Ka(X)hSb(Y, Z) — llba(Y)hSb(X, Z)}— (83)
b=r+1
— 9(X )ha(Y,Z) + O(Y )haa(X,Z) c
= 4{Wa(X)g(JY, Z) — Wa(Y)g(JX, Z) + 2Wa(Z)g(X, jy)}. As Xai = Iab = O(Wa) = 0 and FWb = 0, from (27), we have
F (aeu X ) = —YaFX, F (AEa Wb) = 0. (84)
Applying VX to ha(Y, Ui) = 0 and using (25) and (80)3, we get
(Vxhaa)(Y, Ui) = ha(Y, F(A^X)) — ffha(FX,Y) — O^^X^)
due to pia = 0. Substituting this into (83) with Z = Ui and using the fact that pia = l ab = 0, we have
haa(X)F(Aj Y)) — ha(Y, F(A„ X))+ fi{hSa(X, FY) — ha,(FX,Y)} =
c
= 4{w„(YMX) - wa(XMY)}.
Taking X = and Y = Wa to this equation and using (81), we get
c
hSa(Wa, F(ANt&) - PlhSa(Vl, Wa) = -4. From (5)2, (15), (17)3,4, (22), (84)2 and the fact: pia = 0, we have
hSa(Wa,F (A^&)) = -ta9(ANr &F (AEa Wa))-r r
- Y hSa(Wa, Vk )Vk (A^ Ci) = - Y hSa(Wa, Vk )Vk (A^ Ci ) = k=1 k=l
r
= Y hSa(Wa,Vk ){Vi(ANk Ci) + 2f3k }. k=1
From the last two equations, we see that
r
Y hSa(Wa, Vk ){Vi (ANk Ci) + 2fk} - fihSa(Wa, V) = - C. k=1
Comparing this equation with (82), we obtain
r
Y fk hSa(Wa,Vk ) = fihSa(Wa ,Vi), V i. k=1
It follow that
(r - 1)Y fkhaa(Wa,Vk) = 0. k=1
Assume that Y^k=1 fkha(Wa,Vk) = 0. Then r = 1 and i = j = k =1. Thus, from (17)3, we see that
Vi(AN1 X) = -f3m(X).
From this result and (82), we obtain c = 0. □
Acknowledgment. In this paper, we studied the geometry of generic lightlike submanifolds of an indefinite Kaehler manifold with a semi-symmetric non-metric connection. But the geometry of generic lightlike submanifolds and several CR-type lightlike submanifolds of an indefinite Kaehler manifold with a quarter-symmetric non-metric connection are still open problems. We hope that the publication of this paper will help in solving the above more general cases.
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Received April 12, 2018.
In revised form, July 27, 2018.
Accepted September 27, 2018.
Published online October 11, 2018.
Department of Mathematics Dongguk University Gyeongju 780-714, Republic of Korea E-mail: [email protected]
