Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 42 (2001), No. 2, 583-594.

Geodesic CR-Lightlike Submanifolds

Bayram S.ahin Rifat Günes.

Department of Mathematics, Faculty of Sciences and Arts Inonu University, Malatya, Turkey e-mail:bsahin@inonu.edu.tr

Abstract. Geodesic (totally geodesic, D-geodesic, D′-geodesic and mixed geodesic) CR-lightlike submanifolds in indefinite Kaehler manifold are investigated. Some necessary and sufficient conditions on totally geodesic, D-geodesic, D′-geodesic and mixed geodesic CR-lightlike submanifolds are obtained. We find geometric properties of CR-lightlike submanifolds of an indefinite Kaehler manifold.

1. Introduction

The general theory of a lightlike submanifold has been developed by Kupeli [10] and Bejancu-Duggal [8]. In [9], the authors constructed the principal vector bundles to a lightlike submanifold in semi-Riemann manifold and obtained Gauss-Weingarten formulae as well as other properties of this submanifold.

The study of the geometry of CR-submanifolds in a Kaehler manifold was initiated by Bejancu and has been developed by [2], [4], [5], [6] and others.

In this paper, CR-lightlike submanifolds of indefinite Kaehler manifolds which were defined in [8] are considered. In particular, we study geodesic CR-lightlike submanifolds in indefinite Kaehler manifolds. Some characterizations of totally geodesic, D-geodesic, D′- geodesic and mixed geodesic CR-lightlike submanifolds in indefinite Kaehler manifolds are given.

2. Preliminaries

Let (\overline{M,}\overline{g}) be a real (m + n)-dimensional semi-Riemann manifold, m, n > 1 and \overline{g} be a semi-Riemann metric on \overline{M}. We denote by q the constant index of \overline{g} and we suppose that \overline{M} is not Riemann manifold.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag

584 B. S.ahin, R. Günes.: Geodesic CR-Lightlike Submanifolds

Let M be a lightlike submanifold of dimension m of \overline{M}. In this case there exists a smooth distribution on M, named a radical distribution such that N_{p} = T M_{p} ∩ T M_{p}^{⊥}, ∀p ∈ M. If the rank of RadT M is r > 0, M is called an r-lightlike submanifold of \overline{M}. Then, there are four cases: I. 0 < r < min{m, n}; II. 1 < r = n < m; III. 1 < r = m < n; IV. 1 < r = m = n. In the first case the submanifold is called an r-lightlike submanifold, in the second a coisotropic submanifold, in the third an isotropic submanifold and in the fourth a totally lightlike submanifold.

Let M be an r-lightlike submanifold of \overline{M}. We consider the complementary distribution S(T M) of Rad(T M) on T M which is called a screen distribution. Then, we have the direct orthogonal sum

T M = RadT M ⊥ S(T M).  (2.1)

As S(T M) is a nondegenerate vector subbundle of T \overline{M} |_{M}, we put

T \overline{M} |_{M}= S(T M) ⊥ S(T M)^{⊥},  (2.2)

where S(T M)^{⊥} is the complementary orthogonal vector subbundle of S(T M) in T \overline{M} |_{M}. Moreover, S(T M), S(T M)^{⊥} are non-degenerate, we have the following orthogonal direct decomposition

S(T M)^{⊥} = S(T M^{⊥}) ⊥ S(T M^{⊥})^{⊥}.  (2.3)

Theorem 2.1. [9] Let (M, g, S(T M), S(T M^{⊥})) be an r-lightlike submanifold of a semi-

Riemannian manifold (\overline{M}, \overline{g}). Then, there exists a complementary vector bundle ltr(T M) called a lightlike transversal bundle of Rad(T M) in S(T M^{⊥})^{⊥} and the basis of Γ (ltr(T M) |_{U}) consists of smooth sections {N_{1}, …, N_{r}} of S(T M^{⊥})^{⊥} |_{U} such that

\overline{g} (N_{i}, ξ_{j}) = δ_{ij}, \overline{g} (N_{i}, N_{j}) = 0, i, j = 0, 1 …, r

where {ξ_{1}, …, ξ_{r}} is a basis of Γ (RadT M) |_{U}.

We consider the vector bundle

tr(T M) = ltr(T M) ⊥ S(T M^{⊥}).  (2.4)

Thus

T \overline{M} = T M ⊕ tr(T M) = S(T M) ⊥ S(T M^{⊥}) ⊥ (Rad(T M) ⊕ ltr(T M).  (2.5)

Now, let \overline{∇} be the Levi-Civita connection on \overline{M}, we have

\overline{∇}_{X}Y = ∇_{X}Y + h(X, Y) ∀X, Y ∈ Γ (T M)

and

\overline{∇}_{X}V = −A_{V} X + ∇_{X}^{⊥}V, ∀X ∈ Γ (T M) and V ∈ Γ (tr(T M)) .

Using the projectors L : tr(T M) ⟶ ltr(T M), S : tr(T M) ⟶ S(T M^{⊥}), from [9], we have

\overline{∇}_{X}Y = ∇_{X}Y + h^{l}(X, Y) + h^{s}(X, Y)  (2.6)

and

\overline{∇}_{X}N = −A_{N}X + ∇_{X}^{l}N + D^{s}(X, N)  (2.7)
\overline{∇}_{X}W = −A_{W} X + ∇_{X}^{s}W + D^{l}(X, W)  (2.8)

for any X, Y ∈ Γ (T M) , N ∈ Γ (ltr(T M)) and W ∈ Γ (S(T M^{⊥})), where h^{l}(X, Y) = Lh(X, Y), h^{s}(X, Y) = Sh(X, Y), ∇_{X}^{l}N, D^{l}(X, W) ∈ Γ (ltr(T M)), ∇_{X}^{s}W, D^{s}(X, N) ∈ Γ (S(T M^{⊥})) and ∇_{X}Y, A_{N}X, A_{W} X ∈ Γ (T M). Denote by P the projection morphism of T M to the screen distribution, we consider the decomposition

∇_{X}P Y = ∇_{X}^{∗}P Y + h^{∗}(X, P Y)  (2.9)
∇_{X}ξ = −A_{ξ}^{∗}X + ∇_{X}^{∗^{t}}ξ  (2.10)

for any X, Y ∈ Γ (T M) , ξ ∈ Γ (Rad(T M)) . Then we have the following equations

\overline{g} (h^{l}(X, P Y), ξ) = g (A_{ξ}^{∗}X, P Y ) , \overline{g} (h^{∗}(X, P Y), N) = g (A_{N}X, P Y) ,  (2.11)
g (A_{ξ}^{∗}P X, P Y ) = g (P X, A_{ξ}^{∗}P Y ) , A_{ξ}^{∗}ξ = 0.  (2.12)

Let (\overline{M}, \overline{J}, \overline{g}) be a real 2m-dimensional indefinite almost Hermitian manifold and M be a real n-dimensional submanifold of \overline{M}.

Definition 2.1. [8] A submanifold M of an indefinite almost Hermitian manifold \overline{M} is said to be a CR-lightlike submanifold if the following two conditions are fulfilled:

i) \overline{J} (Rad(T M)) is a distribution on M such that

Rad(T M) ∩ \overline{J}Rad(T M) = \{0\} .

ii) There exist vector bundles S(T M), S(T M^{⊥}), ltr(T M), D_{0} and D′ over M such that

S(T M) = \{\overline{J} (RadT M) ⊕ D′\} ⊥ D_{0}, \overline{J}D_{0} = D_{0}, \overline{J}D′ = L_{1} ⊥ L_{2},

where D_{0} is a nondegenerate distribution on M and L_{1}, L_{2} are vector bundles of ltr(T M) and S(T M^{⊥}), respectively.

From the definition of CR-lightlike submanifold, we have

T M = D ⊕ D′

where

D = RadT M ⊥ \overline{J}RadT M ⊥ D_{0}.

We denote by S and Q the projections on D and D′, respectively. Then we have

\overline{J}X = fX + ωX  (2.13)

for any X, Y ∈ Γ (T M), where fX = \overline{J}SX and ωX = \overline{J}QX. On the other hand, we set

\overline{J}V = BV + CV  (2.14)

for any V ∈ Γ (tr(T M)), where BV ∈ Γ (T M) and CV ∈ Γ (tr(T M)). Unless otherwise stated, M_{1} and M_{2} are supposed to as \overline{J}L_{1} and \overline{J}L_{2}, respectively.

3. Geodesic CR-lightlike submanifolds

Definition 3.1. A CR-lightlike submanifold of an indefinite almost Hermitian manifold is called mixed geodesic CR-lightlike submanifold if its second fundamental form h satisfies

h(X, U) = 0

for any X ∈ Γ(D) and U ∈ Γ(D′).

Definition 3.2. A CR-lightlike submanifold of an indefinite almost Hermitian manifold is called D-geodesic CR-lightlike submanifold if its second fundamental form h satisfies

h(X, Y) = 0

for any X, Y ∈ Γ(D).

Definition 3.3. A CR-lightlike submanifold of an indefinite almost Hermitian manifold is called D′-geodesic CR-lightlike submanifold if its second fundamental form h satisfies

h(U, V) = 0

for any U, V ∈ Γ(D′).

Theorem 3.1. Let \overline{M} be an indefinite Kaehler manifold and M be a CR-lightlike submanifold of \overline{M}. Then, M is totally geodesic if and only if

(L_{ξ}\overline{g}) (X, Y) = 0

and

(L_{W} \overline{g}) (X, Y) = 0

for any X, Y ∈ Γ (T M) , ξ ∈ Γ (Rad(T M)) and W ∈ Γ (S(T M^{⊥})).

Proof. We note that to show M is totally geodesic, it suffices to show that

h(X, Y) = 0

for any X, Y ∈ Γ (T M). On the other hand, by the definition of lightlike submanifolds h(X, Y) = 0 if and only if

\overline{g} (h(X, Y), ξ) = 0 and

\overline{g} (h(X, Y), W) = 0.

From (2.6) and definition of Lie derivative we have

\overline{g} (h(X, Y), ξ) = \overline{g} (\overline{∇}_{X}Y, ξ)
= X\overline{g} (Y, ξ) − \overline{g} (Y, \overline{∇}_{X}ξ)
= −\overline{g} (Y, [X, ξ]) − \overline{g} (Y, \overline{∇}_{ξ}X)
= −\overline{g} (Y, [X, ξ]) − ξ\overline{g} (Y, X) + \overline{g} (\overline{∇}_{ξ}Y, X)
= −\overline{g} (Y, [X, ξ]) − ξ\overline{g} (Y, X) + \overline{g} (X, [ξ, Y]) + \overline{g} (\overline{∇}_{Y} ξ, X)
= − (L_{ξ}\overline{g}) (X, Y) + \overline{g} (\overline{∇}_{Y} ξ, X)
= − (L_{ξ}\overline{g}) (X, Y) − \overline{g} (ξ, \overline{∇}_{Y} X)

or

2\overline{g} (h(X, Y), ξ) = − (L_{ξ}\overline{g}) (X, Y) .  (3.1)

In a similar way we obtain

\overline{g} (h(X, Y), W) = \overline{g} (\overline{∇}_{X}Y, W )
= X\overline{g} (Y, W) − \overline{g} (Y, \overline{∇}_{X}W )
= −\overline{g} (Y, [X, W]) − \overline{g} (Y, \overline{∇}_{W} X)
= −\overline{g} (Y, [X, W]) − W \overline{g} (Y, X) + \overline{g} (\overline{∇}_{W} Y, X)
= −\overline{g} (Y, [X, W]) − W \overline{g} (Y, X) + \overline{g} (X, [W, Y]) + \overline{g} (\overline{∇}_{Y} W, X)
= − (L_{W} \overline{g}) (X, Y) + \overline{g} (\overline{∇}_{Y} W, X)
= − (L_{W} \overline{g}) (X, Y) − \overline{g} (W, \overline{∇}_{Y} X)

or

2\overline{g} (h(X, Y), W) = − (L_{W} \overline{g}) (X, Y)  (3.2)

for any W ∈ Γ (S(T M^{⊥})). Thus, from (3.1) and (3.2), the proof is complete.

It is obvious that from the proof of the theorem, the assertion of the theorem is true for any lightlike submanifold of a semi-Riemann manifold.

Lemma 3.2. Let M be a CR-lightlike submanifold of an indefinite Kaehler manifold \overline{M}. Then

\overline{g} (h (X, Y) , W) = \overline{g} (A_{W} X, Y)

for any X ∈ Γ (D) , Y ∈ Γ (D′) and W ∈ Γ (S(T M^{⊥})) .

Proof. For any X ∈ Γ (D) , Y ∈ Γ (D′) and W ∈ Γ (S(T M^{⊥})) we have

\overline{g} (h (X, Y) , W) = \overline{g} (\overline{∇}_{X}Y, W )
= −\overline{g} (Y, \overline{∇}_{X}W ) .

From (2.8) it follows

\overline{g} (h (X, Y) , W) = −\overline{g} (Y, −A_{W} X + ∇_{X}^{s}W + D^{l}(X, W))
= \overline{g} (Y, A_{W} X) .

Theorem 3.3. Let M be a CR-lightlike submanifold of an indefinite Kaehler manifold \overline{M}. Then M is mixed geodesic if and only if

A_{ξ}^{∗}X ∈ Γ (D_{0} ⊥ \overline{J}L_{1})

and

A_{W} X ∈ Γ (D_{0} ⊥ RadT M ⊥ \overline{J}L_{1})

for any X ∈ Γ (D), ξ ∈ Γ (Rad(T M)) and W ∈ Γ (S(T M^{⊥})).

588 B. S.ahin, R. Günes.: Geodesic CR-Lightlike Submanifolds

Proof. By the definition of CR-lightlike submanifolds, M is mixed geodesic if and only if

\overline{g} (h(X, Y), ξ) = 0

and

\overline{g} (h(X, Y), W) = 0

for any X ∈ Γ(D) and Y ∈ Γ(D′) and W ∈ Γ(S(T M^{⊥})). Thus, from (2.6) and (2.10) we get

\overline{g} (h(X, Y), ξ) = \overline{g} (h^{l}(X, Y), ξ)
= \overline{g} (\overline{∇}_{X}Y, ξ)
= −\overline{g} (Y, \overline{∇}_{X}ξ)
= −\overline{g} (Y, ∇_{X}ξ)

or

\overline{g} (h(X, Y), ξ) = \overline{g} (Y, A_{ξ}^{∗}X) .  (3.3)

Thus assertion of theorem follows from (3.3) and Lemma 3.2. Theorem 3.4. Let M be a CR-lightlike submanifold of an indefinite Kaehler manifold \overline{M}. Then M is D-geodesic if and only if

\overline{g} (Y, A_{W} X) = \overline{g} (D^{l}(X, W), Y )

and

∇_{X}^{∗} \overline{J}ξ ∉ Γ (D_{0} ⊥ \overline{J}L_{1}) , A_{ξ}^{∗}Y ∉ Γ (\overline{J}L_{1})

for any X, Y ∈ Γ (D), ξ, ξ′ ∈ Γ (Rad(T M)) and W ∈ Γ (S(T M^{⊥})) . Proof. By the definition of lightlike submanifolds and Definition 3.2, M is D-geodesic if and only if

\overline{g} (h^{l} (X, Y) , ξ) = 0 and

\overline{g} (h^{s} (X, Y) , W) = 0

for any X, Y ∈ Γ (D), ξ, ξ′ ∈ Γ (Rad(T M)) and W ∈ Γ (S(T M^{⊥})). Thus we have

\overline{g} (h^{s} (X, Y) , W) = \overline{g} (\overline{∇}_{X}Y, W )
= −\overline{g} (Y, \overline{∇}_{X}W )
= −\overline{g} (Y, −A_{W} X + ∇_{X}^{s}W + D^{l}(X, W))
= −\overline{g} (Y, −A_{W} X) − \overline{g} (Y, D^{l}(X, W))

or

\overline{g} (h^{s} (X, Y) , W) = \overline{g} (Y, A_{W} X) − \overline{g} (Y, D^{l}(X, W)) .  (3.4)

In a similar way we get

\overline{g} (h^{l} (X, Y) , ξ) = \overline{g} (\overline{∇}_{X}Y, ξ)
= −\overline{g} (\overline{J}Y, \overline{∇}_{X} \overline{J}ξ)
= −\overline{g} (\overline{J}Y, ∇_{X} \overline{J}ξ + h (X, \overline{J}ξ))
= −\overline{g} (\overline{J}Y, ∇_{X} \overline{J}ξ) − \overline{g} (\overline{J}Y, h (X, \overline{J}ξ))

B. S.ahin, R. Günes.: Geodesic CR-Lightlike Submanifolds 589

or from (2.9) we obtain

\overline{g} (h^{l} (X, Y) , ξ) = −\overline{g} (\overline{J}Y, ∇_{X}^{∗} \overline{J}ξ) − \overline{g} (\overline{J}Y, h (X, \overline{J}ξ)) .  (3.5)

Since Y ∈ Γ (D) for the second expression in the right side of equation (3.5), we have Y ∈ Γ (Rad(T M)), Y ∈ Γ (\overline{J}Rad(T M)) or Y ∈ Γ (D_{0}) . If Y ∈ Γ (Rad(T M)), we get

\overline{g} (h^{l} (X, \overline{J}ξ) , \overline{J}Y ) = 0

and if Y ∈ Γ (D_{0}) then we obtain

\overline{g} (h^{l} (X, \overline{J}ξ) , \overline{J}Y ) = 0.

If Y ∈ Γ (\overline{J}Rad(T M)) then we put Y = \overline{J}ξ′. Hence we derive

−\overline{g} (h^{l} (X, \overline{J}ξ) , ξ′) = −\overline{g} (A_{ξ′}^{∗}X, \overline{J}ξ) .  (3.6)

Thus, from (3.5) follows

\overline{g} (h^{l} (X, Y) , ξ) = −\overline{g} (Y, ∇_{X}^{∗} \overline{J}ξ) − \overline{g} (A_{ξ′}^{∗}X, \overline{J}ξ) .

Hence if ∇_{X}^{∗} \overline{J}ξ ∉ Γ(D_{0} ⊥ M_{1}) and A_{ξ}^{∗}X ∉ Γ(M_{1}) we get h^{l}(X, Y) = 0.

Conversely, if h^{l}(X, Y) = 0 then, for any Y ∈ Γ(\overline{J}RadT M), since

g(h^{l}(X, Y), ξ) = g(A_{ξ}^{∗}X, Y) = 0,

we have A_{ξ}^{∗}X ∉ Γ(M_{1}) and for any Y ∈ Γ(D_{0} ⊥ RadT M), we obtain

\overline{g}(h^{l}(X, Y), ξ) = \overline{g}(\overline{∇}_{X}Y, ξ)
= \overline{g}(\overline{∇}_{X} \overline{J}Y, \overline{J}ξ)
= −\overline{g}(\overline{J}Y, \overline{∇}_{X} \overline{J}ξ)
= −\overline{g}(\overline{J}Y, ∇_{X} \overline{J}ξ)
= −\overline{g}(\overline{J}Y, ∇_{X}^{∗} \overline{J}ξ).

Since M is D-geodesic, we derive ∇_{X}^{∗} \overline{J}ξ ∉ Γ(D_{0} ⊥ M_{1}). Theorem 3.5. Let M be a CR-lightlike submanifold of an indefinite Kaehler manifold \overline{M}. Then M is D′-geodesic if and only if A_{W} Z and A_{ξ}^{∗}Z have no components in M_{2} ⊥ \overline{J}RadT M for any Z ∈ Γ (D′) , ξ ∈ Γ (RadT M) and W ∈ Γ (S(T M^{⊥})) . Proof. From (2.6) we have

\overline{g} (h(Z, V), W) = \overline{g} (\overline{∇}_{Z}V, W )
= −\overline{g} (V, \overline{∇}_{Z}W )

590 B. S.ahin, R. Günes.: Geodesic CR-Lightlike Submanifolds

for any Z, V ∈ Γ (D′) , or

\overline{g} (h(Z, V), W) = \overline{g} (A_{W} Z, V) .  (3.7)

On the other hand we get

\overline{g} (h(Z, V), ξ) = \overline{g} (\overline{∇}_{Z}V, ξ)
= −\overline{g} (V, \overline{∇}_{Z}ξ)

or

\overline{g} (h(Z, V), ξ) = \overline{g} (A_{ξ}^{∗}Z, V ) .  (3.8)

Then our assertion follows from (3.7) and (3.8). Corollary 3.6. Let M be a CR-lightlike submanifold of an indefinite Kaehler manifold \overline{M}. Then M is D′-geodesic if and only if

i) A_{W} Z has no component in M_{2} ⊥ \overline{J}RadT M,

ii) A_{\overline{J}V} Z has no component in M_{1}, for any Z, V ∈ Γ (D′), ξ ∈ Γ (Rad(T M)). Proof. From (2.6)

\overline{g} (h(Z, V), ξ) = \overline{g} (\overline{∇}_{Z}V, ξ)
= \overline{g} (\overline{J}\overline{∇}_{Z}V, \overline{J}ξ)
= \overline{g} (\overline{∇}_{Z} \overline{J}V, \overline{J}ξ)
= −\overline{g} (A_{\overline{J}V} Z, \overline{J}ξ)

for any Z, V ∈ Γ (D′) . Then our assertion follows from Theorem 3.4. Lemma 3.7. Let M be a CR-lightlike submanifold of an indefinite Kaehler manifold \overline{M}. If the distribution D is integrable then the following assertions hold:

i) \overline{g} (D^{l} (\overline{J}X, W ) , Y ) = \overline{g} (D^{l} (X, W) , \overline{J}Y ) ⟺ \overline{g} (A_{W} \overline{J}X, Y ) = \overline{g} (A_{W} X, \overline{J}Y ), ii) \overline{g} (D^{l} (\overline{J}X, W ) , ξ) = \overline{g} (A_{W} X, \overline{J}ξ), iii) \overline{g} (D^{l} (X, W) , ξ) = \overline{g} (A_{W} \overline{J}X, \overline{J}ξ) . Proof. From (2.8) we have

\overline{g} (D^{l} (\overline{J}X, W ) , Y ) = \overline{g} (\overline{∇}_{\overline{J}X}W − ∇_{\overline{J}X}^{s}W + A_{W} \overline{J}X, Y )
= \overline{g} (\overline{∇}_{\overline{J}X}W + A_{W} \overline{J}X, Y )
= −\overline{g} (W, \overline{∇}_{\overline{J}X}Y ) + \overline{g} (A_{W} \overline{J}X, Y )
= −\overline{g} (W, ∇_{\overline{J}X}Y + h (\overline{J}X, Y )) + \overline{g} (A_{W} \overline{J}X, Y )
= −\overline{g} (W, h (\overline{J}X, Y )) + \overline{g} (A_{W} \overline{J}X, Y ) .

B. S.ahin, R. Günes.: Geodesic CR-Lightlike Submanifolds 591

Then, taking in account that D is integrable, we obtain

\overline{g} (D^{l} (\overline{J}X, W ) , Y ) = −\overline{g} (W, h (\overline{J}X, Y )) + \overline{g} (A_{W} \overline{J}X, Y )
= −\overline{g} (W, h (X, \overline{J}Y )) + \overline{g} (A_{W} \overline{J}X, Y )
= −\overline{g} (W, \overline{∇}_{X} \overline{J}Y ) + \overline{g} (A_{W} \overline{J}X, Y )
= \overline{g} (\overline{∇}_{X}W, \overline{J}Y ) + \overline{g} (A_{W} \overline{J}X, Y )
= −\overline{g} (A_{W} X, \overline{J}Y ) + \overline{g} (D^{l} (X, W) , \overline{J}Y ) + \overline{g} (A_{W} \overline{J}X, Y )

or

\overline{g} (D^{l} (\overline{J}X, W ) , Y ) − \overline{g} (D^{l} (X, W) , \overline{J}Y ) = \overline{g} (A_{W} \overline{J}X, Y ) − \overline{g} (A_{W} X, \overline{J}Y ) .

This is proof of (i). Substituting Y = ξ, Y = \overline{J}ξ in (i) we obtain (ii) and (iii). Lemma 3.8. Let M be a mixed geodesic CR-lightlike submanifold of an indefinite Kaehler manifold \overline{M}. Then we have

A_{ξ}^{∗}X ∈ Γ (\overline{J}Rad(T M))

for any X ∈ Γ (D′) . Proof. Using the Kaehler character of \overline{M} and (2.6),

h (\overline{J}ξ, X) = \overline{∇}_{X} \overline{J}ξ − ∇_{X} \overline{J}ξ
= \overline{J} \overline{∇}_{X}ξ − ∇_{X} \overline{J}ξ
= \overline{J}∇_{X}ξ + \overline{J}h(X, ξ) − ∇_{X} \overline{J}ξ

for any X ∈ Γ(D′), Y ∈ Γ(D′). Since M is mixed geodesic, we have

\overline{J}∇_{X}ξ = ∇_{X} \overline{J}ξ.

From (2.9) and (2.10) we derive

−\overline{J}A_{ξ}^{∗}X + \overline{J}∇_{X}^{∗ ^{t}}ξ = ∇_{X}^{∗} \overline{J}ξ + h^{∗}(X, \overline{J}ξ)

or from (2.13)

−fA_{ξ}^{∗}X − ωA_{ξ}^{∗}X + \overline{J}∇_{X}^{∗ ^{t}}ξ = ∇_{X}^{∗} \overline{J}ξ + h^{∗}(X, \overline{J}ξ).

Thus

ωA_{ξ}^{∗}X = 0

or

A_{ξ}^{∗}X ∈ Γ (\overline{J}Rad(T M) ⊥ D_{0}) .

If A_{ξ}^{∗}X ∈ Γ (D_{0}) then since D_{0} is nondegenerate, it must be \overline{g} (A_{ξ}^{∗}X, Z_{0}) ≠ 0.

But from (2.6) and (2.10) we get

\overline{g} (A_{ξ}^{∗}X, Z_{0}) = \overline{g} (−∇_{X}ξ + ∇_{X}^{∗ ^{t}}ξ, Z_{0})
= \overline{g} (−∇_{X}ξ, Z_{0})
= \overline{g} (−\overline{∇}_{X}ξ, Z_{0})
= \overline{g} (ξ, \overline{∇}_{X}Z_{0})
= \overline{g} (ξ, ∇_{X}Z_{0} + h(X, Z_{0}))
= 0.

Thus A_{ξ}^{∗}X ∉ Γ (D_{0}).

From (2.10) and Lemma 3.2, we have the following lemma.

Lemma 3.9. Let M be a mixed geodesic CR-lightlike submanifold of an indefinite Kaehler manifold \overline{M}. Then

\overline{g} (h^{l} (X, Y) , ξ) = 0

for any X ∈ Γ (D′) , Y ∈ Γ (M_{2}) and ξ ∈ Γ (Rad(T M)) .

By the definition of CR-lightlike submanifolds and from (2.11), (2.12) we have the following corollaries.

Corollary 3.10. Let \overline{M} be an indefinite almost complex manifold and M be a mixed CRlightlike submanifold of \overline{M}. Then

A_{ξ}^{∗}X ∈ Γ (\overline{J}Rad(T M) ⊥ M_{2})

for any X ∈ Γ (D′).

Corollary 3.11. Let \overline{M} be an indefinite almost complex manifold and M be a mixed CRlightlike submanifold of \overline{M}. Then

A_{ξ}^{∗}X ∈ Γ (D_{0} ⊥ M_{1})

for any X ∈ Γ (D).

Corollary 3.12. Let \overline{M} be an indefinite almost complex manifold and M be a CR-lightlike submanifold of \overline{M}. If h^{∗}(X, Y) = 0 then we have:

a) A_{N}X has no component in \overline{J}Rad(T M) ⊥ M_{2}

b) A_{N}Y has no component D_{0} ⊥ M_{1} for any X ∈ Γ (D), Y ∈ Γ (D′) .

Corollary 3.13. Let \overline{M} be an indefinite almost complex manifold and M be a mixed CRlightlike submanifold of \overline{M}. Then:

a) A_{W} X has no component in \overline{J}RadT M ⊥ M_{2}

b) A_{W} Y has no component in D_{0} ⊥ M_{1} for any X ∈ Γ(D), Y ∈ Γ (D′) .

Lemma 3.14. Let \overline{M} be an indefinite Kaehler manifold and M be a CR-lightlike submanifold of \overline{M}. Then

\overline{g} (A_{W} \overline{J}X, Y ) = \overline{g} (A_{W} Y, \overline{J}X) − \overline{g} (\overline{J}X, D^{l}(Y, W))

for any X ∈ Γ (D) , Y ∈ Γ (D′) and W ∈ Γ (S(T M^{⊥})).

Proof. From (2.8),

\overline{g} (A_{W} \overline{J}X, Y ) = \overline{g} (h (\overline{J}X, Y ) , W )
= \overline{g} (\overline{∇}_{Y} \overline{J}X, W )
= −\overline{g} (\overline{J}X, \overline{∇}_{Y} W )
= \overline{g} (\overline{J}X, A_{W} Y ) − \overline{g} (D^{l}(Y, W), \overline{J}X)

for X ∈ Γ (D) , Y ∈ Γ (D′) and W ∈ Γ (S(T M^{⊥})) .

Theorem 3.15. Let \overline{M} be an indefinite Kaehler manifold and M be a mixed geodesic CRlightlike submanifold of \overline{M}. Then

A_{V} X ∈ Γ(D)

for any X ∈ Γ(D), V ∈ Γ(L_{1} ⊥ L_{2}).

Proof. Since M is mixed geodesic, h(X, Y) = 0 for any X ∈ Γ(D), Y ∈ Γ(D′). From (2.6) we have

0 = \overline{∇}_{X}Y − ∇_{X}Y

Since D′ is anti-invariant there exists V ∈ Γ(L_{1} ⊥ L_{2}) such that \overline{J}V = Y. Thus, from (2.8), (2.13) and (2.14) we get

0 = \overline{∇}_{X} \overline{J}V − ∇_{X}Y
= \overline{J} \overline{∇}_{X}V − ∇_{X}Y
= \overline{J}(−A_{V} X + ∇_{X}^{t}V) − ∇_{X}Y
= −\overline{J}A_{V} X + \overline{J}∇_{X}^{t}V − ∇_{X}Y
= −fA_{V} X − ωA_{V} X + B∇_{X}^{t}V + C∇_{X}^{t}V − ∇_{X}Y.

Hence

ωA_{V} X = C∇_{X}^{t}V

or

A_{V} X ∈ Γ(D).

Corollary 3.16. Let \overline{M} be an indefinite Kaehler manifold and M be a mixed geodesic CRlightlike submanifold of \overline{M}. Then M is a mixed geodesic CR-lightlike submanifold if and only if

A_{V} X ∈ Γ(D)

and L_{1} ⊥ L_{2} is parallel with respect to D (that is, ∇^{t}_{X}V ∈ Γ(L_{1} ⊥ L_{2}) for any V ∈ Γ(L_{1} ⊥ L_{2}) and X ∈ Γ(D)).

References

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[2] Bejancu, A.: CR-Submanifolds of a Kaehler Manifold II. Trans. Amer. Math. Soc. 250 (1979), 333–345.

[3] Bejancu, A.; Kon, M.; Yano, K.: CR-Submanifolds of Complex Space Form. J. Differential Geom. 16 (1981), 137–145.

[4] Yano, K.; Kon, M.: On CR-submanifolds of a Complex Projective Space. J. Differential Geom. 16 (1981), 431–444.

[5] Chen, B. Y.: CR-Submanifolds of a Kaehler Manifold I. J. Differential Geom. 16 (1981), 305–323.

[6] Chen, B. Y.: CR-Submanifolds of a Kaehler Manifold II. J. Differential Geom. 16 (1981), 493–509.

[7] Duggal, K. L.; Bejancu, A.: Lightlike Submanifolds of Codimension Two. Math. J. Toyama Univ. (1992), 59–82.

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[10] Kupeli, D. N.: Singular Semi-Riemannian Geometry. Kluwer, Dordrecht 1996.

Received October 22, 2000