DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 7–17

2––CODIMENSIONAL LIGHTLIKE SUBMANIFOLDS

OF ALMOST PARA––HERMITIAN MANIFOLDS

Cornelia–Livia BEJAN

0. Introduction. Let (M̄, ḡ) be a semi–Riemannian (pseudo–Riemannian) manifold, [10] and let M be a submanifold of it. If the restriction g = ḡ/M of ḡ to M is still non–degenerate, then (M, g) becomes a semi–Riemannian manifold and it can be studied as the submanifolds of Riemannian manifolds. A different situation appears when g is degenerate and (M, g) is said to be a lightlike (degenerate) submanifold, [5], [6].

Among all semi–Riemannian manifolds, the class of the almost para–Hermitian manifolds is of special interest and it was considered by several authors, as in the list of references of [4].

Bejancu and Etayo initiated in [7] the study of degenerate hypersurfaces of almost para–Hermitian manifolds. Our purpose here is to step from the hypersurfaces to the submanifolds of arbitrary codimension. The present note deals with degenerate submanifolds of codimension two of almost para–Hermitian manifolds, since for higher codimension the results are similar but the calculus is much longer. We classify them, we prove their main properties and we also give some examples.

1. Preliminaries. Let (M̄, ḡ) be a semi–Riemannian manifold endowed with an almost product structure, i.e., a tensor field P̄ of type (1, 1) such that

(1.1)  P̄^{2} = Id; P̄ ≠ ±Id.

The pair (P̄, ḡ) is said to be an almost para–Hermitian (or hyperbolic Hermitian) structure on M̄ if

(1.2)  ḡ(P̄X, P̄Y) = −ḡ(X, Y), ∀X, Y ∈ Γ(T M̄).

These manifolds appeared for the first time in [11].

1991 Mathematics Subject Classification. 53C40; 53C50; 53C15.

Key words and phrases. Degenerate submanifolds; para-Hermitian manifolds.

This paper is in final form and no version of it will be submitted for publication elsewhere.

Dedicated to Professor A. Bejancu on his 50-th birthday.

Remark 1.1 (i) M̄ is of even dimension 2m, m = 1, 2, ... (ii) The eigendistributions P̄^{+} and P̄^{−} of P̄ corresponding to the eigenvalues 1 and (−1) resp., are of rank m:

(1.3)  T M̄ = P̄^{+} ⊕ P̄^{−}.

(iii) P̄^{+} and P̄^{−} are isotropic, i.e. the restriction of ḡ to either P̄^{+} or P̄^{−} is zero.

The notion of the almost para–Hermitian structure given on a differentiable manifold can be extended to an arbitrary vector bundle:

Definition 1. [2] Let (E, π, M̄) be a vector bundle on the base manifold M̄ endowed with a semi–Riemannian structure ḡ. If P̄ : Γ(E)⟶Γ(E) satisfies (1.1) it is said to be a product structure of E and if moreover

(1.4)  ḡ(P̄s, P̄s′) = −ḡ(s, s′), ∀s, s′ ∈ Γ(E),

then (P̄, ḡ) is called a para–Hermitian structure of E.

Notations. [9] Let M be a submanifold of codimension two of an almost para– Hermitian manifold (M̄, P̄, ḡ).

For any q ∈ M, the T_{q}M–perpendicular space [12] is defined by

(1.5)  T_{q}M^{⊥} = {y ∈ T_{q}M̄/ḡ(y, x) = 0, ∀x ∈ T_{q}M}

and the radical of T_{q}M, cf. [1], is denoted by

(1.6)  Δ_{q} = T_{q}M ∩ T_{q}M^{⊥}. We suppose Δ_{q} ≠ {0}, ∀q ∈ M.

The submanifold M is called lightlike (degenerate) if the mapping Δ : q ∈ M⟶Δ_{q} ⊆ T_{q}M is a non–trivial distribution. This notion is motivated by the restriction g = ḡ/M of ḡ to M which is degenerate, i.e. there is locally a non–zero vector field ξ ∈ Γ(T M) :

(1.7)  g(ξ, X) = 0, ∀X ∈ Γ(T M).

Remark 1.2. [9] T_{q}M^{⊥} is a degenerate 2–dimensional subspace of T_{q}M̄ and (T_{q}M^{⊥})^{⊥} = T_{q}M, ∀q ∈ M.

Any orthogonal complement of Δ in T M is called a screen distribution SM and it is proved to be non–degenerate,

(1.8)  T M = SM ⊥ Δ (orthogonal direct sum).

Moreover, there is a degenerate vector bundle NM associated to SM called the canonical affine normal bundle such that

(1.9)  T M̄ = SM ⊥ (T M^{⊥} ⊕ NM)

where the first sum is orthogonal and the second one is only a direct Whitney sum.

The radical Δ being a non–trivial distribution with Δ_{q} ⊆ T_{q}M^{⊥}, q ∈ M, then its dimension can be either one or two. The degenerate submanifold M is said to be half lightlike in the first case and totally lightlike in the second one. Remark that the ambient manifold M̄ does not need to be almost para–Hermitian, but only semi–Riemannian, as in

Proposition 1.1 Let M be a degenerate submanifold of codimension two of a semi–Riemannian manifold (M̄, ḡ). Then the following conditions are equivalent:

(i) M is totally lightlike; (ii) Δ = T M^{⊥}; (iii) T M^{⊥} ⊆ T M; (iv) ḡ restricted to T M^{⊥} is zero.

The proof follows from (1.5), (1.6) and Remark 1.2.

When the ambient manifold M̄ is almost para–Hermitian, then we give an other classification of the degenerate submanifolds, according to the action of P̄ to T M^{⊥}.

2. Classification of degenerate submanifolds. Everywhere in this section we take (M̄, P̄, ḡ) to be a 2m–dimensional almost para–Hermitian manifold and let M be a degenerate submanifold of codimension two. Since all submanifolds of a 4–dimensional almost para–Hermitian manifold were classified by [10] then we consider m > 2.

Case 1. P̄ is invariant on T M^{⊥}, i.e. P̄(T_{q}M^{⊥}) ⊆ T_{q}M^{⊥}, q ∈ M. From (1.1) we get

(2.1)  P̄(T_{q}M^{⊥}) = T_{q}M^{⊥}, q ∈ M.

Theorem 2.1. In the Case 1 we have: (i) T M^{⊥} is locally spanned by two lightlike eigenvector fields of P̄, tangent to M and orthogonal to each other; (ii) M is a totally lightlike submanifold; (iii) M is invariant with respect to P̄, being an almost product manifold.

Proof. First we prove that T M^{⊥} is locally spanned by two lightlike vector fields of P̄. We consider the vector bundle T M^{⊥} on the base manifold M. From (2.1) the restriction of P̄ to T M^{⊥} is either ±id (and therefore we take any frame of T M^{⊥}) or a product structure of T M^{⊥}. In this last situation, as T M^{⊥} is of rank two it follows that locally there are two eigenvector fields ξ_{1}, ξ_{2} of P̄ such that P̄ξ_{i} = (−1)^{i+1}ξ_{i}, i = 1, 2.

From (1.4) we get ḡ(ξ_{i}, ξ_{i}) = 0, that is ξ_{i}, i = 1, 2, are lightlike in both situations. As the radical Δ is a non–trivial distribution and Δ ⊆ T M^{⊥}, then there is locally a non–zero vector field U = αξ_{1} + βξ_{2} ∈ Γ(Δ), with ξ_{1}, ξ_{2} given above and α, β ∈ F(M). Then ḡ(U, Z) = 0, ∀Z ∈ Γ(T M^{⊥}). By taking consequently Z = ξ_{1} and Z= ξ_{2}, we get ḡ(ξ_{1}, ξ_{2}) = 0. Since ξ_{1}, ξ_{2} are lightlike from above, then ḡ restricted to T M^{⊥} is zero. Now (ii) follows from Proposition 1.1. As from (ii), we get that ξ_{1}, ξ_{2} are tangent to M, then (i) is proved. (iii) From (1.2) we obtain

ḡ(P̄X, ξ_{i}) = −ḡ(X, P̄ξ_{i}) = ±ḡ(X, ξ_{i}) = 0, i = 1, 2, for any X ∈ Γ(T M),

where ξ_{1}, ξ_{2} are given above. From Remark 1.2, P̄X ∈ Γ(T M), ∀X ∈ Γ(T M), that is T M is invariant by P̄. We denote by

P = P̄/Γ(T M) : Γ(T M)⟶Γ(T M) the restriction of P̄ to M.

From Remark 1.1 (ii), T M can be contained neither in P̄^{+} nor in P̄^{−} since M is of 2(m − 1)–dimension, m > 2. Then P ≠ ±Id and from (1.1) P is an almost product structure on M.

By Definition 1, we get

Theorem 2.2 In the Case 1, there is in TM a screen distribution SM which inherits a para–Hermitian structure.

Proof. From the proof of Theorem 2.1, P is an almost product structure on M whose eigendistributions P^{+} and P^{−} split T M :

(2.2)  T M = P^{+} ⊕ P^{−}.

From the proof of Theorem 2.1, the restriction P̄/T M^{⊥} of P̄ to T M^{⊥} falls in one of the following three situations:

(a) P̄/T M^{⊥} = Id. Then Δ ⊂ P^{+} and let D_{0} be any supplementary distribution of Δ in P^{+}:

(2.3)  P^{+} = D_{0} ⊥ Δ,

orthogonal direct sum, since Δ is orthogonal to any vector field tangent to M. We take SM = D_{0} ⊕ P^{−}.

(b) P̄/T M^{⊥} = −Id. (analogous).

(c) P̄/T M^{⊥} is a product structure, (1.1).

P^{+} +Δ (resp. P^{−} +Δ) is not a direct sum since from Theorem 2.1 (i), ξ_{1} ∈ Γ(P^{+} ∩ Δ) and ξ_{2} ∈ Γ(P^{−} ∩ Δ). Let D_{1} (resp. D_{2}) be any supplementary distribution of Δ in P^{+} + Δ (resp. P^{−} + Δ)

(2.4)  P^{+} + Δ = D_{1} ⊥ Δ and P^{−} + Δ = D_{2} ⊥ Δ,

ortogonal direct sum since Δ is ortogonal to any vector field tangent to M. We take SM = D_{1} ⊕ D_{2}.

Therefore in all three situations (a), (b), (c), SM satisfies (1.8) and it is invariant by P. From Remark 1.1 (iii), SM being non–degenerate with respect to ḡ, it is contained neither in P^{+} nor in P^{−}, that is P̄/SM ≠ ±Id, i.e. P̄/SM is a product structure. Therefore, from Definition 1, SM inherits a para–Hermitian structure and we complete the proof.

Let SM^{⊥} be the orthogonal supplement of SM in T M̄ which is also non– degenerate with respect to ḡ and it contains T M^{⊥}:

(2.5)  T M̄ = SM ⊥ SM^{⊥}.

From (1.4), (2.5) and Definition 1, we get

Corollary 2.3. In the Case 1, let SM be the screen distribution given by Theorem 2.2. Then SM^{⊥} inherits a para–Hermitian structure.

Remark 2.1. (i) rank SM = dim M̄ − 4; (ii) index SM = index M̄ − 2; (iii) rank SM^{⊥} = 4.

The following result will be used in §4 and it is an improvement of [9, Theorem 1].

Proposition 2.4. In the Case 1, let SM be the screen distribution constructed in Theorem 2.2. Then the local frame {ξ_{1}, ξ_{2}} of T M^{⊥} given by Theorem 2.1 (i), can be extended to a local frame {ξ_{1}, ξ_{2}, N_{1}, N_{2}} of SM^{⊥} such that

(1) N_{1}, N_{2} are two orthogonal lightlike vector fields satisfying

(2.6)  ḡ(ξ_{i}, N_{j}) = (−1)^{i+1}δ_{ij}, i, j = 1, 2 (no sum);

(2) N_{i} is invariant (resp. anti–invariant) by P̄ when ξ_{i} is anti–invariant (resp. invariant) by P̄, i = 1, 2.

Proof. From the proof of Theorem 2.1 (i) the restriction P̄/T M^{⊥} is either ±Id (when the existence of N_{1}, N_{2} follows from [2, Proposition 1.4]) or a product structure (1.1). In this last situation we proceed as follows. As SM^{⊥} is a para– Hermitian vector bundle of rank 4, let δ^{+} and δ^{−} be the eigenvector bundles of P̄/SM^{⊥} corresponding to the eigenvalues 1 and (−1), resp.

(2.7)  SM^{⊥} = δ^{+} ⊕ δ^{−}.

From the proof of Theorem 2.1 we get ξ_{1} ∈ Γ(δ^{+}), ξ_{2} ∈ Γ(δ^{−}). Since δ^{−} is of rank two, let V_{1} ∈ Γ(δ^{−}) be a vector field lin.independent with ξ_{2}. Then P̄V_{1} = −V_{1} and from (1.4) we get ḡ(V_{1}, V_{1}) = ḡ(ξ_{2}, V_{1}) = 0. Since SM^{⊥} is non–degenerate with respect to ḡ and ξ_{1}, ξ_{2}, V_{1} are lin.independent, then ḡ(ξ_{1}, V_{1}) ≠ 0. If we denote

(2.8)  N_{1} = V_{1}/ḡ(ξ_{1}, V_{1}),

we get

(2.9)  P̄N_{1} = −N_{1}; ḡ(N_{1}, N_{1}) = ḡ(ξ_{2}, N_{1}) = 0 and ḡ(ξ_{1}, N_{1}) = 1.

Since δ^{+} is of rank two there is V_{2} ∈ Γ(δ^{+}) lin.independent with ξ_{1}. From (1.4) we have

(2.10)  P̄V_{2} = V_{2} and ḡ(V_{2}, V_{2}) = ḡ(ξ_{1}, V_{2}) = 0.

If we denote W_{2} = −ḡ(V_{2}, N_{1})ξ_{1} + V_{2}, then from (1.4) and (2.10) we get

(2.11)  P̄W_{2} = W_{2} and ḡ(W_{2}, W_{2}) = ḡ(W_{2}, N_{1}) = ḡ(ξ_{1}, W_{2}) = 0.

Since SM^{⊥} is non–degenerate, with respect to ḡ and {ξ_{1}, ξ_{2}, N_{1}, W_{2}} is a local frame of SM^{⊥} then from (2.11) we obtain ḡ(ξ_{2}, W_{2})≠0. We take

(2.12)  N_{2} = W_{2}/ḡ(ξ_{2}, W_{2})

and we complete the proof.

Remark 2.2. There is a normal bundle NM satisfying (1.9) spanned locally by {N_{1}, N_{2}}.

In the Case 1, we assumed the invariance of P̄ on T M^{⊥} that is the intersection P̄(T M^{⊥}) ∩ T M^{⊥} is a vector bundle of rank two on M. Now, we consider

Case 2. E = ⋃_{q∈M} P̄(T_{q}M^{⊥}) ∩ T_{q}M^{⊥} is a vector bundle of rank one on M.

Proposition 2.5. In the Case 2, the vector bundle E is a distribution on M contained in Δ and E = Δ if and only if M is a half lightlike submanifold of M̄.

Proof. As E is of rank one then either E ⊆ Δ or E ∩ Δ = {0}. In the last case T M^{⊥} = E ⊕ Δ, since T M^{⊥} is of rank two. By (1.2), we have

(2.13)  ḡ(σ, σ) = 0, ∀σ ∈ Γ(E),

since E is of rank one and invariant by P̄. From (2.13) and the orthogonality of Δ to any section of T M^{⊥} we get that ḡ restricted to T M^{⊥} is zero. From Proposition 1.1 it follows Δ = T M^{⊥} and therefore E ⊂ Δ.

Case 3. P̄(T_{q}M^{⊥}) ∩ T_{q}M^{⊥} = {0}, q ∈ M.

Proposition 2.6. Let M be a half lightlike submanifold of the almost para–Hermitian manifold (M̄, P̄, ḡ), falling in the Case 3. Then: (i) there is a screen distribution SM containing the tangent part of P̄(Δ), i.e. the projection of P̄(Δ) on TM; (ii) corresponding to SM, there is a unique lightlike normal bundle of TM in T M̄ satisfying (1.9) which is orthogonal to P̄(Δ).

Proof. (i) As Δ is one dimensional it follows from [9] that P̄(Δ) is contained in T M + T M^{⊥} (no direct sum), but not in T M^{⊥}. Then we construct a screen distribution SM which contains the tangent part of P̄(Δ). (ii) follows from (i) and [9, Section 2].

In the Case 1, by virtue of Theorem 2.1, the submanifold M is invariant with respect to P̄, which is not true in all the other cases, as we show next section by examples. However, in the Case 2, E is invariant by P̄ and therefore we give

Definition 2. Any degenerate submanifold of codimension two of an almost para– Hermitian manifold falling in the Case 1 (resp. Case 2, Case 3) is said to be invariant (resp. semi–invariant, non–invariant).

From the theory of degenerate hypersurfaces developed in [7], we get

Proposition 2.7. Let (M̄, P̄, ḡ) be an almost para–Hermitian manifold and let H be an invariant (resp. non–invariant) hypersurface of it. If M is a degenerate submanifold of codimension two of (M̄, P̄, ḡ) embedded in H, then M is either invariant or semi–invariant (resp. non–invariant).

Corollary 2.8. Let (N, G) be a Riemannian manifold and T N its tangent bundle endowed with the almost para–Hermitian structure introduced in [8]. The unit tangent bundle T′N is considered as an invariant hypersurface of TN, [10]. Then no 2–codimensional non–invariant submanifold of TN lies in T′N.

3. Examples. Let R_{m}^{2m} be the semi–Euclidean space of dimension 2m and of index m, endowed with the semi–Euclidean metric

(3.1)  ḡ = ∑_{i=1}^{m}[(dx^{m+1})^{2} + (dx^{i})^{2}] of signature (m, m).

Let P̄ be a product structure defined on R_{m}^{2m} by

(3.2)  P̄ (\frac{∂}{∂x^{i}} ) = \frac{∂}{∂x^{m+i}} and P̄ (\frac{∂}{∂x^{m+i}} ) = \frac{∂}{∂x^{i}} , i = \overline{1, m}.

Then (R_{m}^{2m}, P̄, ḡ) is just a para–Hermitian space, as P̄ is integrable. The next result given for codimension two only to fix the ideas, shows how [7, Proposition 1] can be generalized for any codimension.

Proposition 3.1. Let M be a submanifoldof codimension two of (R_{m}^{2m}, P̄, ḡ) given by x^{2m−1} = F (x^{1}, ..., x^{2m−2}); x^{2m} = G(x^{1}, ..., x^{2m−2}), where F,G are differentiable functions. Then:

(i) M is a degenerate submanifold if and only if

(3.3)  AC = B^{2}, with
(3.4)  A = 1 − ((F, F)); B = ((F, G)); C = 1 − ((G, G)), where

(3.5)  ((h, k)) = ∑_{i=1}^{m} \frac{∂h}{∂x^{i}} \frac{∂k}{∂x^{i}} − ∑_{a=1}^{2m−2} \frac{∂h}{∂x^{a}} \frac{∂k}{∂x^{a}} for any functions h, k;

(ii) under the degeneration assumption, M is totally lightlike (resp. half lightlike) if and only if A, B, C are (resp. are not) all zero.

Proof. (i) In the natural parametrization

(3.6)  \begin{matrix}x^{1} = u^{1}; ...; x^{2m−2} = u^{2m−2}; \\ x^{2m−1} = F (u^{1}, ..., u^{2m−2}); x^{2m} = G(u^{1}, ..., u^{2m−2}) \end{matrix}

the natural local frame on M is

(3.7)  \frac{∂}{∂u^{α}} = \frac{∂}{∂x^{α}} + \frac{∂F}{∂u^{α}} \frac{∂}{∂x^{2m−1}} + \frac{∂G}{∂u^{α}} \frac{∂}{∂x^{2m}} , α = \overline{1, 2m − 2}.

From (1.7) and (3.1), M is degenerate if and only if there is a non–zero vector

field ξ = ∑_{α=1}^{2m−2} ξ^{α} \frac{∂}{∂u^{α}} whose components satisfy the homogeneous system:

(3.8)  \{ \begin{matrix}ξ^{i} − \frac{∂F}{∂u^{i}} ξ(F) − \frac{∂G}{∂u^{i}} ξ(G) = 0, i = \overline{1, m} \\ ξ^{a} + \frac{∂F}{∂u^{a}} ξ(F) + \frac{∂G}{∂u^{a}} ξ(G) = 0, a = \overline{m + 1, 2m − 2} \end{matrix}

ξ is a non–zero solution of (3.8) if and only if ξ(F) and ξ(G) are not simultaneously

zero. Now, by multiplying the α equation of (3.8) with \frac{∂G}{∂u^{α}} (resp. \frac{∂F}{∂u^{α}}), α =

\overline{1, 2m − 2}, and then by adding altogether we get

(3.9)  Cξ(G) = Bξ(F) (resp. Aξ(F) = Cξ(G)).

We denote by (S) the homogeneous linear system of both equations (3.9). Therefore ξ(F) and ξ(G) are not simultaneously zero if and only if the determinant of (S) is zero, i.e. (3.3) is satisfied.

(ii) Suppose M is degenerate. Then M is totally lightlike (resp. half lightlike) if and only if the solutions space of the system (3.8) is two (resp. one) dimensional, that is (S) is of rank zero (resp. one), i.e., A, B, C are (resp. are not) all zero.

Remark. Let M be a half lightlike submanifold of (R_{m}^{2m}, P̄, ḡ). Since A, B, C are not all zero, we may suppose C ≠ 0, otherwise is similar. From (3.7), (3.8), (3.3),

(3.9) the vector field ξ = ∑_{α=1}^{2m} \frac{∂}{∂u^{α}} from the proof of Propostion 3.1 is

(3.10)  \begin{matrix}ξ = ∑_{i=1}^{m} [C \frac{∂F}{∂x^{i}} + B \frac{∂G}{∂x^{i}} ] \frac{∂}{∂x^{i}} − ∑_{a=m+1}^{2m−2} [C \frac{∂F}{∂x^{a}} + B \frac{∂G}{∂x^{a}} ] \frac{∂}{∂x^{a}} \\ +C \frac{∂}{∂x^{2m−1}} + B \frac{∂}{∂x^{2m}} . \end{matrix}

The next three examples follows from Proposition 3.1 and (3.2).

Example 1. (Case 1) The submanifold M of (R_{m}^{2m}, P̄, ḡ) given by x^{2m−1} = x^{m−1}; x^{2m} = x^{m} is a degenerate submanifold falling in the invariant case whose Δ = T M^{⊥} is spanned by

(3.11)  ξ_{1} = \frac{∂}{∂x^{m−1}} + \frac{∂}{∂x^{2m−1}} ; ξ_{2} = \frac{∂}{∂x^{m}} + \frac{∂}{∂x^{2m}} .

Example 2. (Case 2) The submanifold M of (R_{m}^{2m}, P̄, ḡ) given by (3.12)

x^{2m−1} = 2x^{m} + ∑_{k=1}^{m−2}(x^{m+k} − x^{k})^{2}; x^{2m} = 2x^{m−1} − 3x^{m} + ∑_{k+1}^{m−2}(x^{m+k} − x^{k})^{2}.

Since from (3.4) A = −3, B = −6; C = −12, then M is half lightlike falling in the semi–invariant case whose radical distribution Δ is spanned by

(3.13)  \begin{matrix}ξ = 36 ∑_{k=1}^{m−2}(x^{m+k} − x^{k}) (\frac{∂}{∂x^{k}} + \frac{∂}{∂x^{m+k}} ) \\ −12 (\frac{∂}{∂x^{m−1}} + \frac{∂}{∂x^{2m−1}} ) − 6 (\frac{∂}{∂x^{m}} + \frac{∂}{∂x^{2m}} ) \end{matrix}

which verifies (3.10).

Example 3. (Case 3, totally lightlike) The submanifold M of (R_{m}^{2m}, P̄, ḡ) given by x^{2m−1} = x^{1}; x^{2m} = x^{2} is totally lightlike, falling in the non–invariant case whose radical distribution Δ = T M^{⊥} is spanned by

(3.14)  σ = \frac{∂}{∂x^{1}} + \frac{∂}{∂x^{2m−1}} ; ζ = \frac{∂}{∂x^{2}} + \frac{∂}{∂x^{2m}} .

Besides all examples given above which are applications of Proposition 3.1, the next example is different, being a submanifold of an other para–Hermitian manifold.

Example 4. (Case 3, half lightlike) Let T^{3} = S^{1} × S^{1} × S^{1} be the 3–dimensional torus with the Riemannian metric G which is the product of the standard metrics on the factors S^{1} and let \{X_{i}\}_{i=\overline{1,3}} be an orthonormal basis giving a parallelization of T^{3}. M̄ = T^{3} × T^{3} is endowed with the almost para–Hermitian structure

(3.15)  ḡ = ( \begin{matrix}−I & 0 \\ 0 & I \end{matrix} ); P̄ = ( \begin{matrix}0 & I \\ I & 0 \end{matrix} );

with respect to the frame \{(X_{i}, 0), i = \overline{1, 3}; (0, X_{b}), b = \overline{1, 3}\} of M̄.

Then γ : x ∈ S^{1}⟶((0, 0, x), (x, 0, 0)) ∈ T^{3} × T^{3} = M̄ is a lightlike curve on M̄. By identifying γ with its image in M̄, we take

(3.16) M = T^{2} ×γ ×S^{1}; i.e. M = T^{2} ×γ ×(S^{1} ×{0}) ⊂ T^{2} ×(S^{1} ×S^{1})×T^{2} = M̄.

Then {(X_{1}, 0), (X_{2}, 0), (X_{3}, X_{1}), (0, X_{2})} is a basis on T M and Δ is spanned by {(X_{3}, X_{1})}. Therefore M is a half lightlike submanifold of M̄ which provides an example of a degenerate non–invariant submanifold.

4. Invariant submanifolds of para–Kähler manifolds. Let (M̄, P̄, ḡ) be a 2m–dimensional para–Kähler manifold (m > 2), i.e. an almost para–Hermitian manifold such that P̄ is parallel with respect to the Levi–Civita connection \overline{∇} of ḡ:

(4.1)  \overline{∇}P̄ = 0

Let M be a degenerate submanifold of codimension two falling in the invariant case. Now, we take the screen distribution SM given by Theorem 2.2 and the local frame {ξ_{1}, ξ_{2}, N_{1}, N_{2}} of Proposition 2.4. The Gauss–Weingarten formulae for M are

(4.2)  \overline{∇}_{X}Y = \overline{∇}_{X}Y + B_{1}(X, Y)N_{1} + B_{2}(X, Y)N_{2} and

(4.3)  \overline{∇}_{X}N = −A_{N}X + ∇_{X}^{⊥}N, ∀X, Y ∈ Γ(T M), N ∈ Γ(NM),

where \overline{∇}_{X}Y and A_{N}X are tangent to M and ∇^{⊥} denotes the linear normal connection of Γ(NM). Similarly, the Gauss–Weingarten formulae for SM are

(4.4)  ∇_{X}hY =∇^{∗}_{X} hY + C_{1}(X, hY)ξ_{1} + C_{2}(X, hY)ξ_{2} and

(4.5)  ∇_{ξ}Y = −A_{ξ}X+ ∇^{∗}_{X}^{⊥} ξ, ∀X, Y ∈ Γ(T M), ξ ∈ Γ(T M^{⊥}),

where h denotes the projection of T M on SM; ∇^{∗}^{⊥} denotes the metric linear

connection of T M^{⊥} and ∇^{∗}_{X} hY, A_{ξ}X ∈ Γ(SM), [9].

Now, we apply (4.1) to (4.2) and (4.4) and we take the components from Γ(T M^{⊥}) and Γ(NM), respectively, to obtain:

(4.6)  C_{i}(X, P hY) = ±C_{i}(X, hY); B_{i}(X, P hY) = ±B_{i}(hX, hY);
(4.7)  B_{i}(P hX, P hY) = B_{i}(hX, hY), i = 1, 2, ∀X, Y ∈ Γ(T M).

Proposition 4.1. The screen distribution SM is integrable if and only if

(4.8)  C_{i}(P hX, P hY) = C_{i}(hX, hY), i = 1, 2, ∀X, Y ∈ Γ(T M).

Proof. SM is integrable if and only if the forms C_{i}, i = 1, 2, are symmetric, [9], which is equivalent to (4.8).

M is said to be totally umbilical if

(4.9)  B_{i}(X, Y) = f_{i}g(X, Y), i = 1, 2, ∀X, Y ∈ Γ(T M),

where g = ḡ/M and f_{1}, f_{2} are differentiable functions.

M is said to be totally geodesic when f_{1} = f_{2} = 0.

Since SM is a non–degenerate distribution, from Remark 2.1, there is locally an orthonormal basis {E_{j}} of SM where E_{1}, ..., E_{m−2} are timelike and E_{m−1}, ..., E_{2m−2} are spacelike. The mean vector field

(4.10)  H = (1/2(m − 2)) ∑_{i=1}^{2} ∑_{k=1}^{m−2}{B_{i}(E_{m−2+k′}E_{m−2+k}) − B_{i}(E_{k}, E_{k})}N_{i}

does not depend on the basis.

M is said to be minimal if H = 0.

Theorem 4.2. Any invariant submanifold M of a para–Kähler manifold is minimal.

Proof. From Theorem 2.2 and [2, Proposition 1.4] there is a local frame {E_{j}} of SM such that E_{m−2+k} = P E_{k}, k = \overline{1, m − 2}. From (4.7) we get H = 0.

Theorem 4.3. Any totally umbilical invariant submanifold of a para–Kähler manifold is totally geodesic.

Proof. From (4.9) it follows B_{i}(X, ξ) = 0, i = 1, 2, ∀X ∈ Γ(T M), ξ ∈ Γ(T M^{⊥}). By (4.7), (4.9) and (1.2), we get f_{i}g(hX, hY) = 0, i = 1, 2, ∀X, Y ∈ Γ(T M). Since SM is non–degenerate then f_{1}, f_{2} vanish identically.

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Seminar Matematic Universitatea ”Al. I. Cuza” Iasi 6600, ROMANIA E-mail : bejan@math.tuiasi.ro