The theory of warped product manifolds is very important in geometry. For example, in Riemannian geometry, warped product manifolds in their generic forms have been used to construct new examples with interesting curvature properties [1] - [3] . Some solutions to Einstein’s field equations can be expressed in terms of warped product in Lorentzian geometry [4] [5] . These solutions provide many clues and in-sights into astrophysical and cosmological questions. Submanifolds theory is a very active research field and plays an important role in the development of modern differential geometry. Its application to a variety of subjects in mathematics and physics has attracted the interest of considerable group of researchers.

A basic problem in theory of submanifolds is to provide conditions which imply that an isometric immersion of a warped product manifold must be a warped product of isometric immersions. In [6] under purely intrinsinc assumptions, Moore showed that an isometric immersion of a product of Riemannian manifolds in euclidean space must be a product of hypersurfaces if the codimension equals the number of factors and no factor has an open subset at which all sectional curvatures vanish. In [7] Nölker proved that isometric immersion of warped product of connected Riemannian manifolds whose second fundamental form satisfies a certain condition splits in warped product of isometric immersions. Marcos Dajczer and Ruy Tojeiro provided a local classification of isometric immersions in codimension 1 and 2 of warped products of Riemannian manifolds into space forms under assumption that the warped product has no points with the same constant sectional curvature as the ambient space form.

In differential geometry, some factorisation results such as the de Rham decomposition theorem [8] , the Moore’s theorem [6] about the decomposition of isometric immersion into Riemannian product of isometric immersions are essential tools in handling certain important aspects of the geometry of Riemannian manifolds.

Let M be a connected, simply connected and complete Riemannian manifold. Assume that M is reducible. Let $T_{x}M={T^{\prime }}_{x}M+{T^{″}}_{x}M$ ( $x\in M$) be a decomposition into subspaces invariant by the linear holonomy group $Ho{l}^{\nabla }$ and let $T^{\prime }$ and ${T^{\prime }}^{\prime }$ be the parallel distributions defined by ${T^{\prime }}_{x}M$ and ${T^{″}}_{x}M$ respectively. We fix a point $0\in M$ and let $M^{\prime }$ and ${M^{\prime }}^{\prime }$ be the maximal integral manifolds of $T^{\prime }$ and ${T^{\prime }}^{\prime }$ trough 0 respectively. Both $M^{\prime }$ and ${M^{\prime }}^{\prime }$ are complete, totally geodesic submanifolds of M.

Theorem 1.1 [8] M is isometric to the direct product $M^{\prime }\times {M^{\prime }}^{\prime }$.

Theorem 1.2 (G. de Rham) [8] A connected, simply connected and complete Riemannian manifold M is isometric to the direct product $M_0\times M_1\times M_2\times \cdots \times M_l$ where M₀ is a euclidean space (possibly of dimension 0) and $M_1\mathrm{,}{M}_2\mathrm{,}\cdots \mathrm{,}{M}_l$ are all simply connected, complete, irreducible Riemannian manifolds. Such a decomposition is unique up to an order.

The main goal of this paper is to study isometric immersibility of globally null warped product manifolds into pseudo-Riemannian manifold. Taking into account that a globally null manifold is of lightlikeity degree $r=1$, it can be isometrically immersed in semi-Riemannian manifold with index $q\ge 1$ [9] . The important results provided by K. L. Duggal on Globally null manifolds and the physical applications known in lorentzian geometry motivated us to consider, in our study, isometric immersion of globally null warped product manifolds in lorentzian ambient with constant sectional curvature c. Let M₁ be a globally null manifold and M₂ a complete Riemannian manifold. Considering a positive function ρ on M₁, $M_1{\times }_{\rho }{M}_2$ is a globally null warped product manifold. We show that $f\mathrm{<mo>\; :\; </mo>}{M}_1{\times }_{\rho }{M}_2\to Q_{\left(c\right)}$ is an isometric immersion which splits into warped product of isometric immersions whenever $Q_{\left(c\right)}$ is simply connected and complete reducible with respect to its holonomy group.

Consider $f:M=M_1{\times }_{\rho }{M}_2\to Q_{\left(c\right)}^{m+k}$ an isometric immersion of lightlike warped product manifold M where M₁ is ( $m_1=n_1+1$)-dimensional globally null manifold, M₂ is a (m₂)-dimensional complete Riemanniann manifold and $Q_{\left(c\right)}^{m+k}$ is a space form of constant sectional curvature c endowed with a Lorentzian metric, $m=m_1+m_2$. In [13] Duggal have shown that M splits as $M=L\times B{\times }_{\rho }{M}_2$ where L is one-dimensional integral manifold of a global null vector field in M and B is (n₁)-dimensional complete Riemannian hypersurface of M₁. Since B and M₂ are both complete Riemannian manifolds, the warped product $M^{\prime }=B{\times }_{\rho }{M}_2$ is a complete Riemannian manifold. Let i and j be the inclusion maps of L and $M^{\prime }$ in M respectively; then the isometric immersion f induces isometric immersions $f^{\prime }=f\circ i$ and ${f^{\prime }}^{\prime }=f\circ j$ of L and $M^{\prime }$ in $Q_{\left(c\right)}^{m+k}$ respectively.

Our objective is to understand the possible cases in which the isometric immersion f is a product of isometric immersions $f^{\prime }$ and ${f^{\prime }}^{\prime }$. This situation supposes the splitting of $Q_{\left(c\right)}^{m+k}$ into two factors such that $f^{\prime }$ is an isometric immersion of L in one part of the product and ${f^{\prime }}^{\prime }$ is an isometric immersion of $M^{\prime }$ in the other one. Taking into account $Q_{\left(c\right)}^{m+k}$ is a Lorentz manifold, if it is reducible, it splits into product of Lorentz manifold and Riemannians manifolds. It is obvious that L, being one-degenerate, it must be isometrically immersed into the Lorentzian part and ${f^{\prime }}^{\prime }$ is reduced to the isometric immersion of the complete Riemannian manifold $M^{\prime }$ in a Riemannian submanifold of $Q_{\left(c\right)}^{m+k}$. It is now important to study the reductibility of $Q_{\left(c\right)}^{m+k}$.

Let N be a n-dimensional smooth manifold equipped with a Levi-civita connexion $\nabla$, i.e. a connexion on the tangent bundle TN. If $X\in \Gamma \left(TN\right)$ is a tangent vector at a point $x\in N$, $\nabla$ allows us to parallel translate this vector along any given curve $\gamma \mathrm{<mo>\; :\; </mo>}\left[\mathrm{0,1}\right]\to M$. The holonomy group $Ho{l}_x(\nabla )$ of $\nabla$ at x is the group defined by parallel displacement along loops about this point that is subgroup of invertible linear transformations of $T_{x}N$ i.e. $GL\left(T_{x}N\right)\simeq GL\left(n\mathrm{,}ℝ\right)$. The holonomy group $Ho{l}_x(\nabla )$ acts as group of orthogonal mapping on the tangent space $T_{x}N$. The holonomy representation $\delta \mathrm{<mo>\; :\; </mo>}Ho{l}_x(\nabla )\to O\left(T_{x}N\mathrm{,}{g}_x\right)$ is called reducible if there is proper holonomy invariant subspace $D_x$ of $T_{x}N$ i.e. a subspace such that $\delta \left(H\right)D_x\subset D_x$ $\forall H\in Ho{l}_x(\nabla )$. If $D_x$ is non degenerate and holonomy invariant, then $D_x^{\perp }$ is also holonomy invariant and $T_{x}N$ is direct sum of these holonomy invariant subspaces: $T_{x}N=D_x\oplus D_x^{\perp }$. If γ is a picewise smooth curve from x to y, then

$D\mathrm{<mo>\; :\; </mo>}y\in M\mapsto D_y\mathrm{<mo>\; :\; </mo>}=P_{\gamma }\left(D_x\right)\subset T_{y}M$

where $P_{\gamma }$ is a parallel displacement along γ is an involutive distribution on M, the holonomy distribution defined by D_x [14] . The maximal connected integral manifold N₁ of D is a totally geodesic submanifold of N which is complete whenever $\left(N\mathrm{,}g\right)$ is. In the same sense, $D_x^{\perp }$ define an involutive distribution whose connected integral manifold N₂ is also totally geodesic submanifold of N. If N is simply connected and complete, it is globally isometric to the product $N_1\times N_2$.

Definition 3.1 Let N be a connected Riemannian manifold with metric g and $Ho{l}_x(\nabla )$ the linear holonomy group of the Riemannian connection $\nabla$ with reference point $x\in N$. Then N is said to be reducible or irreducible according as $Ho{l}_x(\nabla )$ is reducible or irreducible as a linear group acting on $T_{x}N$.

The following decomposition theorem is the analogous of the de Rham decomposition theorem 1.2 in case of semi-Riemannian manifold.

Theorem 3.1 (Decomposition theorem of de Rham and Wu) [15]

Any simply-connected, complete semi-Riemannian manifold $\left(M\mathrm{,}g\right)$ is isometric to a product of simply-connected, complete semi-Riemannian manifolds one of which can be flat and the others have an indecomposably acting holonomy group and the holonomy group of $\left(M\mathrm{,}g\right)$ is the product of these indecomposably acting holonomy groups.

In case of a n-dimensional Lorentzian manifold, its holonomy group is a subgroup of $O\left(\mathrm{1,}n-1\right)$ and it is known that the only subgroup of $O\left(\mathrm{1,}n-1\right)$ that is invariant is $S{O}_0\left(\mathrm{1,}n-1\right)$. The decomposition due to theorem 3.1 gives following result of Lorentzian manifolds.

Corollary 3.1 [15] Any simply-connected, complete Lorentzian manifold is isometric to the following product of simply-connected, complete semi-Riemannian manifolds

$\left(M\mathrm{,}h\right)\times \left(M_1\mathrm{,}{g}_1\right)\times \cdots \times \left(M_k\mathrm{,}{g}_k\right)$

where $\left(M_i\mathrm{,}{g}_i\right)$ are either flat or irreducible Riemannian manifolds and $\left(M\mathrm{,}h\right)$ is either $\left(ℝ\mathrm{,}-d{t}^2\right)$ or an indecomposable Lorentzian manifold, the holonomy of which is either $S{O}_0\left(\mathrm{1,}n-1\right)$ or contained in the stabiliser of a lightlike line.

If the Lorentzian factor is not flat, it is indecomposable in which case it can be irreducible or not. In the first case, it is well known in [16] that the only connected lie subgroup of $O\left(\mathrm{1,}n-1\right)$ which acts irreducibly is the connected component of the identity $S{O}_0\left(\mathrm{1,}n-1\right)$. The latter case means that there exists a degenerate invariant subspace whose intersection with its orthogonal complement yields a lightlike line invariant by holonomy and the holonomy group of the Lorentzian part is contained in the stabiliser of this lightlike line.

We already saw that in the study of the isometric immersion f as product of isometric immersions, the totally degenerate manifold L must be isometrically immersed in a Lorentzian submanifold of $Q_{\left(c\right)}^{m+k}$ with dimension > 1. We have thus to define conditions under which a Lorentzian manifold admits a global decomposition into two factors such that the Lorentzian factor is of dimension > 1. We explore the preceding paragraph on holonomy group to give the following

Lemma 3.1 Let N be a n-dimensional simply-connected, complete Lorentzian manifold and let $Ho{l}_x(\nabla )$ its holonomy group at $x\in N$. If $T_{x}N$ admits a ( $n_1\ge 2$)-dimensional proper non-degenerate subspace E_x invariant by $Ho{l}_x(\nabla )$ such that the involutive holonomy distribution defined by E_x is of index 1, then N is isometric to the product $N_1\times N_2$ where N₁ is a maximal integral Lorentzian manifold with respect to the distribution defined by E_x and N₂ is the maximal integral Riemannian manifold with respect to the distribution defined by $E_x^{\perp }$; both N₁ and N₂ are totally geodesic simply-connected complete submanifolds of N.

Definition 3.2 Let $f:M=M_0^{m_0}{\times }_{{\rho }_1}{M}_1^{m_1}{\times }_{{\rho }_2}{M}_2^{m_2}\times \cdots {\times }_{{\rho }_l}{M}_l^{m_l}\to Q_{\left(c\right)}^{m+k},m={\displaystyle {\sum }_{i=0}^l}{m}_i$ be an isometric immersion of a lightlike warped product M in a semi-Riemannian space form $Q_{\left(c\right)}$. If there exist an isometry $\Theta :Q_{\left(c_0\right)}^{n_0}{\times }_{{\sigma }_1}\times \cdots {\times }_{{\sigma }_l}{Q}_{\left(c_l\right)}^{n_l}\to Q_{\left(c\right)}^{m+k},m+k={\displaystyle {\sum }_{i=0}^l}{n}_i$ and isometric immersions $f_i\mathrm{<mo>\; :\; </mo>}{M}_i^{m_i}\to Q_{\left(c_i\right)}^{n_i}$ such that $f=\Theta \circ \left(f_0\times f_1\times \cdots \times f_l\right)$, then f is called a lightlike warped product of isometric immersion $f_i$.

In the following, we suppose that the Lorentzian space form $Q_{\left(c\right)}^{m+k}$ admits the global decomposition $Q_{\left(c\right)}^{m+k}\simeq H^2{\times }_{\omega }{N}_{\left(\bar{c}\right)}^{m+k-2}$ which means $Q_{\left(c\right)}^{m+k}$ is reducible where $H^2$ is an indecomposable Lorentzian plane and $N_{\left(\bar{c}\right)}^{m+k-2}$ is an irreducible complete Riemannian mannifold.

Lemma 3.2 Each h-level set $\left\{h\right\}{\times }_{\omega }N$ of $H^2{\times }_{\omega }{N}_{\left(\bar{c}\right)}^{m+k-2}$ is a Riemannian space form with constant sectional curvature $\bar{c}=c+{\Vert \frac{grad\omega }{\omega }\Vert }^2$.

Proof. $\left\{h\right\}{\times }_{\omega }N$ being submanifol of $H^2{\times }_{\omega }{N}^{m+k-2}$, $\forall U\mathrm{,}V\mathrm{,}W\mathrm{,}{U}^{\prime }\in \Gamma \left(\mathcal{V}\right)$, if $\bar{R}$ and R are the Riemannian curvature of $H^2{\times }_{\omega }{N}^{m+k-2}$ and $\left\{h\right\}{\times }_{\omega }N$ respectively, the Gauss equation is given by

$\begin{array}{c}\langle \bar{R}\left(U\mathrm{,}V\right)W\mathrm{,}{U}^{\prime }\rangle =\langle R\left(U\mathrm{,}V\right)W\mathrm{,}{U}^{\prime }\rangle +\langle \alpha \left(U\mathrm{,}W\right)\mathrm{,}\alpha \left(V\mathrm{,}{U}^{\prime }\right)\rangle \\ -\langle \alpha \left(V\mathrm{,}W\right)\mathrm{,}\alpha \left(U\mathrm{,}{U}^{\prime }\right)\rangle \end{array}$(6)

where

$\alpha \left(U\mathrm{,}V\right)=-\langle U\mathrm{,}V\rangle \frac{grad\omega }{\omega }$(7)

and

$\bar{R}\left(U\mathrm{,}V\right)W=c\left\{\langle V\mathrm{,}W\rangle U-\langle U\mathrm{,}W\rangle V\right\}\mathrm{.}$(8)

Put (7) and (8) in (6) we have

$\begin{array}{l}c\langle V\mathrm{,}W\rangle \langle U\mathrm{,}{U}^{\prime }\rangle -c\langle U\mathrm{,}W\rangle \langle V\mathrm{,}{U}^{\prime }\rangle \\ =\langle R\left(U\mathrm{,}V\right)W\mathrm{,}{U}^{\prime }\rangle +\langle U\mathrm{,}W\rangle \langle V\mathrm{,}{U}^{\prime }\rangle \langle \frac{grad\omega }{\omega }\mathrm{,}\frac{grad\omega }{\omega }\rangle \\ -\langle V\mathrm{,}W\rangle \langle U\mathrm{,}{U}^{\prime }\rangle \langle \frac{grad\omega }{\omega }\mathrm{,}\frac{grad\omega }{\omega }\rangle \end{array}$

$\begin{array}{l}\iff \langle R\left(U\mathrm{,}V\right)W\mathrm{,}{U}^{\prime }\rangle =\left(c+{\Vert \frac{grad\omega }{\omega }\Vert }^2\right)\left(\langle V\mathrm{,}W\rangle \langle U\mathrm{,}{U}^{\prime }\rangle \right)\\ -\left(c+{\Vert \frac{grad\omega }{\omega }\Vert }^2\right)\left(\langle U\mathrm{,}W\rangle \langle V\mathrm{,}{U}^{\prime }\rangle \right)\end{array}$

$\iff R\left(U\mathrm{,}V\right)W=\left(c+{\Vert \frac{grad\omega }{\omega }\Vert }^2\right)\left\{\langle V\mathrm{,}W\rangle U-\langle U\mathrm{,}W\rangle V\right\}\mathrm{.}$(9)

Since ${\omega |}_{\left\{h\right\}\times N}$ is constant, we conclude. ■

For $Q_{\left(c\right)}^{m+k}$, to admit the global decomposition $Q_{\left(c\right)}^{m+k}\simeq H^2{\times }_{\omega }{N}_{\left(\bar{c}\right)}^{m+k-2}$ means that there exists a global isometry $\Theta \mathrm{<mo>\; :\; </mo>}{H}^2{\times }_{\omega }{N}_{\left(\bar{c}\right)}^{m+k-2}\to Q_{\left(c\right)}^{m+k}$ and we give the following:

Theorem 3.2 Let $f\mathrm{<mo>\; :\; </mo>}M=L\times B^{n_1}{\times }_{\rho }{M}_2^{m_2}\to Q_{\left(c\right)}^{m+k}$ be an isometric immersion of a global null warped product manifold in a simply-connected, complete Lorentzian manifold $Q_{\left(c\right)}^{m+k}$ reductive to $H^2{\times }_{\omega }{N}_{\left(\bar{c}\right)}^{m+k-2}$, $f^{\prime }$ and ${f^{\prime }}^{\prime }$ the isometric immersions induced by f on L and $M^{\prime }=B^{n_1}{\times }_{\rho }{M}_2^{m_2}$ in $Q_{\left(c\right)}^{m+k}$ respectively. Assume that ${\alpha }_f$ the second fundamental form of M satisfy

${\pi }_{N_{\mathrm{<mo>\; *\; </mo>}}}\left[{\alpha }_f\left(X\mathrm{,}Y\right)\right]={\alpha }_{{f^{\prime }}^{\prime }}\left({\pi }_{{M^{\prime }}_{\mathrm{<mo>\; *\; </mo>}}}X\mathrm{,}{\pi }_{{M^{\prime }}_{\mathrm{<mo>\; *\; </mo>}}}Y\right)\forall X\mathrm{,}Y\in \Gamma \left(TM\right)\mathrm{.}$

Then there exists a global isometry $\Theta \mathrm{<mo>\; :\; </mo>}{H}^2{\times }_{\omega }{N}_{\left(\bar{c}\right)}^{m+k-2}\to Q_{\left(c\right)}^{m+k}$ and a warped product representation $\psi \mathrm{<mo>\; :\; </mo>}{V}^{n_1+k_1}{\times }_{\sigma }{N}_{\left(\bar{c}\right)}^{m_2+k_2}\to N_{\left(\bar{c}\right)}^{m+k-2}$, $k_1+k_2=k-1$ such that f is a null warped product of isometric immersions $f^{\prime }$ and ${f^{\prime }}^{\prime }$ i.e.

2) f is a composition of isometric immersions $\Theta \circ \left[f^{\prime }\times \left(\tau \circ \eta \right)\right]$ where Θ and $f^{\prime }$ are defined as in 1) and $\eta =\psi \circ \left(h_1\times h_2\right)$ is a warped product of isometric immersions with respect to a warped product representation $\psi :V^{n_1+k_1}{\times }_{\sigma }{N}_{\left(\tilde{c}\right)}^{m_2+k_2}\to N_{\left(\bar{c}\right)}^{n_1+m_2+1}$, $k_1+k_2=1$ and $\tau \mathrm{<mo>\; :\; </mo>}W\to N_{\left(\bar{c}\right)}^{n_1+m_2+1}$ is an isometric immersions of an open subset $W\supset \eta \left(U\right)$ of $N_{\left(\bar{c}\right)}^{n_1+m_2+1}$.

onto open subsets such that $f=\Theta \circ \left\{f^{\prime }\times \left[{\psi }_1\circ \left(i_1\times \left({\psi }_2\circ \left(g\times i_2\right)\right)\right)\right]\right\}$, $\bar{\rho }={\sigma }_2\circ g$ and ${\rho }_1={\sigma }_1\circ i_1$.

Case $c\ne 0$.

1) There exists an embedding $\theta \mathrm{<mo>\; :\; </mo>}{N}_{\left({\bar{c}}_1\right)}^{n_1+k_1}\times N_{\left({\bar{c}}_2\right)}^{m_2+k_2}\to N_{\left(\bar{c}\right)}^{n_1+m_2+2}$ as an extrinsinc Riemannian product with $k_1+k_2=1$ and isometric immersions $f^{\prime }\mathrm{<mo>\; :\; </mo>}L\to H^2$, $h_1\mathrm{<mo>\; :\; </mo>}{B}_0^{n_1}\to N_{\left({\bar{c}}_1\right)}^{n_1+k_1}$ and $h_2\mathrm{<mo>\; :\; </mo>}{M}_{2_0}^{m_2}\to N_{\left({\bar{c}}_2\right)}^{m_2+k_2}$ such that $f=\Theta \circ \left\{f^{\prime }\times \left[\theta \circ \left(h_1\times h_2\right)\right]\right\}$.

We considered in this paper isometric immersion of globally null warped product manifolds. This subject has been motived by the important applications highlighted by K. L. Duggal works on globally null warped product geometry. The immersibility of such null manifolds has been explored. In the continuation of our research, we will be investigating physical applications of our results given that we have considered isometric immersions in Lorentzian ambient space form.