PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 89(103) (2011), 69–75 DOI: 10.2298/PIM1103069U

SEMI-PARALLEL LIGHTLIKE HYPERSURFACES OF INDEFINITE COSYMPLECTIC SPACE FORMS

Abhitosh Upadhyay and Ram Shankar Gupta

Communicated by Darko Milinković

Abstract. We study the semiparallel lightlike hypersurface of an indefinite cosymplectic space forms which are tangent to the structure vector field.

1. Introduction

In the theory of submanifolds of semi-Riemannian manifolds it is interesting to study the geometry of lightlike submanifolds due to the fact that the intersection of normal vector bundle and the tangent bundle is nontrivial making it more interesting and remarkably different from the study of nondegenerate submanifolds. The geometry of lightlike hypersurfaces and submanifolds of indefinite Kaehler manifolds was studied by Duggal and Bejancu [5]. On the other hand, lightlike hypersurfaces of indefinite Sasakian manifolds was studied in [3, 6], whereas lightlike hypersurfaces in indefinite cosymplectic space form was studied in [7].

The basic Gauss, Codazzi–Mainardi and Ricci equations give that the extrinsic conditions parallel, semiparallel and pseudo-parallel imply the correspondent intrinsic conditions symmetry, semisymmetry and pseudo-symmetry, respectively [1].

We study semiparallel lightlike hypersurface of an indefinite cosymplectic space form and we prove:

Theorem 1.1. Let M be a semiparallel lightlike hypersurface of an indefinite cosymplectic space form M̄(c) of constant curvature c, with ξ ∈ T M. Then either c = 0, or M is (ϕ̄(T M^{⊥}), D ⊕ D′) mixed totally geodesic. Moreover, if c = 0, then either M is totally geodesic or C(E, A_{E}^{∗}P X) = 0, for any X ∈ Γ(T M).

2010 Mathematics Subject Classification: Primary 53C15, 53C40, 53C50, 53D15.

Key words and phrases: degenerate metric, cosymplectic manifold.

Acknowledgement: This research is partly supported by the University Grants Commission (UGC), India, under a Major Research Project No. SR. 36-321/2008. The second author would like to thank the UGC for providing the financial support to pursue this research work.

2. Preliminaries

An odd-dimensional semi-Riemannian manifold M̄ is said to be an indefinite almost contact metric manifold if there exist structure tensors {ϕ̄, ξ, η, ḡ}, where ϕ̄ is a (1,1) tensor field, ξ a vector field, η a 1-form and ḡ is the semi-Riemannian metric on M̄ satisfying

ϕ̄^{2} X̄ = −X̄ + η(X̄)ξ, η ∘ ϕ̄ = 0, ϕ̄ξ = 0, η(ξ) = 1 (2.1)

ḡ(ϕ̄X̄, ϕ̄Ȳ) = ḡ(X̄, Ȳ) − εη(X̄)η(Ȳ), η(X̄) = εḡ(X̄, ξ), ḡ(ξ, ξ) = ε, ε = ±1

for any X̄, Ȳ ∈ Γ(T M̄), where Γ(T M̄) denotes the Lie algebra of vector fields on M̄.

An indefinite almost contact metric manifold M̄ is called an indefinite cosymplectic manifold if [4] ( ¯∇X̄ ϕ̄)Ȳ = 0, and ¯∇X̄ξ = 0 for any X̄, Ȳ ∈ T M̄, where ¯∇ denotes the Levi–Civita connection on M̄.

A plane section Π in T_{x}M̄ of a cosymplectic manifold M̄ is called a ϕ̄-section if it is spanned by a unit vector X̄ orthogonal to ξ and ϕ̄X̄, where X̄ is a non-null vector field on M̄. The sectional curvature K(Π) with respect to Π determined by X̄ is called a ϕ̄-sectional curvature. If M̄ has a ϕ̄-sectional curvature c which does not depend on the ϕ̄-section at each point, then c is a constant in M̄ and M̄ is called an indefinite cosymplectic space form, which is denoted by M̄(c). The curvature tensor R̄ of M̄(c) is given by [4]

R̄(X̄, Ȳ)Z̄ = \frac{c}{4}\{ḡ(Ȳ , Z̄)X̄ − ḡ(X̄, Z̄)Ȳ + η(X̄)η(Z̄)Ȳ − η(Ȳ)η(Z̄)X̄ (2.2)

+ḡ(X̄, Z̄)η(Ȳ)ξ − ḡ(Ȳ , Z̄)η(X̄)ξ + ḡ(ϕ̄Ȳ , Z̄)ϕ̄X̄
−ḡ(ϕ̄X̄, Z̄)ϕ̄Ȳ − 2ḡ(ϕ̄X̄, Ȳ)ϕ̄Z̄ }

for any X̄, Ȳ , Z̄ ∈ Γ(T M̄).

Let (M, g) be a hypersurface of a (2n + 1)-dimensional semi-Riemannian manifold (M̄, ḡ) with index s, 0 < s < 2n + 1 and g = ḡ_{|M}. Then M is a lightlike hypersurface of M̄ if g is of constant rank (2n − 1) and the normal bundle T M^{⊥} is a distribution of rank 1 on M [5]. A nondegenerate complementary distribution S(T M) of rank (2n − 1) to T M^{⊥} in T M, that is, T M = T M^{⊥}⊥S(T M), is called screen distribution. The following result (cf. [5, Theorem 1.1, p. 79]) has an important role in studying the geometry of lightlike hypersurfaces.

Theorem 2.1. Let (M, g, S(T M)) be a lightlike hypersurface of M̄. Then, there exists a unique vector bundle tr(T M) of rank 1 over M such that for any nonzero section E of T M^{⊥} on a coordinate neighbourhood U ⊂ M, there exists a unique section N of tr(T M) on U satisfying ḡ(N, E) = 1 and ḡ(N, N) = ḡ(N, W) = 0, for each W ∈ Γ(S(T M)|_{U}).

Then, we have the following decomposition:

(2.3)  T M = S(T M)⊥T M^{⊥},  T M̄ = S(T M)⊥(T M^{⊥} ⊕ tr(T M)).

Throughout this paper, all manifolds are supposed to be paracompact and smooth. We denote by Γ(E) the smooth sections of the vector bundle E, and by ⊥ and ⊕ the orthogonal and the nonorthogonal direct sum of two vector bundles, respectively.

Let ¯∇, ∇ and ∇^{t} denote the linear connections on M̄, M and vector bundle tr(T M), respectively. Then, the Gauss and Weingarten formulae are given by

(2.4)  ¯∇_{X}Y = ∇_{X}Y + h(X, Y), for all X, Y ∈ Γ(T M),
(2.5)  ¯∇_{X}V = −A_{V} X + ∇_{X}^{t}V, for all V ∈ Γ(tr(T M)),

where {∇_{X}Y, A_{V} X} and {h(X, Y), ∇_{X}^{t}V } belong to Γ(T M) and Γ(tr(T M)), respectively and A_{V} is the shape operator of M with respect to V. Moreover, in view of decomposition (2.3), equations (2.4) and (2.5) take the form

(2.6)  ¯∇_{X}Y = ∇_{X}Y + B(X, Y)N
(2.7)  ¯∇_{X}N = −A_{N}X + τ(X)N

for any X, Y ∈ Γ(T M) and N ∈ Γ(tr(T M)), where B(X, Y) and τ(X) are local second fundamental form and a 1-form on U, respectively. It follows that

B(X, Y) = ḡ( ¯∇_{X}Y, E) = ḡ(h(X, Y), E), B(X, E) = 0, τ(X) = ḡ(∇_{X}^{t}N, E).

Let P denote the projection of T M on S(T M) and ∇^{∗}, ∇^{∗t} denote the induced linear connections on S(T M) and T M^{⊥}, respectively. Then from the decomposition of tangent bundle of lightlike hypersurface, we have

(2.8)  ∇_{X}P Y = ∇_{X}^{∗}P Y + h^{∗}(X, P Y)
(2.9)  ∇_{X}E = −A_{E}^{∗}X + ∇_{X}^{∗t}E

for any X, Y ∈ Γ(T M) and E ∈ Γ(T M^{⊥}), where h^{∗}, A^{∗} are the second fundamental form and the shape operator of distribution S(T M) respectively.

By direct calculations using Gauss–Weingarten formulae (2.8) and (2.9), we find

(2.10)  g(A_{N}Y, P W) = ḡ(N, h^{∗}(Y, P W)),  ḡ(A_{N}Y, N) = 0;
(2.11)  g(A_{E}^{∗}X, P Y) = ḡ(E, h(X, P Y)),  ḡ(A_{E}^{∗}X, N) = 0;

for any X, Y, W ∈ Γ(T M), E ∈ Γ(T M^{⊥}) and N ∈ Γ(tr(T M)).

Locally, we define on U

(2.12)  C(X, P Y) = ḡ(h^{∗}(X, P Y), N), and λ(X) = ḡ(∇_{X}^{∗t}E, N).

Hence, h^{∗}(X, P Y) = C(X, P Y)E, and ∇_{X}^{∗t}E = λ(X)E. On the other hand, by using (2.6), (2.7), (2.9) and (2.12), we obtain

λ(X) = ḡ(∇_{X}E, N) = ḡ( ¯∇_{X}E, N) = −ḡ(E, ¯∇_{X}N) = −τ(X).

Thus, locally (2.8) and (2.9) become

∇_{X}P Y = ∇_{X}^{∗}P Y + C(X, P Y)E, and ∇_{X}E = −A_{E}^{∗}X − τ(X)E.

Finally, (2.10) and (2.11), locally become

g(A_{N}Y, P W) = C(Y, P W), ḡ(A_{N}Y, N) = 0;
g(A_{E}^{∗}X, P Y) = B(X, P Y), ḡ(A_{E}^{∗}X, N) = 0.

In general, the induced connection ∇ on M is not a metric connection. Since ¯∇ is a metric connection, we have

0 = ( ¯∇_{X} ḡ)(Y, Z) = X(ḡ(Y, Z)) − ḡ( ¯∇_{X}Y, Z) − ḡ(Y, ¯∇_{X}Z).

If R̄ and R are the curvature tensors of ¯∇ and ∇, then using (2.6) in the equation R̄(X, Y)Z = ¯∇_{X} ¯∇_{Y} Z − ¯∇_{Y} ¯∇_{X}Z − ¯∇_{[X,Y]}Z, we obtain

(2.13) R̄(X, Y)Z = R(X, Y)Z + B(X, Z)A_{N}Y − B(Y, Z)A_{N}X

+ {(∇_{X}B)(Y, Z) − (∇_{Y} B)(X, Z) + τ(X)B(Y, Z) − τ(Y)B(X, Z)}N.

A hypersurface M is semiparallel if its second fundamental form h satisfies,

(2.14) (R(X, Y) · h)(X_{1}, X_{2}) = −h(R(X, Y)X_{1}, X_{2}) − h(X_{1}, R(X, Y)X_{2}) = 0

for any X, Y, X_{1}, X_{2} ∈ Γ(T M), where R is the curvature tensor field of M.

3. Proof of the theorem

Let (M̄, ϕ̄, ξ, η, ḡ) be an indefinite cosymplectic manifold and (M, g) be its lightlike hypersurface, tangent to the structure vector field ξ with ḡ(ξ, ξ) = ε = +1. If E is a local section of T M^{⊥}, then ḡ(ϕ̄E, E) = 0 implies that ϕ̄E is tangent to M. Thus ϕ̄(T M^{⊥}) is a distribution on M of rank 1 such that ϕ̄(T M^{⊥}) ∩ T M^{⊥} = {0}. This enables us to choose a screen distribution S(T M) such that it contains ϕ̄(T M^{⊥}) as vector subbundle.

Now, we consider a local section N of tr(T M). Then ϕ̄N is tangent to M and belongs to S(T M) as ḡ(ϕ̄N, E) = −ḡ(N, ϕ̄E) = 0 and ḡ(ϕ̄N, N) = 0. From (2.1), we have

ḡ(ϕ̄N, ϕ̄E) = ḡ(N, E) − η(N)η(E) = ḡ(N, E) = 1.

Therefore, ϕ̄(T M^{⊥}) ⊕ ϕ̄(tr(T M)) is a direct sum but not orthogonal and is a nondegenerate vector subbundle of S(T M) of rank 2.

It is known [2] that if M is tangent to structure vector field ξ, then ξ belongs to S(T M). Since ḡ(ϕ̄E, ξ) = ḡ(ϕ̄N, ξ) = 0, there exists a non degenerate invariant distribution D_{0} of rank (2n − 4) on M such that

(3.1)  S(T M) = {ϕ̄(T M^{⊥}) ⊕ ϕ̄(tr(T M))}⊥D_{0}⊥⟨ξ⟩ and ϕ̄(D_{0}) = D_{0},

where ⟨ξ⟩ = span ξ. Moreover, from (2.3) and (3.1), we obtain

T M = {ϕ̄(T M^{⊥}) ⊕ ϕ̄(tr(T M))}⊥D_{0}⊥⟨ξ⟩⊥T M^{⊥}.

Now, we consider the distributions D and D′ on M as follows

D = T M^{⊥}⊥ϕ̄(T M^{⊥})⊥D_{0}, D′ = ϕ̄(tr(T M)).

Then D is invariant under ϕ̄ and T M = D ⊕ D′⊥⟨ξ⟩.

If P_{1} and Q denote the projection morphisms of T M on D and D′ and U = −ϕ̄N, V = −ϕ̄E are local lightlike vectors, respectively, then we write

(3.2)  X = P_{1}X + QX + η(X)ξ

for X ∈ Γ(T M), where QX = u(X)U, and u is a differential 1-form locally defined on M by u(·) = g(V, ·). From (3.1) and (3.2), we obtain ϕ̄X = ϕX + u(X)N and ϕ^{2}X = −X + η(X)ξ + u(X)U, for each X ∈ Γ(T M), where ϕ is a tensor field of type (1, 1) defined on M by ϕX = ϕ̄P_{1}X.

Putting (2.2), (2.6), (2.13) into (2.14), by a straightforward calculation we obtain

0 = \frac{c}{4}[ g(Y, X_{1})B(X, X_{2}) − g(X, X_{1})B(Y, X_{2}) + η(X)η(X_{1})B(Y, X_{2})
− η(Y)η(X_{1})B(X, X_{2}) + ḡ(ϕ̄Y, X_{1})B(ϕX, X_{2})
− ḡ(ϕ̄X, X_{1})B(ϕY, X_{2}) − 2ḡ(ϕ̄X, Y)B(ϕX_{1}, X_{2})]

− B(X, X_{1})B(A_{N}Y, X_{2}) + B(Y, X_{1})B(A_{N}X, X_{2})

+ \frac{c}{4}[ g(Y, X_{2})B(X, X_{1}) − g(X, X_{2})B(Y, X_{1}) + η(X)η(X_{2})B(Y, X_{1})

− η(Y)η(X_{2})B(X, X_{1}) + ḡ(ϕ̄Y, X_{2})B(ϕX, X_{1})
− ḡ(ϕ̄X, X_{2})B(ϕY, X_{1}) − 2ḡ(ϕ̄X, Y)B(ϕX_{2}, X_{1})]

− B(X, X_{2})B(A_{N}Y, X_{1}) + B(Y, X_{2})B(A_{N}X, X_{1}).

Putting above X = E and using the fact that B(E, ·) = 0, we get

(3.3) 0 = \frac{c}{4}[ḡ(ϕ̄Y, X_{1})B(ϕE, X_{2}) + u(X_{1})B(ϕY, X_{2}) + 2u(Y)B(ϕX_{1}, X_{2})]

+ B(Y, X_{1})B(A_{N}E, X_{2}) + \frac{c}{4}[ḡ(ϕ̄Y, X_{2})B(ϕE, X_{1}) + u(X_{2})B(ϕY, X_{1})

+ 2u(Y)B(ϕX_{2}, X_{1})] + B(Y, X_{2})B(A_{N}E, X_{1}).

Putting X_{2} = E into (3.3) we get \frac{3}{4}cu(Y)B(V, X_{1}) = 0. If we put here Y = U, we find

(3.4)  \frac{3}{4}cB(V, X_{1}) = 0.

From (3.4), we get c = 0 as B(V, X_{1}) ≠ 0, for each X_{1} ∈ Γ(D ⊕ D′). If c ≠ 0, then (3.4) implies that B(V, X_{1}) = 0, for each X_{1} ∈ Γ(D ⊕ D′). Hence M is (ϕ̄(T M^{⊥}), D ⊕ D′)-mixed totally geodesic.

On the other hand, suppose that c = 0; then from (3.3), by putting X_{1} = X_{2}, we obtain

(3.5)  B(Y, X_{1})B(A_{N}E, X_{1}) = 0.

If B(Y, X_{1}) = 0 for each Y, X_{1} ∈ Γ(T M), then M is totally geodesic. If B(Y, X_{1}) ≠ 0, then (3.5) imply that B(A_{N}E, X_{1}) = 0, that is C(E, A_{E}^{∗}P X_{1}) = 0, for any X_{1} ∈ Γ(T M). This finishes the proof of our Theorem. □

From Theorem 1.1 and the fact that C(E, A_{E}^{∗}P X) = Ric(E, X), X ∈ Γ(T M), where Ric denotes the Ricci tensor of M, we have the following characterization (cf. [5, Theorem 2.2, p. 88]):

Corollary. Let (M, g, S(T M)) be a semiparallel lightlike hypersurface of an indefinite cosymplectic space form M̄(c) of constant curvature c = 0, with ξ ∈ T M, such that Ric(E, X) ≠ 0, for any X ∈ Γ(T M) and E ∈ Γ(T M^{⊥}). Then following assertions are equivalent:

(i) A_{W}^{∗} X = 0, for any W ∈ Γ(T M^{⊥}) and X ∈ Γ(T M).

(ii) There exists a unique torsion free metric connection ∇ induced by ¯∇ on M.

(iii) T M^{⊥} is a parallel distribution with respect to ∇.

(iv) T M^{⊥} is a killing distribution on M.

Hereafter, (R_{q}^{2m+1}, ϕ̄, ξ, η, ḡ) will denote the manifold R_{q}^{2m+1} with its usual cosymplectic structure given by

η = dz, ξ = ∂z,

ḡ = η ⊗ η − ∑_{i=1}^{q/2} dx^{i} ⊗ dx^{i} + dy^{i} ⊗ dy^{i} + ∑_{i=q}^{m} dx^{i} ⊗ dx^{i} + dy^{i} ⊗ dy^{i},

ϕ̄(∑_{i=1}^{m}(X_{i}∂x^{i} + Y_{i}∂y^{i}) + Z∂z) = ∑_{i=1}^{m}(Y_{i}∂x^{i} − X_{i}∂y^{i}),

where (x^{i}, y^{i}, z) are Cartesian coordinates.

Example. Let M̄ = (R_{2}^{7}, ḡ) be a semi-Euclidean space, where ḡ is of signature (−, +, +, −, +, +, +) with respect to the canonical basis

{∂x_{1}, ∂x_{2}, ∂x_{3}, ∂y_{1}, ∂y_{2}, ∂y_{3}, ∂z}.

Consider the hypersurface M of R_{2}^{7}, defined by

X(u, v, θ_{1}, θ_{2}, s, t) = (u, u, v, θ_{1}, θ_{2}, s, t).

Then a local frame of T M is given by

Z_{1} = ∂x_{1} + ∂x_{2}, Z_{2} = ∂x_{3}, Z_{3} = ∂y_{1},
Z_{4} = ∂y_{2},  Z_{5} = ∂y_{3}, Z_{6} = ξ = ∂z.

Hence, T M^{⊥} = span{Z_{1}} and ϕ̄(T M^{⊥}) = span{−Z_{3} − Z_{4}} which implies that ϕ̄(T M^{⊥}) ∈ Γ(S(T M)). Thus D = T M^{⊥}⊥ϕ̄(T M^{⊥})⊥D_{0} is invariant under ϕ̄, where D_{0} = span\{Z_{2}, Z_{5}\}. Now, tr(T M) is spanned by N = \frac{1}{2}(−∂x_{1} + ∂x_{2}) and D′ = ϕ̄(tr(T M)) = span\{\frac{1}{2}(Z_{3} − Z_{4})\}. Hence T M = D ⊕ D′⊥⟨ξ⟩.

Using (2.4) and (2.5) we obtain

(3.6)  h(Z_{i}, Z_{j}) = 0, and ¯∇_{Z_{i}}N = 0, for i, j = 1, …, 6.

From (3.6) and (2.14), it is easy to see that M is semiparallel hypersurface of M̄. Moreover, using (3.6), M is totally geodesic hypersurface and c = 0 as M̄ = R_{2}^{7} is a semi-Euclidean space, which supports Theorem 1.1.

References

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University School of Basic and Applied Sciences (Received 21 11 2009) Guru Gobind Singh Indraprastha University (Revised 02 12 2010) Sector-16C, Dwarka, Delhi-110075 India

abhi.basti.ipu@gmail.com
ramshankar.gupta@gmail.com