HALF LIGHTLIKE SUBMANIFOLDS OF A SEMI-RIEMANNIAN SPACE FORM WITH A SEMI-SYMMETRIC NON-METRIC CONNECTION

Abstract

In this paper, we study screen quasi-conformal irrotational half lightlike submanifolds M of a semi-Riemannian space form $\tilde{M}(c)$ admitting a semi-symmetric non-metric connection, whose structure vector field ${\zeta}$ is tangent to M. The main result is a classification theorem for such Einstein half lightlike submanifolds of a Lorentzian space form admitting a semi-symmetric non-metric connection.

1. INTRODUCTION

The theory of lightlike submanifolds is indeed important for both the geometry of submanifolds to mathematics and its applications to physics. The study of such notion was initiated by Duggal and Bejancu [3] and later studied by many authors (see up-to date in [4, 5]). The notion of a semi-symmetric non-metric connection on a Riemannian manifold was introduced by Ageshe and Cha°e [1]. Although now we have lightlike version of a large variety of Riemannian submanifolds, the geometry of lightlike submanifolds of semi-Riemannian manifolds admitting semi-symmetric non-metric connections has been few known. Recently Yasar, Cöken and Yücesan [15] and Jin [6, 7] studied lightlike hypersurfaces in a semi-Riemannian manifold admitting a semi-symmetric non-metric connection. Jin [10] and Jin-Lee [11] studied general lightlike submanifolds and half lightlike submanifolds of a semi-Riemannian manifold with a semi-symmetric non-metric connection.

The objective of this paper is to study screen quasi-conformal irrotational half lightlike submanifolds M of a semi-Riemannian space form (c) admitting a semi-symmetric non-metric connection, whose structure vector field ζ of (c) is tangent to M but it does not belongs to S(TM ). The reason for this geometric restriction on M is due to the fact that such a class admits an integrable screen distribution and a symmetric induced Ricci tensor of M. Our main result is a classification theorem for such Einstein half lightlike submanifolds M of a Lorentzian space form admitting a semi-symmetric non-metric connection.

 

2. SEMI-SYMMETRIC NON-METRIC CONNECTION

Let (, ) be a semi-Riemannian manifold. A connection on is called a semi-symmetric non-metric connection [1] if and its torsion tensor satisfy

for any vector fields X, Y and Z on , where 𝜋 is a 1-form associated with a non-vanishing vector field ζ, which is called the structure vector field, by

𝜋(X ) = (X, ζ ).

A submanifold (M, g) of codimension 2 is called half lightlike submanifold if the radical distribution Rad(TM ) = TM ∩ TM⊥ is a vector subbundle of the tangent bundle TM and the normal bundle TM⊥ of M, with rank 1. In this case, there exists complementary non-degenerate distributions S(TM ) and S(TM⊥) of Rad(TM ) in TM and TM⊥ respectively, which are called the screen and co-screen distributions on M respectively, such that

where ⊕orth denotes the orthogonal direct sum. We denote such a half lightlike submanifold by M = (M, g, S(TM )). Denote by F(M ) the algebra of smooth functions on M, by 𝚪(E ) the F(M ) module of smooth sections of a vector bundle E over M and by (2.3)i the i-th equation of (2.3). We use same notations for any others. Choose L ∈ 𝚪(S(TM⊥)) as a spacelike unit vector field, without loss of generality, i.e., (L, L ) = 1. We call L the canonical normal vector field of M. Consider the orthogonal complementary vector bundle S(TM)⊥ to S(TM ) in T . Certainly Rad(TM) and S(TM⊥) are vector subbundles of S(TM )⊥. As S(TM⊥) is non-degenerate, we have

S(TM)⊥ = S(TM⊥) ⊕orth S(TM⊥)⊥,

where S(TM⊥)⊥ is the orthogonal complementary to S(TM⊥ in S(TM)⊥. It is well-known [3] that, for any null section ξ of Rad(TM) on a coordinate neighborhood 𝒰 ⊂ M, there exists a uniquely defined lightlike vector bundle ltr(TM) and a null vector field N of ltr(TM) on 𝒰 satisfying

(ξ,N) = 1, (N,N) = (N,X) = (N,L) = 0, ∀X ∈ 𝚪(S(TM)).

We call N, ltr(TM) and tr(TM) = S(TM⊥)⊕orth ltr(TM) the lightlike transversal vector field, lightlike transversal vector bundle and transversal vector bundle of M with respect to S(TM) respectively [11]. Then T is decomposed as

In the entire discussion of this article, we shall assume that the structure vector field ζ of to be spacelike unit tangent vector field of M. In the sequel, we take X, Y, Z, W ∈ 𝚪(TM), unless otherwise specified. Let P be the projection morphism of TM on S(TM) with respect to the decomposition (2.3)1. Then the local Gauss and Weingartan formulas of M and S(TM) are given respectively by

where 𝛁 and 𝛁* are induced linear connections on TM and S(TM) respectively, B and D are called the local second fundamental forms of M, C is called the local second fundamental form on S(TM). AN, and AL are linear operators on TM, which are called the shape operators, and 𝜏, 𝜌 and 𝜙 are 1-forms on TM. We say that h(X, Y) = B(X, Y)N +D(X, Y)L is the global second fundamental form tensor of M. Using (2.1), (2.2) and (2.5), we have

and B and D are symmetric on TM, where T is the torsion tensor with respect to the induced connection 𝛁 and 𝜂 is a 1-form on TM such that

𝜂(X) = (X, N )

From the facts B(X, Y) = (XY,ξ) and D(X, Y) = (XY, L), we know that B and D are independent of the choice of S(TM) and satisfy

The above three local second fundamental forms M and S(TM) are related to their shape operators by

where f is the smooth function given by f = 𝜋(N). From (2.12) and (2.13), we show that is S(TM)-valued self-adjoint and satisfies

Denote by , R and R* the curvature tensors of the semi-symmetric non-metric connection of , the induced connection 𝛁 on M and the induced connection 𝛁* on S(TM). Using the Gauss-Weingarten formulas for M and S(TM), we obtain the Gauss-Codazzi equations for M and S(TM) :

A complete simply connected semi-Riemannian manifold of constant curvature c is called a semi-Riemannian space form and denote it by (c). For any X, Y, Z ∈ 𝚪(T ), the curvature tensor of (c) is given by

Taking the scalar product with ξ and L to (2.22), we get

From this results and (2.17), for all X, Y, Z ∈ 𝚪(TM), we obtain

 

3. CHARACTERIZATION THEOREMS

Definition. A half lightlike submanifold M of a semi-Riemannian manifold is said to be irrotational [12] if X ξ ∈ 𝚪(TM) for any X ∈ 𝚪(TM).

From (2.5) and (2.12), we show that the above definition is equivalent to the condition: D(X, ξ) = 0 = 𝜙(X) for all X ∈ 𝚪(TM).

Lemma 1 ([8, 11]). Let M be an irrotational half lightlike submanifold of a semi-Riemannian manifold admitting a semi-symmetric non-metric connection such that the structure vector field ζ of is tangent to M. Then ζ is conjugate to any vector field X on M, i.e., ζ satisfies h(X, ζ) = 0.

Note that h(X, ζ) = 0 is equivalent to the following two equations:

Definition. A half lightlike submanifold M of a semi-Riemannian manifold admitting a semi-symmetric non-metric connection is called screen quasi-conformal [9, 13] if the second fundamental forms B and C satisfy

where 𝜑 is a non-vanishing function on a coordinate neighborhood 𝒰 in M.

Due to (2.13) and (2.15), we show that M is screen quasi-conformal if and only if the shape operators AN and are related by

We quote the following results for irrotational screen quasi-conformal half lightlike submanifold due to Jin [9]:

Theorem 3.1. Let M be an irrotational screen quasi-conformal half lightlike sub-manifolds M of a semi-Riemannian space form (c) admitting a semi-symmetric non-metric connection. If the structure vector field ζ is tangent to M but it does not belong to S(TM), then we have c = 1.

Let be the Ricci curvature tensor of and R(0, 2) the induced Ricci type tensor on M given respectively by

(X, Y) = trace{Z → (Z,X)Y}, ∀X, Y ∈ 𝚪(T),

R(0, 2)(X, Y) = trace{Z → R(Z,X)Y}, ∀X, Y ∈ 𝚪(TM).

Consider a quasi-orthonormal frame field {ξ, Wa} on M, where Rad(TM) = Span{ξ} and S(TM) = Span{Wa} and let E = {ξ,N,Wa} be the corresponding frame field on . Using this quasi-orthonormal frame field, we obtain

R(0, 2)(X, Y) = (X, Y) + B(X, Y)tr AN + D(X, Y)trAL - g(ANX,Y) - g(ALX, ALY) + 𝜌(X)𝜙(Y) - ((ξ, Y)X, N) - ((L, X)Y, L),

This shows that R(0, 2) is not symmetric. The tensor field R(0, 2) is called its induced Ricci tensor [4, 5], denoted by Ric, of M if it is symmetric. It is known [11] that R(0, 2) is an induced Ricci tensor of M if and only if the 1-form 𝜏 is closed, i.e., d𝜏 = 0, for any coordinate neighborhood 𝒰 ⊂ M.

Remark 1. If R(0, 2) is symmetric, then there exists a null pair {ξ , N} such that the corresponding 1-form 𝜏 satisfies 𝜏 = 0 [11], which called a canonical null pair of M. Although S(TM) is not unique, it is canonically isomorphic to the factor vector bundle S(TM)# = TM/Rad(TM) [12]. This implies that all screen distribution are mutually isomorphic. For this reason, in case d𝜏 = 0 we consider only lightlike hypersurfaces M endow with the canonical null pair.

We say that M is an Einstein manifold if the Ricci tensor of M satisfies

It is well-known that if dim M > 2, then κ is a constant. For dim M = 2, any manifold M is Einstein but κ is not necessarily constant.

In case the ambient manifold is a space form (c), R(0, 2) is given by

Taking the scalar product with ξ to (2.17) and using (2.22), we have

Definition. A vector field X on is said to be conformal Killing [8] if

x = -2𝛿

for any non-vanishing smooth function 𝛿, where denotes the Lie derivative on , that is, for all Y, Z ∈ 𝚪(T),

(x)(Y,Z) = X((Y,Z)) - ([X,Y],Z) - (Y,[X,Z]).

In particular, if 𝛿 = 0, then X is called a Killing vector field on .

Theorem 3.2 ([8, 11]). Let M be a half lightlike submanifold of admitting a semi-symmetric non-metric connection. If the canonical normal vector field L is conformal Killing, then L is a Killing vector field.

Proof. Using (2.1) and (2.2), for any X, Y, Z ∈ 𝚪(T ), we have

(x)(Y,Z) = (YX,Z) + (Y, ZX) - 2𝜋(X)(Y,Z).

As L is conformal Killing, we have (XL, Y ) = -D(X, Y ) by (2.9) and (2.16). This implies ( L )(X, Y ) = -2D(X, Y ) for any X, Y ∈ 𝚪(TM ). Thus we have

D(X, Y) = 𝛿g(X, Y ), ∀X, Y ∈ 𝚪(TM).

Taking X = Y = ζ to this and using (3.1)2, we get 𝛿 = 0 and L is Killing. □

Theorem 3.3 ([11]). Let M be a half lightlike submanifold of a semi-Riemannian manifold admitting a semi-symmetric metric connection. Then the following assertions are equivalent :

(1) The screen distribution S(TM) is an integrable distribution. (2) C is symmetric, i.e., C(X, Y) = C(Y, X) for all X, Y ∈ 𝚪(S(TM)). (3) The shape operator AN is self-adjoint with respect to g, i.e.,

g(ANX, Y ) = g(X,AN Y ), ∀X, Y ∈ 𝚪(S(TM )).

Remark 2. Just as in the well-known case of locally product Riemannian or semi-Riemannian manifolds [3, 4, 5, 14], if S(TM ) is an integrable distribution, then M is locally a product manifold C × M* where C is a null curve tangent to Rad(TM ) and M* is a leaf of the integrable distribution S(TM ).

Theorem 3.4. Let M be a screen quasi-conformal irrotational Einstein half lightlike submanifold of a Lorentzian space form (c) with a semi-symmetric non-metric connection. If ζ is tangent to M but it does not belong to S(TM), the canonical normal vector field is conformal Killing and the mean curvature of M is constant, then M is locally a product manifold M = C × M1 × M2, where C is a null curve, M1 is an Euclidean space and M2 is a totally umbilical Riemannian space.

Proof. As L is Killing, we get D = 𝜙 = 0 and g(ALX, Y ) = 0 for any X, Y ∈ 𝚪(TM ). From (3.3), (3.5) and the fact is self-adjoint, we show that R(0, 2) is a symmetric induced Ricci tensor Ric and S(TM ) is an integrable distribution. As g(ζ, X ) = B(ζ,X ) = 0 and S(TM ) is non-degenerate, we have

Using (2.13), (3.3), (3.4) and the fact c = 1, from (3.5) we have

for all X, Y ∈ 𝚪(TM ) due to c = 1, where 𝛼 = tr - fm𝜑-1. Taking X = Y = ζ to (3.8) and using (3.7), we have 𝜅 = m. Thus (3.8) becomes

As is Lorentzian manifold, S(TM ) is a Riemannian. Since ξ is an eigenvector field of corresponding to the eigenvalue 0 due to (2.16) and is S(TM )-valued real self-adjoint operator, have m real orthonormal eigenvector fields in S(TM ) and is diagonalizable. Consider a frame field of eigenvectors {ξ,E1, . . . , Em} of such that {E1, . . . , Em} is an orthonormal frame field of S(TM ) and Ei = 𝜆iEi. Put X = Y = Ei in (3.9), each eigenvalue 𝜆i is a solution of the equation

x2 - 𝛼x = 0.

As this equation has at most two distinct solutions 0 and 𝛼, there exists p ∈ {0, 1, . . . , m} such that 𝜆1 = . . . = 𝜆p = 0 and 𝜆p+1 = . . . = 𝜆m = 𝛼(≠ 0), by renumbering if necessary. As tr = 0p + ( m - p )𝛼, we have

(m - p - 1)𝛼 = fm𝜑-1.

Consider four distributions Do, D𝛼, and on S(TM ) given by

Do = {X ∈ 𝚪(TM) | X = 0}, = Do ∩ S(TM), D𝛼 = {U ∈ 𝚪(TM) | U = 𝛼PU}, = D𝛼 ∩ S(TM ).

Clearly we show that Do ∩ D𝛼 = Rad(TM ), ∩ = {0} as 𝛼 ≠ 0 and = PDo, = D𝛼 In the sequel, we take the vector fields X, Y ∈ 𝚪(Do), U, V ∈ 𝚪(D𝛼) and Z, W ∈ 𝚪(TM ). Denote X *= PX, Y * = PY, U * = PU and V *= PV . Then X *, Y * ∈ 𝚪() and U *, V * ∈ 𝚪(). Since X * and U * are eigenvector fields of the real self-adjoint operator corresponding to the different eigenvalues 0 and 𝛼 respectively, X * ⊥ U * and g(X,U ) = g(X *, U *) = 0, that is, Do ⊥ g D𝛼. Also, since B(X,U ) = g(X,U ) = 0, we show that D𝛼 ⊥ B Do. Since {Ei}1≤i≤p and {Ea}p+1≤a≤m are vector fields of and respectively and and are mutually orthogonal, and are non-degenerate distributions of rank p and rank (m - p) respectively. Thus S(TM ) is decomposed as S(TM ) = ⊕orth .

From (3.9), we get ( - 𝛼P ) = 0. Let W ∈ Im. Then there exists Z ∈ 𝚪(TM ) such that W = Z. Then ( - 𝛼P )W = 0 and W ∈ 𝚪(D𝛼). Thus Im ⊂ 𝚪(D𝛼). By duality, we have Im( - 𝛼P) ⊂ 𝚪(Do).

Applying 𝛁X to B(Y,U ) = 0 and using (2.13) and Y = 0, we obtain

(𝛁XB )(Y,U ) = -g(𝛁XY,U ).

Substituting this into (3.6) and using (2.11) and X = Y = 0, we get

g([X, Y], U) = 0.

As Im ⊂ 𝚪(D𝛼) and D𝛼 is non-degenerate, we get [X, Y] = 0. Thus [X, Y] ∈ 𝚪(Do) and Do is integrable. This result implies [X *, Y *] ∈ 𝚪(Do). On the other hand, since S(TM) is integrable, [X *, Y *] ∈ 𝚪(S(TM )). Thus [X *, Y *] ∈ 𝚪(). Thus is also an integrable distribution.

Applying 𝛁V to B(U, Y ) = 0 and using Y = 0 and U = 𝛼PU, we get

(𝛁V B)(U, Y ) = -𝛼g(𝛁V Y,U ).

Substituting this into (3.6) and using the fact 𝛼 ≠ 0, we obtain

g(𝛁V Y,U ) = g(V, 𝛁UY ).

Applying 𝛁V to g(Y,U ) = 0 and using (2.10), we have

𝜋(Y )g(U, V ) - B(V,U )η(Y ) - g(𝛁V Y,U ) = g(Y,𝛁V U ).

Taking the skew-symmetric part of this equation and using (2.11), we have

g([V,U], Y) = 0, ∀ Y ∈ 𝚪(Do) and U, V ∈ 𝚪(D𝛼).

From this, we get g([V *,U *], Y *) = 0 for all Y * ∈ 𝚪() and U *, V * ∈ 𝚪(). As and are mutually orthogonal non-degenerate distributions, we show that [V *, U *] ∈ 𝚪(). Thus is also an integrable distribution.

Applying 𝛁U to B(X, Y ) = 0 and 𝛁X to B(U, Y ) = 0, we have

(𝛁UB )(X, Y ) = 0, (𝛁XB )(U, Y ) = -𝛼g(𝛁XY,U ).

Substituting this two equations into (3.6), we have 𝛼g(𝛁XY, U ) = 0. As

g(𝛁XY, U ) = B(𝛁XY, U ) = 𝛼g(𝛁XY, U ) = 0

and Im ⊂ 𝚪(D𝛼) and D𝛼 is non-degenerate, we get 𝛁XY = 0. This implies 𝛁XY ∈ 𝚪(Do). Thus Do is an auto-parallel distribution on S(TM ). This implies that 𝛁X *Y * ∈ 𝚪(Do) for any X *, Y * ∈ 𝚪(). As C(X *, Y *) = 𝜑B(X *, Y *) + η(X *)𝜋(Y *) = 0, we have 𝛁X *Y * = 𝛁*X *Y * ∈ 𝚪(S(TM )). Thus 𝛁X *Y * ∈ 𝚪() and is also an auto-parallel distribution.

As ζ = 0, ζ belongs to Do. Thus 𝜋(U ) = 0 for any U ∈ 𝚪(D𝛼). Applying 𝛁X to g(U, Y ) = 0 and using (2.10) and the fact Do is auto-parallel, we get g(𝛁XU, Y ) = 0. This implies 𝛁XU ∈ 𝚪(D𝛼).

Assume that the mean curvature vector field

of M is constant. Then 𝛼 is a constant. Applying 𝛁X to B(U, V ) = 𝛼g(U, V ) and 𝛁U to B(X, V ) = 0 and using the fact 𝛼 is constant, we have

(𝛁XB )(U, V ) = 0, (𝛁UB )(X, V ) = -𝛼g(𝛁UX, V ).

Substituting this two equations into (3.6) and using Do ⊥B D𝛼, we have

g(𝛁UX, V ) = 𝜋(X )g(U, V ).

Applying 𝛁U to g(X, V ) = 0 and using (2.10), we obtain

g(X,𝛁UV ) = 0.

From this, we get g(X *, 𝛁U *V *) = 0 for all X * ∈ 𝚪() and U *, V * ∈ 𝚪(). As and are mutually orthogonal non-degenerate distributions, we show that 𝛁U *V * ∈ 𝚪(). Thus is auto-parallel distribution.

Since the leaf M * of S(TM ) is a Riemannian manifold and S(TM ) = ⊕orth , where and are auto-parallel distributions of M *, by the decomposition theorem of de Rham [2] we have M * = M1 × M2, where M1 is a totally geodesic leaf of and M2 is a totally umbilical leaf of . Consider the frame field of eigenvectors {ξ, E1, . . . , Em} of such that {Ei}i is an orthonormal frame field of S(TM ), then B(Ei, Ej) = C(Ei, Ej) = 0 for 1≤ i < j ≤ m and B(Ei, Ei) = C(Ei, Ei) = 0 for 1 ≤ i ≤ m - 1. From (2.17) and (2.20), we have ((Ei, Ej)Ej , Ei) = g(R *(Ei, Ej)Ej, Ei) = 0. Thus the sectional curvature K of M2 is given by

Thus M is a locally product C × M1 × M2, where C is a null curve, M1 is an Euclidean space and M2 is a totally umbilical Riemannian space. □