TWO CHARACTERIZATION THEOREMS FOR HALF LIGHTLIKE SUBMANIFOLDS OF AN INDEFINITE KENMOTSU MANIFOLD

Abstract

In this paper, we study the curvature of locally symmetric or semi-symmetric half lightlike submanifolds M of an indefinite Kenmotsu manifold $\bar{M}$, whose structure vector field is tangent to M. After that, we study the existence of the totally geodesic screen distribution of half lightlike submanifolds of indefinite Kenmotsu manifolds with parallel co-screen distribution subject to the conditions: (1) M is locally symmetric, or (2) the lightlike transversal connection is flat.

1. INTRODUCTION

The theory of lightlike submanifolds is an important topic of research in differential geometry due to its application in mathematical physics, especially in the electromagnetic field theory. The study of such notion was initiated by Duggal and Bejancu [2] and later studied by many authors (see up-to date results in two books [4, 5]). The class of lightlike submanifolds of codimension 2 is compose of two classes by virtue of the rank of its radical distribution, which are called the half lightlike and coisotropic submanifolds [3]. Half lightlike submanifold is a special case of r-lightlike submanifold such that r = 1 and its geometry is more general form than that of coisotrophic submanifold. Much of the works on half lightlike submanifolds will be immediately generalized in a formal way to general r-lightlike submanifolds of arbitrary codimension n and arbitrary rank r.

In the theory of Sasakian manifolds, the following result is well-known [9]: If a Sasakian manifold is locally symmetric, then it is of constant positive curvature 1. In 1971, K. Kenmotsu proved the following result [8]: If a Kenmotsu manifold is locally symmetric, then it is of constant negative curvature –1.

In this paper, we study the curvature of locally symmetric or semi-symmetric half lightlike submanifolds of an indefinite Kenmotsu manifold , whose structure vector fild is tangent to M. After that, we study the existence of the totally geodesic screen distribution of half lightlike submanifolds of indefinite Kenmotsu manifolds with parallel co-screen distribution subject such that either M is locally symmetric or the lightlike transversal connection is flat. We prove the following results:

Theorem 1.1. Let M be a half lightlike submanifold of an indefinite Kenmotsu manifold , whose structure vector field is tangent to M. If M is locally symmetric or semi-symmetric, then M is a space of constant negative curvature –1. In this case, the induced connection on M is a torsion-free metric connection and the lightlike transversal connection is flat.

Theorem 1.2. Let M be a half lightlike submanifold of an indefinite Kenmotsu manifold with parallel co-screen distribution. If either M is locally symmetric or the lightlike transversal connection is flat, then the screen distribution S(TM) of M is never totally geodesic in M.

 

2. HALF LIGHTLIKE SUBMANIFOLDS

An odd dimensional semi-Riemannian manifold is said to be an indefinite Kenmotsu manifold [7, 8, 10] if there exist a structure set , where J is a

(1, 1)-type tensor field, 𝜁 is a vector field and θ is a 1-form such that

for any vector fields X, Y on , where is the Levi-Civita connection of .

A submanifold (M, g) of a semi-Riemannian manifold of codimension 2 is called a half lightlike submanifold if the radical distribution Rad(TM) = TM ∩ TM⊥ of M is a vector subbundle of the tangent bundle TM and the normal bundle TM⊥ of rank 1. Then there exist 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, 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 and by 𝚪(E) the F(M) module of smooth sections of a vector bundle E over M. Choose L ∈ 𝚪(S(TM⊥)) as a unit vector field with . In this paper we may assume that , without loss of generality. Consider the orthogonal complementary distribution S(TM)⊥ to S(TM) in . For any null section 𝜉 of Rad(TM), certainly 𝜉 and L belong to 𝚪(S(TM)⊥). Thus we have

S(TM)⊥ = S(TM⊥) ⊕orth S(TM⊥)⊥,

where S(TM⊥)⊥ is the orthogonal complementary to S(TM⊥) in S(TM)⊥. For any null section 𝜉 of Rad(TM) on a coordinate neighborhood U ⊂ M, there exists a uniquely defined null vector field N ∈ 𝚪(ltr(TM)) satisfying

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. Therefore T is decomposed as

Let P be the projection morphism of TM on S(TM). Then the local Gauss and Weingarten 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 and τ, ρ and ϕ are 1-forms on TM.

Since is torsion-free, ∇ is also torsion-free and both B and D are symmetric. From the facts , we know that B and D are independent of the choice of S(TM) and satisfy

The induced connection ∇ of M is not metric and satisfies

for all X, Y, Z ∈ 𝚪(TM), where 𝜂 is a 1-form on TM such that

But the connection ∇* on S(TM) is metric. The above three local second fundamental forms are related to their shape operators by

In case C = 0 on any coordinate neighborhood U, we say that S(TM) is totally geodesic in M. From (2.10), we show that S(TM) is totally geodesic in M if and only if S(TM) is a parallel distribution on M, i.e.,

∇XY ∈ 𝚪(S(TM)), ∀X ∈ 𝚪(TM) and Y ∈ 𝚪(S(TM)).

In the sequel, we let X, Y, Z, U, … be the vector fields of M, unless otherwise specified. Denote by and R the curvature tensors of and ∇ respectively. Using (2.7)~(2.11), we have the Gauss-Codazzi equations for M and S(TM):

A half lightlike submanifold M = (M, g, ∇) equipped with a degenerate metric g and a linear connection ∇ is said to be of constant curvature c if there exists a constant c such that the curvature tensor R of ∇ satisfies

For any X ∈ 𝚪(TM), let where Q is the projection morphism of on 𝚪(ltr(TM)) with respect to (2.6). Then ∇𝓁 is a linear connection on the lightlike transversal vector bundle ltr(TM) of M. We say that ∇𝓁 is the lightlike transversal connection of M. We define the curvature tensor R𝓁 on ltr(TM) by

If R𝓁 vanishes identically, then the transversal connection is said to be flat.

From (2.8) and the definition of ∇𝓁, we get for all X ∈ 𝚪(TM). Substituting this equation into the right side of (2.24), we get

R𝓁(X, Y)N = 2dτ(X, Y)N.

From this result we deduce the following theorem:

Theorem 2.1 ([6]). Let M be a half lightlike submanifold of a semi-Riemannian manifold . Then the lightlike transversal connection of M is flat, if and only if the 1-form τ is closed, i.e., dτ = 0, on any 𝒰 ⊂ M.

Note 1. We know that dτ is independent of the choice of the section 𝜉 on Rad(TM), where τ is given by . In fact, if we take and , it follows that . If we take the exterior derivative d on the last equation, then we have .

 

3. PROOF OF THEOREM 1.1

Assume that 𝛇 is tangent to M. It is well known [1] that if 𝛇 is tangent to M, then it belongs to S(TM). Replacing Y by 𝛇 to (2.7) and using (2.2), we have

Substituting (3.1)1 into R(X, Y)𝛇 = ∇X∇Y𝛇−∇Y∇X𝛇−∇[X, Y]𝛇 and using (2.19), (3.1) and the fact that ∇ is torsion-free, we have

Taking the scalar product with 𝛇 to this and using the fact and (2.1), we show that θ is closed, i.e., dθ = 0 on TM. Thus we obtain

Applying to θ(Y) = g(Y, 𝛇) and using (2.2), (2.5) and , we have

Case 1. Assume that M is locally symmetric, i.e., ∇R = 0. Applying ∇Z to (3.2) and using the first equation of (3.1)[denote by (3.1)1], (3.2) and (3.3), we have

Thus M is a space of constant curvature −1. Applying ∇U to (3.4), we have

(∇Ug)(X,Z)Y = (∇Ug)(Y,Z)X.

Taking Z = Y = 𝜉 to this and using (2.12)1 and (2.13), we get B = 0. Thus ∇ is a torsion-free metric connection on M by (2.13). As B = 0, we have by (2.15). From (2.22), we get R(X, Y)𝜉 = −2dτ(X,Y )𝜉. On the other hand, replacing Z by 𝜉 to (3.4), we have R(X, Y)𝜉 = 0. These two results imply dτ = 0. Thus the lightlike transversal connection ∇𝓁 is flat.

Case 2. Assume that M is semi-symmetric, i.e., R(X, Y)R = 0. Applying ∇Z to (3.2) and using (3.1)1, (3.2) and (3.3), we have

Substituting (3.5) into (R(U,Z)R)(X, Y)𝜉 = 0 and using (3.1)1, we have

Replacing U by 𝛇 to (3.6) and using (∇𝛇R)(X, Y)𝛇 = 0 due to (3.2) and (3.5), we have (∇ZR)(X, Y)𝛇 = 0. From this and (3.5), we show that

Thus M is a space of constant negative curvature −1. Replacing U by 𝜉 to (3.6) and using (2.12)1, (3.7) and (∇ZR)(X, Y)𝛇 = 0, we have

B(Y,Z)X = B(X,Z)Y.

Replacing Y by 𝜉 to this and using (2.12)1, we get B = 0. Thus, by (2.13), ∇ is a torsion-free metric connection on M. Using (2.22), (3.7) and the method of Case 1, we see that the lightlike transversal connection is flat. □

 

4. PROOF OF THEOREM 1.2

From the decomposition (2.6) of T, the vector field 𝛇 is decomposed as

where W is a smooth vector field on M and m = θ(𝜉) and n = θ(L) are smooth functions. Substituting (4.1) in (2.2) and using (2.8) and (2.9), we have

Substituting (4.3) and (4.4) into the following two equations

[X, Y]m = X(Y m) − Y (Xm), [X, Y ]n = X(Y n) − Y (Xn),

and using (2.19), (2.20), (2.21), (4.1), (4.3), (4.4), we have respectively

Substituting (4.2) into R(X, Y)W = ∇X∇YW − ∇Y∇XW − ∇[X, Y]W and using (2.19)~(2.21), (4.2)~(4.5) and the fact ∇ is torsion-free, we have

Taking the scalar product with 𝛇 to (4.6) and using (2.1), we show that the structure 1-form θ is closed, i.e., dθ = 0 on TM.

Assume that S(TM) is totally geodesic in M. In this case, 𝛇 is not tangent to M and l = θ(N) ≠ 0. In fact, if 𝛇 is tangent to M or l = 0, then . Applying to and using (2.2) and (2.8), we have η(X) = 0 for all X ∈ 𝚪(TM). It is a contradiction as η(𝜉) = 1. Thus 𝛇 is not tangent to M and l ≠ 0. As 𝛇 is not tangent to M, we see that (m, n) ≠ (0, 0). As S(TM⊥) is a parallel distribution, we have AL = ϕ = 0 due to (2.9). From (2.17) and (2.18), we also have D = ρ = 0.

Substituting (2.19)~(2.21) into (4.6) and using (4.5), we get

Applying to θ(Y) = g(Y, 𝛇) and using (2.2) and (2.6), we have

Case 1. Assume M is locally symmetric. Applying ∇Z to (4.7), we have

R(X, Y)∇ZW = (∇Zθ)(X)Y − (∇Zθ)(Y)X.

Substituting (4.2) and (4.8) in this equation and using (4.7), we obtain

Replacing Z by 𝜉 to (4.9) and using (2.12)1, we have R(X, Y)𝜉 = 0. Comparing the Rad(TM)-components of this and (2.22), we have dτ = 0. Thus by Theorem 2.1 the lightlike transversal connection is flat. From (2.19), (2.20) and (4.9), we have

Replacing Y by 𝜉 to (4.10) and using (2.12)1, we get

From (4.9) and (4.11), we show that R = 0. From this and (4.7), we have

θ(X)Y = θ(Y )X.

Replacing Y by 𝜉 to this equation and using X = PX + η(X)𝜉, we have

mPX = g(X,W)𝜉.

As the left term of this equation belongs to S(TM) and the right term belongs to Rad(TM), we have mPX = 0 and g(X,W)𝜉 = 0 for all X ∈ 𝚪(TM). Thus m = 0 and g(X,W) = 0 for all X ∈ 𝚪(TM). This imply W = l𝜉 and

From this and the fact , we show that n2 = 1.

It is known [6] that, for any half lightlike submanifold of an indefinite almost contact metric manifold , J(Rad(TM)), J(ltr(TM)) and J(S(TM⊥)) are vector subbundles of S(TM) of rank 1 respectively. Applying to and using (2.1), (2.3), (2.8) and (2.9), we have

Replacing X by J𝜉 to (4.13) and using (2.1)6, we have n = 0. It is a contradiction as n2 = 1. Thus S(TM) is not totally geodesic in M.

Case 2. Assume that the transversal connection is flat. We have dτ = 0. Substituting (4.1) into (4.6) with dθ = 0 and using (2.19)~(2.21) and (4.5), we have

Taking the scalar product with W to this and using the facts θ(X) − m𝜂(X) = g(X,W) and , we have

θ(Y)𝜂(X) − θ(X)𝜂(Y) = 0.

Replacing Y by 𝜉 to this equation, we have g(X,W) = 0 for all X ∈ 𝚪(TM). This implies W = l𝜉. Thus 𝛇 is decomposed as

From the fact and (4.14), we show that 2lm = 1 − n2. Applying to (4.14) and using (2.2), (2.8), (2.9) and (2.11), we have

Taking the scalar product with 𝜉, N and L to this result by turns, we get

respectively. From (2.15) and (4.15), we have

Applying to and using (2.1), (2.3), (2.8), (2.9) and the fact S(TM) is non-degenerate, we have

Taking the scalar product with J𝜉 to this and using (2.1)6, we have n(1 − ml) = −lmn. This implies n = 0. As (m, n) ≠ (0, 0) and n = 0, we have m ≠ 0 and 2lm = 1. Consequently we get JL = 0 by (4.17). It is a contradiction as

Thus S(TM) is not totally geodesic in M.

Corollary 1. Let M be a half lightlike submanifold of an indefinite Kenmotsu manifold . Then the structure 1-form θ, given by (2.1), is closed on TM.