Global Embedding of Analytic Branes into Einstein MD Bulk Cosmology
Nikolaos I. Katzourakis\footnote{Address: Troias 6, 18541, Kaminia, Pireaus, Greece}
\noindent In recent work \cite{Katz 1}, \cite{Katz 2} we have shown that any $n$-dimensional analytic semi-Riemannian manifold $\mathcal{M$ has a global analytic and isometric embedding into an Einstein manifold $\mathcal{E}$ of dimension $n+d$, $d \geqq 1$} ($\Lambda \in \R$). In addition, if $\mM$ is an analytic spacetime manifold, there exists an Einstein global embedding space $\mE^{\Sigma$, admitting unsmoothable differential-topological singularities ($0-D$ points or extended $(d-k)-D$ spatial "anomalies") living into the complement $\mE^{\Sigma} setminus \mM$}. They were expounded as wormholes. \noindent In this paper we continue extending this result on Cosmological Brane embeddings: Any finite-many analytic semi-Riemannian manifolds $\mM^{(n_1)_1$, ..., $\mM^{(n_m)}_m$ have all a global isometric analytic embedding into an $(\underset{1 \leq k \leq m}{\sum n_k} + 1)$-Einstein Bulk $\mE$ as disjoint submanifolds, while $\pi_m(\mE)$ splits}. \medskip {Keywords:} Spacetime Embedding, Einstein space, Campbell-Magaard Theorem; Brane-World, MD Gravity, Brane Cosmology \medskip \noindent {MSC Subject Classification $2000$:} Primary 53C20, 53C25, 32C09, 53Cxx, Secondary 83C75, 83C55, 83E15, 83F05.
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\begin{document}

\pagestyle{myheadings} \markboth{\centerline{\small{\sc Nikolaos
I. Katzourakis}}}
         {\centerline{\small{\sc Global Embedding of Analytic Branes in
          Einstein Bulk Cosmology}}}

\title{Global Embedding of Analytic Branes into Einstein MD Bulk Cosmology}

\author{ Nikolaos I. Katzourakis\footnote{Address: Troias 6, 18541, Kaminia, Pireaus,
Greece}\\[8pt]
         {University of Athens, Department of Mathematics}\\[8pt]
         { \small e-mail: nkatzourakis@math.uoa.gr}}

\date{}

\maketitle

\begin{abstract}
\noindent In recent work \cite{Katz 1}, \cite{Katz 2} we have
shown that \textit{any $n$-dimensional analytic semi-Riemannian
manifold $\mathcal{M}$ has a global analytic and isometric
embedding into an Einstein manifold $\mathcal{E}$ of dimension
$n+d$, $d \geqq 1$} ($\Lambda \in \R$).

In addition, \emph{if $\mM$ is an analytic spacetime manifold,
there exists an Einstein global embedding space $\mE^{\Sigma}$,
admitting unsmoothable differential-topological singularities
($0-D$ points or extended $(d-k)-D$ spatial "anomalies") living
into the complement $\mE^{\Sigma} \smallsetminus \mM$}. They were
\emph{expounded as wormholes}.

\noindent In this paper we continue extending this result on
Cosmological Brane embeddings:

\emph{Any finite-many analytic semi-Riemannian manifolds
$\mM^{(n_1)}_1$, ..., $\mM^{(n_m)}_m$ have all a global isometric
analytic embedding into an $(\underset{1 \leq k \leq m}{\sum n_k}
+ 1)$-Einstein Bulk $\mE$ as disjoint submanifolds, while
$\pi_m(\mE)$ splits}.

\medskip {\it Keywords:} Spacetime Embedding, Einstein space,
Campbell-Magaard Theorem; Brane-World, MD Gravity, Brane Cosmology

\medskip \noindent
{\it MSC Subject Classification $2000$:} Primary 53C20, 53C25,
32C09, 53Cxx, Secondary 83C75, 83C55, 83E15, 83F05.
\end{abstract}


\section*{Introduction}

Multi-dimensional theories of gravity and, in particular, the
recent Braneworld Cosmological scenario, in which the Universe of
our perception is a submanifold (viz. \textit{(mem)-Brane})
embedded in a multi-dimensional (MD) manifold (viz.
\textit{Bulk}), have generated a marvelous interest in
mathematical and physical aspects of embedding theories based on
the \textit{$4$-spacetime} of \textit{General Relativity}
(\textit{G.T.R.}). The theme of the present paper is devoted to
the \emph{mathematical} perspective of this subject.

According to the G.T.R., the model we adopt to describe the large
scale structure of the Universe is a \textit{4-manifold} equipped
with a \textit{semi-Riemannian metric} structure, the solution of
the \textit{field equations} which (the latter) correlate the
\textit{Curvature} of spacetime with the distribution of
\textit{matter}, the \textit{energy} and the \textit{momentum}
(\cite{OG-SH}, \cite{Hawk}, \cite{Sa-Wu}, \cite{Stef}).

The possibility that there may exist more than $3+1$ space-time
dimensions was taken into account not long after the final
formulation of G.T.R.. \textit{Kaluza} and \textit{Klein} (K.-K.)
\textit{unified} $4D$ \textit{vacuum Relativity} and classical
\textit{electromagnetism} (\cite{W}) on a \textit{$5$-manifold}
that satisfies the vacuum field equations (curved by the
contribution of the photons). The extra dimension was considered
topologically \textit{compact} (\textit{homeomorphic to
$S^{1}$}), \textit{infinitesimal} (thus invisible) and the
$4$-spacetime was assumed not to depend on the 5th dimension. In
fact, the Universe was considered as a \textit{hyper-cylinder},
locally \textit{homeomorphic to $\R^{4} \times S^{1}$}.

In a modern approach called \textit{Space-Time-Matter} or
\textit{Induced Matter Theory} (S.T.M.) (\cite{D-R2},
\cite{D-R3}, \cite{W} and references therein) these assumptions
are removed and the result is the \textit{unification of
Relativity and electromagnetism with their sources} (\emph{mass}
and \emph{charge}), expounded (the latter) as an artifact induced
on the $4$-manifold, generated by the (non-linear) geometry of
invisible extra spatial dimensions.

One of the most recent Cosmological scenarios concerning the
geometric structure of the (classical) Universe introduced by L.
Randall and R. Sundrum (\cite{RS-1}, \cite{RS-2}) claims that the
Universe of our perception is a (mem-)Brane of $3+1$ dimensions
embedded in a multi-dimensional Bulk of at least $5$ dimensions.
The main ansatz for this (possibly moving in the Bulk) Brane is
that \textit{matter is confined on the Brane} and \textit{only
gravity and electromagnetic fields can propagate on the bulk}.

Mathematically, these theories are modeled on a local embedding
theorem of Riemannian Geometry, due to Campbell who was the first
who stated it \cite{Camp}, and Magaard (\cite{Maa},
\cite{L-R-T-R}) who presented a strict proof (see \cite{Pavisic},
\cite{Seahra-1}, \cite{Wands}): \textit{any analytic n-manifold
can be locally and isometrically embedded into some other
Ricci-flat of n+1 dimensions} ($R_{AB}=0$). In a sense, this is
equivalent to the \emph{local} \textit{embedding of Einstein 4D
Relativity}, \textit{into} some \textit{5D Relativity in the
vacuum} where \textit{no matter exists} ("wood", according to
Einstein), but only \emph{geometrical properties} ("marble").

It becomes clear that in the light of these theories that are
entirely based on differential-geometric (though \emph{local}
only) embeddings, new \emph{global results are needed}, in order
to provide the \emph{supporting mathematical framework for
Brane-Worlds}. This owes to the fact that the starting point of
an MD gravitational-cosmological theory is the $4$-dimensional
G.T.R. that has to be incorporated in the new context.

In recent work (\cite{Katz 1}, \cite{Katz 2}) the author proved
that \textbf{the} \textbf{Campbell-Magaard} (C.-M.)
\textbf{Theorem} (extended by \emph{Dahia \& Romero} in
\cite{D-R2} on \emph{Einstein from Ricci-flat spaces})
\textbf{holds true globally}:

\textit{Any $n$-dimensional analytic semi-Riemannian manifold
$\mathcal{M}$ has a global analytic isometric embedding into an
Einstein manifold $\mE$ of dimension $n+d$, $d \geqq 1$} ($\Lambda
\in \R$).

\emph{This $\mE$ may admit (if desired) $1$ for embedding
codimension $d = 1$ \& countable many for $d > 1$ unsmoothable
differential-topological singularities, which may be distinct
points into spacetime or extended spatial anomalies} (\cite{Katz
2}). These pathologies living into the complement $\mE
\smallsetminus \mM$ were expounded as \emph{wormholes connecting
distant points of spacetime}.

In this sequel we continue showing that \emph{any finite many
semi-Riemannian manifolds $\mM_1$, . . ., $\mM_m$ have all a
global isometric analytic embedding into an Einstein space $\mE$
as disjoint analytic submanifolds. The minimum dimensionality of
$\mE$ is the sum of the respective of the $\mM_k$'s plus $1$ and
this is optimum}.

The process in the proof is a slight modification of that
presented in \cite{Katz 1}. We form a bulk that contains all the
$\mM_k$'s as disjoint submanifolds and using the known local
embedding results and certain differential-geometric \&
topological arguments we obtain the requested global extension.

In \emph{Section 1} we give the necessary results we will use
throughout the rest of the paper. \emph{Section 2} contains the
proof of the basic result stated above. In \emph{Section 3} we
study the topology of such embedding Einstein spaces, giving a
Homotopic description of the Einstein manifolds $\mE$ we
construct, which proves to be quite simple, due to direct sum
decomposition: $\pi_q(\mE)$ can be completely determined if
$\pi_q(\mM_1)$, . . . , $\pi_q(\mM_m)$ are known.

\medskip \noindent {\bf Notations:} In the sequel, the terms
\textbf{Spacetime}, \textbf{Brane}, \textbf{Bulk} will be used in
the place of \emph{paracompact}, \emph{Hausdorff},
\emph{connected manifolds (finite dimensional)}, equipped with a
\emph{semi-Riemannian metric structure} and the
\emph{corresponding Levi-Civita linear connection}.


\section{Preliminaries}

In this section we collect for the readers convenience some
fundamental notions retrieved from the references that will be
employed in the rest of the paper.

\begin{definition} \label{GTR f eq}
The \textit{G.T.R. field equations} on a semi-Riemannian manifold
($\mathcal{M}^{(n)}; \nabla, g $) in presence of
\textit{Cosmological Constant} $\Lambda \in \R$ are:

         $$G \equiv Ric^{\nabla}(g) - \frac{1}{2} R^{(n)}g = k T
          - \Lambda g $$

where $Ric,\;g,\;T \;\in \Gamma(T^* \mathcal{M} \bigotimes T^*
\mathcal{M})$, $ k \in \R$, $G$ is the \textit{Einstein tensor of
Curvature} and $R^{(n)}$ is the \textit{scalar Curvature},
defined as the double contraction with the metric. $T$ is called
the \textit{Stress-Energy-Momentum tensor}.
\end{definition}

\begin{definition} \label{Einst}
A semi-Riemannian manifold ($\mM^{(n)}; \nM, \gM $) is said to be
an \emph{Einstein space} if there exists a $\Lambda \in \R$ such
that:

               $$Ric^{\nM}(\gM)=\frac{2 \Lambda}{n-2}\;\gM$$

That is, its metric is a solution of the previous equations with
$T=0$.

\noindent If $\Lambda = 0$ the manifold is said to be
\emph{Ricci-flat}.
\end{definition}

The following has been originally retrieved from reference
\cite{D-R2} and used also in \cite{Katz 1}, \cite{Katz 2}, as the
local analogues of the basic results.

\begin{theorem} \label{C-M-D-R}(\textbf{Campbell-Magaard-Dahia-Romero
 local embedding into Einstein Spaces})

\noindent Let $(\mM^{(n)},\;\gM)$ be a semi-Riemannian analytic
manifold and

   $$g|_{U}=g_{ij}\;dx^{i} \otimes dx^{j}\;\;\in \; \Gamma_{\alpha_{0}}
   (T^*\mathcal{M} \bigotimes T^*\mathcal{M})$$

a local analytic at $0 \equiv \phi(p) \in \R^n$ presentation of
its metric, with respect to $(U,\phi)\;\in \mA_{\mM}$.

\noindent Then, there exists a local isometric embedding
(analytic at $\phi(p)$) of $U$ into some (coordinated)
semi-Riemannian manifold $(\mathcal{N}^{(n+1)}; \;g_{\mathcal{N}}
, \nabla^{\mathcal{N}})$:

   $$\varepsilon:\;\;U \longrightarrow \varepsilon(U) \subseteq \mathcal{N}$$

which is an \emph{Einstein space} in the local embedding domain:

    $$Ric^{\nabla^{\varepsilon(U)}}(g_{\mathcal{N}}|_{\varepsilon(U)})=
    \frac{2 \Lambda}{n-1} \;g_{\mathcal{N}}|_{\varepsilon(U)}\;\;\;\in
    \Gamma(T^*\varepsilon(U) \bigotimes T^* \varepsilon(U))$$
that is
    $$R_{AB}\;dx^{A} \otimes dx^{B}=\frac{2 \Lambda}{n - 1}\; g_{AB}
    \;dx^{A} \otimes dx^{B}$$
\end{theorem}

\noindent Setting $\Lambda = 0$, \emph{Th.} \ref{C-M-D-R} reduces
to the Campbell-Magaard \emph{Theorem} of embedding into
Ricci-flat spaces (\cite{Maa},\cite{Katz 1}, \cite{Katz 2} etc).

The following can be easily deduced from \emph{Th.} \ref{C-M-D-R}
and the Criterion of local isometric embeddings (see \cite{Maa},
\cite{D-R2}, \cite{Katz 1}, \cite{Katz 2}):

\begin{corollary} \label{inclusion embedding plus criterion}
Let $(U , \chi = \{x^A\}_{1 \leq A \leq n+1})$ be a patch of an
analytic $(n+1)$-manifold $\mM$ with local metric of the diagonal
form $(g_{a (n+1)} \equiv 0)$:

             $$g_{U}\;=\;g_{ab}\;dx^a \otimes dx^b \;+\; \psi\;dx^{n+1}
                  \otimes dx^{n+1}$$

Then, if the functions $g_{ab}$ are analytic at $0 \in \chi(U)
\subseteq \R^{n+1}$, there exists an analytic at $0$ function
          $$\psi\;\; \in \; C^{\infty}_{\alpha_0}(U \larrow \R)$$

such that the coordinated domain $U$ to be a Einstein space, with
respect to $g_{U}$ and the respective Levi-Civita metric
connection $\nabla^{U}$.
\end{corollary}


\section{Global multi-Brane Embedding into Einstein Manifolds}

\noindent The manifolds we employ in this section are considered
\underline{real analytic}. Notwithstanding, it follows from
results of Whitney \cite{Whit 2} and Grauert \cite{Grauert} that
this is not restrictive, provided that any $C^k$ or smooth
($C^{\infty}$) manifold has a $C^k$ (resp. $C^{\infty}$)
diffeomorphism with an analytic one, that is, it \textit{carries
a unique compatible real analytic structure}. See also \cite{Nar}.

\noindent (\textit{This Theorem \ref{Katz 3} is almost a replica
of the respective embedding results of \cite{Katz 1} \&
\cite{Katz 2}. Notwithstanding, for complementary reasons, we
reformulate a self contained proof from the very beginning,
instead of giving citations to the prequels.})

\noindent We state now our basic result:

\begin{theorem}\label{Katz 3}(\textbf{Global Isometric Embedding
of Branes into Einstein manifolds in codimension 1})

Any $n_k$-dimensional analytic semi-Riemannian manifolds
$(\mathcal{M}_k; \nabla^{\mM_k}, g_{\mM_k})$, $1 \leq k \leq m$
(Branes) have all a global isometric analytic embedding into an
$\bigl( \overset{m}{\underset{k = 1}{\sum}} n_k + 1
\bigr)$-dimensional Einstein manifold $(\mathcal{E};
\nabla^{\mathcal{E}}, g_{\mathcal{E}})$ satisfying:
       $$Ric^{\nabla^{\mathcal{E}}}(g_{\mathcal{E}})
        =\frac{2 \Lambda}{\bigl( \overset{m}{\underset{k = 1}{\sum}}
         n_k \bigr) -1}\;g_{\mathcal{E}}$$

as disjoint submanifolds for any $\Lambda \in \R$.
\end{theorem}

\begin{proof}

\textbf{Step 1: Construction of the Bulk $\mE$ that contains
disjoint copies of $\mM_1$, . . ., $\mM_m$.}


Let $\mF$ be an abstract analytic curve ($1$-dimensional
manifold). We set
            $$\mM\;:=\;\mM_1 \times \ldots \times \mM_m$$

and $\mA_{\mM}$ will denote the Atlas of differential structure of
this product manifold. This product contains as submanifolds the
given $\mM_k$'s (each inclusion map is an embedding) in the
induced structure. For convenience, we set

      $$n\;:=\;\overset{m}{\underset{k = 1}{\sum}}\; n_k \; \in \N$$

We define the fibre bundles over $\mM$ \& $\mF$:
            $$\pi\;\;:\;\;\mE \;{\xrightarrow{\hspace{25pt}}}\; \mM$$
            $$pr_{\mF}\;\;:\;\;\mE \;{\xrightarrow{\hspace{25pt}}}\; \mF$$
where
           $$\mE :=\mM_1 \times \ldots \times \mM_m \times \mF
            \equiv \mM \times \mF$$
and
     $$\pi:=pr_{\mM}\;:\;\;\{(p_1,...,p_m)\} \times
     \mF \lmapsto (p_1,...,p_m) \equiv p \in \mM $$

This construction gives a Bulk  $\mE$ that contains all the
$\mM_k$'s as submanifolds. We shall treat $\mM$ as if it was a
single manifold ignoring the product structure.

The local (coordinated) trivializations have the form:

  $$\pi^{-1}(U)|_{\;V}\;
  {\underset{id}{\xrightarrow{\;\;\;\;\cong\;\;\;\;\;}}}
   \;U \times (\mathcal{F}|_{\;V}) \equiv W \in \mA_{\mE}$$
where $(U;\{x^a\}_{1 \leq a \leq n}) \in \mA_{\mM}$, $(V;y) \in
\mA_{\mF}$, the atlases of differential structure.

\noindent In addition, the $\mM_k$'s are all disjoint
submanifolds of $\mE$: to see this, it suffices to chose $m$-many
distinct points $f_1$, . . ., $f_m$ of the fibre $\mF$ and
identify each of the $\mM_k$'s with a copy living into the bulk
$\mE$

     $$\mM_k\;\equiv\;\underbrace{\emptyset \times \ldots \times \emptyset
      \times \mM_k
     \times \emptyset \ldots \times \emptyset} \times \{f_k\}\;\;,
     \;\;\;\;\;\;1 \leq k \leq m$$
     $$k-position \hspace{50pt}$$


\medskip \noindent \textbf{Step 2: The topological covers $\mW$,
$\mW_{\;\B}$ \& $\mQ$ of $\mE$}.

Paracompactness \& product structure of $\mE$ imply that there
exist natural covers constituted of product charts of $\mA_{\mM}$
\& $\mA_{\mF}$:

If $p \in \mE$, choose a $(U;x^1, \ldots , x^n)$ at $\pi(p)$ and
a $(V,y)$ at $pr_{\mF}(p)$. Thus, the product
           $$(W \;; \;\{x^A\}_{1 \leq A \leq  n+1}) \equiv
            \{(U \times V \;; \{x^a\}_{1 \leq a \leq n} \times y)
             \in \mA_{\mE}$$

is a patch at $p$.

At each $p \in \mE$ choose $N+1$ distinct patches $(W_{i_a}, \chi
_{(i_a)})$, $0 \leq a \leq N$. The number $N \in \N $ will be
specified later as will. Such arbitrarily but finite many patches
with different coordinate functions can be easily illustrated to
exist on $\mM$ \& $\mF$: The $U$-patches of $\mM$ can be chosen
distinct with "geodesical-coordinates" arguments (diagonalizing
$\gM|_{U}$ in different coordinates, if we choose different
initial curves with respect to which we construct the geodesical
orbit of the patch). The $V$-patches of the $1$-manifold $\mF$
can be chosen using the induced structure of an analytic embedding
of $\mF$ into some $\mathcal{G}$ of higher dimension restricted
on the curve $\mF$.

Consider the restriction of each $W_{i_a}$ on their common
intersection

     $$W_{i_a}\; \equiv \; \overset{N}{\bigcap_{b = 0}} \; W_{i_b}\;,
     \;\;\;\;\;\;\;\;\;\;a=0,1,\ldots ,N$$

Thus, we obtain $(N+1)$-patches at $p$ covering the same domain
on $\mE$, with different coordinate functions. Put $N$ of them
together to make an $N$-element class of patches at each $p \in
\mE$:

    $$[W_i]\;:=\;\{((W_{i_a}, \chi_{(i_a)}))\;\;/\;\;a = 1, \ldots, N\}$$

Use now a Euclidean transfer to identify the distinct points
$\chi_{(i_1)}(p), \ldots, \chi_{(i_N)}(p)$ as the origin $0 \in
\R^{n+1}$. Consider the intersection $\overset{N}{\underset{a =1
}{\bigcap}} \; \chi_{(i_a)}(W_{i_a})$ , which is an open set in
$\R^{n+1}$. Within this set lives an open $(n+1)$-ball $\B(0,R_i)$
of maximum radius $R_i > 0$. Choose an $r_i < R_i $. Then

    $$\mB \; \subseteq \; \overset{N}
    {\underset{a =1}{\bigcap}} \; \chi_{(i_a)}(W_{i_a})$$

Consider now the inversion of $\mB$ via $1$ of the coordinates,
say the $1$st, $\chi_{(i_1)}|_{\mB}$. This implies that within
each $W_{i_a}$ lives an analytically diffeomorphic copy of the
ball $\mB$:

   $$\chi_{(i_1)}^{-1}(\mB) \; \subseteq \;W_{i_a}\;=\; \overset{N}
    {\underset{b =1}{\bigcap}} \; \chi_{(i_b)}(W_{i_b})\;,\;\;\;\;\;
    a = 1,\ldots , N$$

We may also define a class of $N$-elements of inversed balls at
each $p \in \mE$, distinguished by the different induced
coordinates on each of them:

    $$[\B_i]\;:=\;\{(\chi_{i_1}^{-1}(\mB), \chi_{(i_a)}|_{\chi_{i_1}^{-1}(\mB)})
    \;\;/\;\;a = 1, \ldots, N\}$$

Take now the remaining $W_{i_0}$ patch at $p$ not included in
$[W_i]$ and restrict it on $\chi_{i_1}^{-1}(\mB)$:

       $$(W_{i_0} \; \equiv \; W_{i_0}|_{\;\chi_{i_1}^{-1}(\mB)} ; \;
       \chi_{(i_0)}|_{\;\chi_{i_1}^{-1}(\mB)}) \in \mA_{\mE}$$

Rename it to $Q_j$ and the coordinate functions to $y^1 _{(j)}$,
. . . , $ y^{n+1}_{(j)}$:

  $$(Q_j,\{y^A_{(j)}\}_{1 \leq A \leq
  n+1})\;:=\;(W_{i_0}|_{\;\chi_{i_1}^{-1}(\mB)}\;,
  \;\chi_{(i_0)}|_{\;\chi_{i_1}^{-1}(\mB)})$$

We form now $3$ different topological covers of $\mE$, as follows:

Consider the above $(Q_j ,\{y^A_{(j)}\}_{1 \leq A \leq
  n+1})$ patch at each $p \in \mE$ \&
refine to a locally finite cover:

   $$\mQ\;:=\;\{(Q_j, \{y^A_{(j)}\}_{1 \leq A \leq n+1})
   \;\;\;/\;\;\;j\; \in I\}$$

\emph{Dimension theory} (see e.g. \cite{Mun} for an elementary
introduction) implies that the \emph{Lebesgue topological
dimension} $td(\mE) \in  \N $ of the paracompact manifold $\mE$
equals its geometrical dimension

     $$td(\mE)\;=\;dim(\mE)\;=\;n+1 \in \N$$

This means that no point of $\mE$ lies in more than $(n+1)+1=n+2$
elements of the refinement $\mQ$.

The previous cover implies the existence of $2$ more locally
finite covers, which we obtain considering the classes $[\B_i]$
and $[W_i]$ at each $p \in \mE$, refined by the way prescribed by
the construction of $\mQ$:

   $$\mW_{\B}\;:=\;\{[\B_i]\;\;\;/\;\;\;i \in \{1 , \ldots , N\}
    \times I\}\;\equiv\;\hspace{100pt}$$

   $$\hspace{50pt} \equiv \;\{(\chi_{i_1}^{-1}(\mB), \chi_{(i_a)}
   |_{\chi_{i_1}^{-1}(\mB)})
    \;\;/\;\;a = 1, \ldots, N\;/\;i_a \in I\}$$
and
  $$\mW \; := \; \{[W_i] \;\;\;/\;\;\;i \in \{1 , \ldots , N\} \times I\}
   \;\equiv\;  \{(W_{i_a} \;; \chi_{(i_a)})\;\;/\;\;a = 1, \ldots,
    N\;\;/\;\; i_a \in I\}$$

We note that if some $p \in \mE$ lies in $1$ chart $(Q_j, \{y^A
_{(j)} \}_{1 \leq A \leq n+1}) \in \mQ$, then it lies in
$N-W_{i_a}$'s and $N-\chi_{(i_a)}^{-1}(\mB)$'s of $\mW$ \&
$\mW_{\B}$ respectively.

\medskip \noindent \textbf{Step 3: The family $\{f_{i_a}\}_{i_a \in \{1,
 \ldots, N\} \times I}$ of smooth "Bell" functions on $\mE$}.

In the sequel we shall need to use the partition functions' tool
to obtain a global metric structure. Deviating from the standard
approach, we do \underline{not} introduce a partition subordinate
to a locally finite cover, but instead we present constructively a
family of functions with similar properties (except for the
normalization $\sum f_i=1$).

This novelty owes to the fact that we will need to use the
explicit form of these functions and their crucial property of
being real analytic within the interior of their supports.

The "bell functions" characterization owes to the form of their
graph.

\begin{statement} \label{Partitions} There exists a family $\{f_{i_a}\}
_{i_a \in \{1, \ldots, N\} \times I}$ of $C^{\infty}$
non-negative functions on $\mE$, with properties:
                    $${f}_{i_a}\;\in \;C^{\infty}
            (\mE \larrow \R \cap [0,+\infty))\;\;\;\;\;\&\;\;\;\;\;
 supp({f}_{i_a}) \subseteq W_{i_a} \;,\;\;\;\;\forall\;\;\;[W_i]\; \in \; \mW$$

  $$and\;\;\;\;\;\;\;\;\;\;\; \forall\;\;p
  \in \mE,\;\;\;\;\sum_{i_a \in \{1, \ldots, N\} \times I} {f}_{i_a}(p)\;>\;0
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$

and the $f_{i_a}$'s are \underline{real analytic} within the set
they are strictly positive
             $$\{p \in \mE \;/\; f_{i_a}(p) > 0 \} \equiv \{f_{i_a} > 0 \}$$

\noindent (which is open \& coincides with
$int(supp(f_{i_a}))\equiv \overset
{\circ}{\widehat{supp(f_{i_a})}}$):

     $${f}_{i_a}|_{\{f_{i_a} > 0\}}=f_{i_a}|_{\overset{\circ}
     {\widehat{supp(f_{i_a})}}} \;\in \;C^{\infty}
      _{\alpha}(\mE \cap \{f_{i_a} > 0\} \larrow \R)$$
\end{statement}

\begin{remark}
The normalization condition $\underset{i_a \in \{1, \ldots, N\}
\times I}{\sum} {f}_{i_a}(p) = 1$ is not necessary to be
requested, since we do not employ convex functional combinations
in the sequel. We are only interested in the existence of a
collection of smooth functions, locally analytic within their
supports.
\end{remark}

\begin{proof}
\noindent Consider the $C^{\infty}$ non-negative function on
$\R^{n+1}$ :

   $$f \;\; \in \; C^{\infty}(R^{n+1}\larrow [0,+\infty))$$

\[
    f(x)\;:= \left\{
    \begin{array}{lll}
    \exp \bigl( \frac{1}{\|x\|^2 - r^2} \bigr), &
    for\;\;\;
    \|x\| < r\;\\
    \;\;\; \\
    0, &  for\;\;\;\|x \| \geq r .
    \end{array}
    \right.
\]

\noindent ($r > 0$) which satisfies:

      $$\{f > 0\}\;\equiv\;\{x \in \R^{n+1}\;/
      \;f(x)>0\}
       \;=\; \mathbb{B}(0,r) \subsetneqq supp(f)$$
and
      $$supp(f)\;=\;\overline{\mathbb{B}(0,r)}$$

and is real analytic in the interior of its support as
composition of analytic functions enjoying the same property:

    $$f|_{\;\overset{\;\;}{\{f>0\}}}\;=\;f|_{\;\overset{\;\;}{\mathbb{B}
    (0,r)}}\;\; \in \; C^{\infty}_{\alpha}(\mathbb{B}(0,r) \larrow \R)$$

(the \emph{Analytic Continuation Principle} implies that $f$ can
\underline{never} be analytic out of $\mathbb{B}(0,r)$, provided
that

   $$(D^{a_1+...+a_s}f)(x)\;|_{x \rightarrow
    \partial( \mathbb{B}(0,r))}
   \;=\;0\;,\;\;\;\;\;\forall\;a=(a_1,...,a_s) \in \N^s$$

see also \cite{Nar}.)

\noindent Recall now the cover $\mW$ of $W$-patches and its
refinement $\mW_{\;\mathbb{B}} \subset \mW$ of inversed images of
open balls and define a family of smooth real functions on (the
analytic manifold) $\mE$ as:

    $$f_{i_a}\;\; \in \;C^{\infty}(\mE \larrow \R)\;,
    \;\;\;\;\;\;\;\;\forall\;\;\; i_a\; \in \{1, \ldots, N\} \times I\;\;\;\;\;\;:$$

\begin{align} \label{Partition Functions}
    p \; \longmapsto \; f_{i_a}(p) := \left\{
    \begin{array}{lll}
    \exp \bigl( \frac{1}{\|\chi_{i_1}(p)\|^2 - r_i^2} \bigr)\;, &
    \;\;\;\forall\;\;\;\;\;p \; \in \; \chi_{i_1}^{-1}(\mB
    \;\subseteq W_{i_a}\\
    \;\;\; \\
    0\;, & \;\;\; \forall\;\;\;\;\;p \; \in \;  \mE \smallsetminus
     \chi_{i_1}^{-1}(\mB)\; .
    \end{array}
    \right.
\end{align}

\noindent Since $\mE$ is real analytic, we have

    $$f_{i_a}|_{\overset{\;\;}{\{f_{i_a} > 0\}}}\;=\;f_{i_a}
    |_{\;\chi_{i_a}^{-1}(\mB)}\;\;\in \;C^{\infty}_{\alpha}
    (\chi_{i_a}^{-1}(\mB) \larrow \R)$$

and the supports of the $f_{i_a}$'s (as well as their interior)
form a locally finite cover of $\mE$:

   $$supp(f_{i_a})\;=\;\chi_{i_1}^{-1}\bigl(\overline{\mB)}\bigr)\;
    \subseteq \; W_{i_a}\;,$$

    $$\bigcup_{i_a \in \;\{1, \ldots, N\} \times I} supp(f_{i_a})
    \;=\;\bigcup_{i_a \in \;\{1, \ldots, N\} \times I} \{f_{i_a} > 0\}\;=\;
    \mE\;\;\;\;\;\;\;\;\; \&$$

    $$\sum_{i_a \in \;\{1, \ldots, N\} \times I}f_{i_a}\;>\;0$$

and this ends the proof of the \emph{Statement}.
\end{proof}



\noindent \textbf{Step 4: The global smooth \& locally analytic
metric structure $g_{\mE}$ on $\mE$}.


\noindent We can define a global section $g_{\mE} \in
\Gamma(T^*\mE \bigotimes T^*\mE)$ as the product metric on $\mM_1
\times \ldots \times \mM_m \times \mF$: let $\psi^{(i_a)} \in
C^{\infty}_{\alpha}(W_{i_a} \larrow \R)$ be any analytic function
on $W_{i_a}$ (the $\psi^{(i_a)}$'s will be specified later so
that this metric to give an Einstein space geometry, using the
local \ref{C-M-D-R} \emph{Theorem}). First, we define a metric on
$\mM$ as the product metric on $\mM_1 \times \ldots \times \mM_m$:

   $$\gM\;:=\;g_{\mM_1} \circ (d pr_{\mM_1} \times d pr_{\mM_1}) +
   \ldots + \;g_{\mM_m} \circ (d pr_{\mM_m} \times d pr_{\mM_m})$$

where $pr_{\mM_k}\;:\;\mE \larrow \mM_k$ is the projection on the
$k$-factor. Now, we may set

  $$g_{\mE}\;:=\gM \circ (d\pi \times d\pi)\;+\;\Bigl(\underset{1 \leq
   a \leq N}{\sum_{i_a \in I}}f_{i_a}\; \psi^{(i_a)}\;dx_{(i_a)}^{n+1}
     \otimes dx_{(i_a)}^{n+1}\Bigr) \circ (d pr_{\mF} \times d pr_{\mF})$$

\noindent Introducing the functions:

   $$\breve{\psi}^{(i_a)}:=\;f_{i_a}\;\psi^{(i_a)}\;\;\;\in\;
   C^{\infty}(\mE \larrow \R)$$

(each $f_{i_a}$ given by \eqref{Partition Functions}) the metric
on $\mE$ can be rewritten as

\begin{align} \label{Global metric}
    g_{\mE}\;=\gM \circ (d\pi \times d\pi)\;+\;\Bigl(\underset{1 \leq
   a \leq N}{\sum_{i_a \in I}} \breve{\psi}^{(i_a)}\;dx_{(i_a)}^{n+1}
     \otimes dx_{(i_a)}^{n+1}\Bigr) \circ (d pr_{\mF} \times d pr_{\mF})
\end{align}
or
  $$ g_{\mE} = \underset{1 \leq
   k \leq m}{\sum} g_{\mM_k} \circ (dpr_{\mM_k} \times dpr_{\mM_k})\;
    +\;\Bigl(\underset{1 \leq
   a \leq N}{\sum_{i_a \in I}} \breve{\psi}^{(i_a)}\;dx_{(i_a)}^{n+1}
     \otimes dx_{(i_a)}^{n+1}\Bigr) \circ (d pr_{\mF} \times d pr_{\mF})$$

\noindent Recall now the \emph{Fundamental Theorem of Riemannian
Geometry} and equip $(\mE\;,g_{\mE})$ with the respective unique
Levi-Civita linear connection which we denote by $\nE$. Thus, the
the triple
           $$(\mE\;;\nE,\gE)$$
is an analytic semi-Riemannian manifold with smooth (\& locally
analytic) metric and the respective metric connection.


If $(D\;;z^A) \in \mA_{\mE}$ is any chart of $\mE$ (not
necessarily product), the presentation $g_{\mE}$ in the local
coordinates of this patch is:

  $$g_{\mE}|_{D} = \{\gM|_{D}(\frac{\partial \pi}{\partial z^A},
   \frac{\partial \pi}{\partial z^B}) + \Bigl(\underset{1 \leq
   a \leq N}{\sum_{i_a \in I}} \breve{\psi}^{(i_a)}
   \frac{\partial (x_{(i_a)}^{n+1}  \circ pr_{\mF}) }{\partial z^A}
  \frac{\partial (x_{(i_a)}^{n+1}  \circ pr_{\mF}) }{\partial z^B} \Bigr)|_{D}\}
     dz^{A} \otimes dz^{B}$$

having used that
              $$d\pi(\frac{\partial}{\partial z^A})
               \;=\;\frac{\partial \pi}{\partial z^A}$$
and
     $$dx_{(i_a)}^{n+1}\bigl(d  pr_{\mF}(\frac{\partial}{\partial z^A})\bigr)
     \;=\;d(x_{(i_a)}^{n+1} \circ pr_{\mF})(\frac{\partial}{\partial z^A})
     \;=\;\frac{\partial(x_{(i_a)}^{n+1} \circ pr_{\mF})}{\partial z^A}$$

Taking $D = Q_j$, the form of the product patch implies the
following diagonalized form of $\gE|_{Q_j}$ in the respective
coordinates:

  $$g_{\mE}|_{Q_j} \;=\; \{\gM(\frac{\partial \pi}{\partial y_{(j)}^a},
   \frac{\partial \pi}{\partial y_{(j)}^b})|_{Q_j}\}\;dy_{(j)}^{a} \otimes
   dy_{(j)}^{b}\;+\hspace{140pt}$$

   $$\;\hspace{50pt} +\; \{\Bigl(\underset{1 \leq
   a \leq N}{\sum_{i_a \in I}} \breve{\psi}^{(i_a)} \Bigl(
   \frac{\partial (x_{(i_a)}^{n+1}  \circ pr_{\mF}) }{\partial y_{(j)}^{n+1}}
   \Bigr)^2 \Bigr)|_{Q_j}\}\; dy_{(j)}^{n+1} \otimes dy_{(j)}^{n+1}\;\;,
   \;\;\;\;\;1 \leq a, b \leq n$$

for all $Q_j$'s in $\mQ$. Thus, the components of $g_{\mE}
|_{Q_j}$ are

\begin{align} \label{metric components 1}
            [\gE^{(j)}]_{ab}\;:=\;\gM(\frac{\partial \pi}{\partial y_{(j)}^a},
   \frac{\partial \pi}{\partial y_{(j)}^b})|_{Q_j}\;\;,
   \;\;\;\;\;1 \leq a, b \leq n
\end{align}
and
\begin{align}  \label{metric components 2}
    [\gE^{(j)}]_{(n+1)(n+1)}\;:=\;\{\underset{1 \leq
   a \leq N}{\sum_{i_a \in I}} \breve{\psi}^{(i_a)}
   \Bigl( \frac{\partial (x_{(i_a)}^{n+1}
      \circ pr_{\mF}) }{\partial y_{(j)}^{n+1}}
       \Bigr)^2\} |_{Q_j}
\end{align}
and they are all analytic functions, under the assumption for the
$\psi^{(i_a)}$'s:

   $$ [\gE^{(j)}]_{(n+1)(n+1)}\;\;\;\& \;\;\; [\gE^{(j)}]_{ab} \;\;\in
   \;C^{\infty}_{\alpha}(Q_j \larrow \R)$$

\noindent \textbf{Step 5: The requested Curvature condition on
$\mE$ expressed in terms of local coordinates of the cover $\mQ$
using the local results}.

\noindent The condition $\mE$ to be an $n+1$-dimensional Einstein
manifold, with respect to the semi-Riemannian structure
$(\nE,\gE)$ (Def. \ref{Einst})

\begin{align} \label{Globally Einstein}
    Ric^{\nE}(\gE)=\frac{2 \Lambda}{n-1}\;\gE
\end{align}

is equivalent to the condition this equation to hold within all
the local coordinated open submanifolds of a cover of $\mE$, say
$\mQ$:

\begin{align} \label{Locally Einstein}
    \;\;\;\;\;\;\;Ric^{\nE|_{Q_j}}(\gE|_{Q_j})=
    \frac{2 \Lambda}{n-1}\;(\gE|_{Q_j})
    \;\;,\;\;\;\;\;\;\;\forall\;\;j  \in I
\end{align}

and in the local coordinates $\{y_{(j)}^A\}_{1 \leq A \leq n+1}$
of each $Q_j$ the previous relation reads

   $$[R_{\mE}^{(j)}]_{AB}\;dy_{(j)}^A \otimes dy_{(j)}^B=
   \frac{2 \Lambda}{n-1}\;[\gE^{(j)}]_{AB}\;dy_{(j)}^A \otimes
   dy_{(j)}^B
    \;\;,\;\;\;\;\;\;\;\;\forall\;\;j  \in I$$

where $[\gE^{(j)}]_{AB}$ are given by \eqref{metric components 1}
\& \eqref{metric components 2} and $[R_{\mE}^{(j)}]_{AB}$ are the
components of the Ricci tensor of curvature, calculated by (due
to the vanishing \emph{Torsion} of $\nE$):
              $$[{R_{\mE}^{(j)}}]_{AB}\;=\;
     \partial_{M} {\Gamma}_{(j)\;AB}^{M} + {\Gamma}_{(j)\;AB}^{M}
      {\Gamma}_{(j)\;MN}^{N} -\partial_{B}
     {\Gamma}_{(j)\;AM}^{M}-{\Gamma} _{(j)\;AN}^{M}{\Gamma}_{(j)\;MB}^{N}$$
and
       $${R}ic^{\nE|_{Q_{j}}}(\gE|_{Q_j})\;=\;{R}ic^{\nE|_{Q_{j}}}
       (\frac{\partial}{\partial y_{(j)}^{A}},\;\frac{\partial}
       {\partial y_{(j)}^{B}})\;dy_{(j)}^A \otimes dy_{(j)}^B
        \equiv [{R_{\mE}^{(j)}}]_{AB} \;dy_{(j)}^A \otimes
   dy_{(j)}^B$$

where the Christoffel symbols are given by

      $${\Gamma}_{(j)\;AB}^{C}\;=\;\frac{1}{2}\;[\gE^{(j)}]^{CM}\bigl(
    -\;\partial_{M} ([\gE^{(j)}]_{AB})\;+\;\partial_{A}([\gE^{(j)}]_{BM})\;+\;
     \partial_{B} ([\gE^{(j)}]_{MA})\bigr)$$

\noindent \emph{Corollary} \ref{inclusion embedding plus
criterion} implies that, since the functions $[\gE^{(j)}]_{ab}$ of
\eqref{metric components 1} are analytic, there \underline{do}
\underline{exist} \underline{analytic} \underline{functions}
$[\gE^{(j)}]_{(n+1)(n+1)},\; \forall\; j\;\in I$ (at $1$ point at
least, say $0\equiv y^A_{(j)}(p) \in \R^{n+1}$) such that
equation \eqref{Locally Einstein} to hold on every $Q_j$.

\begin{remark}
\emph{The objective pursued to the rest of the proof is to show
that $\gE$ can indeed be fixed through the undetermined
$\psi^{(i_a)}$'s living in the $W_{i_a}$'s of the underlying cover
$\mW$, so that the local embeddings on every $Q_j$ of $\mQ$ to
hold, and therefore the whole $\mE$ to be an Einstein manifold.}
\end{remark}


\noindent \textbf{Step 6: Fixing the Einstein space metric
$g_{\mE}$ through the unspecified functions $\psi^{(i_a)} \in
C^{\infty}_{\alpha}(W_{i_a} \larrow \R)$.}

The topological dimensionality arguments of \emph{Step 2} imply
that no point of $\mE$ lies in more than $n+2$ charts $(Q_j,
\{y^A_{(j)}\}_{1 \leq A \leq n+1})$ of $\mQ$.

On every $Q_j$ relation \eqref{metric components 2} gives an
equation that will be used to determine the $\psi^{(i_a)}$'s in
terms of the $[\gE^{(j)}]_{(n+1)(n+1)}$'s.

    $$\underset{\underbrace{\hspace{65pt}}}{[\gE^{(j)}]_{(n+1)(n+1)}}
    \;=\;\underbrace{\{\underset{1 \leq
   a \leq N}{\sum_{i_a \in I}}\;f_{i_a}
   \Bigl( \frac{\partial (x_{(i_a)}^{n+1}
      \circ pr_{\mF}) }{\partial y_{(j)}^{n+1}}
       \Bigr)^2\} |_{Q_j}}\; \psi^{(i_a)}$$
       $${\Phi_{(j)}}\hspace{120pt}\Theta^{(i_a)}_{(j)}\hspace{60pt}$$

This is a linear functional equation with analytic coefficients:

\begin{align} \label{linear eq}
             \Phi_{(j)}\; =\; \underset{i_a \in I}{\sum}\{
           \Theta^{(i_1)}_{(j)}\;\psi^{(i_1)}+ \ldots + \Theta^{(i_a)}_{(j)}
           \;\psi^{(i_a)}+\ldots + \Theta^{(i_N)}_{(j)}\;\psi^{(i_N)}\}
\end{align}

and the $\psi^{(i_a)}$-variables are all linearly independent,
since the coordinates of the patches are chosen distinct,
although in the coefficients there exist the same $f_{i_a}$'s.
Notwithstanding, these functions are distinguished by the
different local presentations on the patches.

\noindent Consider the domain of $\mE$ that appear the maximum
$n+2$ finite-many overlaps of the $Q_j$'s of the $\mQ$-cover:

  $$\overset{n+2}{\underset{r =1}{\bigcap}}Q_{j_r}\;=\;Q_{j_1} \bigcap \; \ldots
   \; \bigcap Q_{j_{n+2}} \;\subseteq \; \mE$$

If we wish eq. \eqref{Locally Einstein} to hold on this
intersection, we must request the equations \eqref{linear eq} to
hold on all the $Q_j$'s simultaneously. This gives rise to a
system of linear (in the $\psi^{(i_a)}$'s) equations

\[
    (\Sigma1) \;\; \left\{
    \begin{array}{lll}
     \Phi_{(j_1)}\; =\; \overset{n+2}{\underset{i_a = 1}{\sum}}\{
           \Theta^{(i_1)}_{(j_1)}\;\psi^{(i_1)}+ \ldots + \Theta^{(i_a)}_{(j_1)}
       \;\psi^{(i_a)}+\ldots + \Theta^{(i_N)}_{(j_1)}\;\psi^{(i_N)}\}\\
     \;\;\; \vdots \hspace{180pt} \vdots \\
    \Phi_{(j_{n+2})}\; =\; \overset{n+2}{\underset{i_a = 1}{\sum}}\{
           \Theta^{(i_1)}_{(j_{n+2})}\;\psi^{(i_1)}+ \ldots +
           \Theta^{(i_a)}_{(j_{n+2})}
           \;\psi^{(i_a)}+\ldots + \Theta^{(i_N)}_{(j_{n+2})}\;\psi^{(i_N)}\}
    \end{array}
    \right.
\]

which admits \underline{infinite} $N$-tuples of solutions
$\psi^{(i_a)}$ on each $Q_j \bigcap \overset{n+2}{\underset{r'
=1}{\bigcap}}Q_{j_{r'}}$, if we set $N \geq 2$.

\noindent We wish to extend the $\psi^{(i_a)}$-solutions outside
of $\overset{n+2}{\underset{r =1}{\bigcap}}Q_{j_{r}}$ on
$\overset{n+2}{\underset{r =1}{\bigcup}}Q_{j_{r}}$ and finally on
the whole $\mE$. We follow a stepwise procedure, to illustrate
the method.

First, we extend the $\psi^{(i_a)}$'s on the sets $$Q_{j_2}
      \bigcap \; \ldots \; \bigcap Q_{j_{n+2}}\;,\;\;\;\; \ldots\;\;,
       \;\;\;\; Q_{j_1} \bigcap \; \ldots \; \bigcap \widehat{Q_{j_s}}
        \bigcap \ldots \; \bigcap Q_{j_{n+2}}\;,\;\; \ldots\;,$$
         $$Q_{j_1} \bigcap \; \ldots \; \bigcap Q_{j_{n+1}}$$

where the $(\widehat{\;\;})$-notation indicates that the
$Q_{j_s}$ is omitted. These are

   $$\Bigl(^{n+2}_{n+1}\Bigr)\;=\;\frac{(n+1)!\;(n+2)}{(n+1)!}\;=\;n+2$$

many sets. If we wish the $\psi^{(i_a)}$'s to be definable on
these, we must additionally to $(\Sigma1)$ request the following
equations to hold (setting any sufficient value $N \in \N$ so
that to have more $\psi^{(i_a)}$'s than equations)


\[
    (\Sigma2) \;\; \left\{
    \begin{array}{lll}
     \Phi_{(j_1)}\; =\; \underset{[\B_i] \nsubseteq Q_{j_s}}{\overset{n+2}{\underset
     {i_a = 1}{\sum}}}\{
           \Theta^{(i_1)}_{(j_1)}\;\psi^{(i_1)}+ \ldots + \Theta^{(i_a)}_{(j_1)}
       \;\psi^{(i_a)}+\ldots + \Theta^{(i_N)}_{(j_1)}\;\psi^{(i_N)}\}\\
     \;\;\; \vdots \hspace{180pt} \vdots \\
     \Phi_{(j_{s-1})}\; =\; \underset{[\B_i] \nsubseteq Q_{j_{s}}}{\overset{n+2}
     {\underset{i_a = 1}{\sum}}}\{
           \Theta^{(i_1)}_{(j_{s-1})}\;\psi^{(i_1)}+ \ldots + \Theta^{(i_a)}_{(j_{s-1})}
       \;\psi^{(i_a)}+\ldots +
       \Theta^{(i_N)}_{(j_{s-1})}\;\psi^{(i_N)}\}\\
     \Phi_{(j_{s+1})}\; =\; \underset{[\B_i] \nsubseteq Q_{j_s}}
     {\overset{n+2}{\underset{i_a = 1}{\sum}}}\{
           \Theta^{(i_1)}_{(j_{s+1})}\;\psi^{(i_1)}+ \ldots + \Theta^{(i_a)}_{(j_{s+1})}
       \;\psi^{(i_a)}+\ldots +
       \Theta^{(i_N)}_{(j_{s+1})}\;\psi^{(i_N)}\}\\
     \;\;\; \vdots \hspace{180pt} \vdots\\
    \Phi_{(j_{n+2})}\; =\;\underset{[\B_i] \nsubseteq Q_{j_s}}
    {\overset{n+2}{\underset{i_a = 1}{\sum}}}\{
           \Theta^{(i_1)}_{(j_{n+2})}\;\psi^{(i_1)}+ \ldots +
           \Theta^{(i_a)}_{(j_{n+2})}
           \;\psi^{(i_a)}+\ldots + \Theta^{(i_N)}_{(j_{n+2})}\;\psi^{(i_N)}\}
    \end{array}
    \right.
\]

\noindent The extension of solutions $\psi^{(i_a)}$ from
$\overset{n+2}{\underset{r =1}{\bigcap}}Q_{j_r}$ to $Q_{j_1}
\bigcap \; \ldots \; \bigcap \widehat{Q_{j_s}} \bigcap \ldots \;
\bigcap Q_{j_{n+2}}$ ($1 \leq s \leq n+2$) has the following
meaning:

Requesting the system $(\Sigma1)$ to hold on $\overset{n+2}
{\underset{r =1}{\bigcap}}Q_{j_r}$, nothing guarantees that there
exist solutions $\psi^{(i_a)}$ given by $(\Sigma2)$ on the
$Q_{j_1} \bigcap \; \ldots \; \bigcap \widehat{Q_{j_s}} \bigcap
\ldots \; \bigcap Q_{j_{n+2}}$'s, which, if restricted, coincide
with that of $\overset {n+2}{\underset{r =1}{\bigcap}}Q_{j_r}$.
Consequently, requesting both $(\Sigma1)$ \& $(\Sigma2)$ to hold
on $\overset {n+2}{\underset{r =1}{\bigcap}}Q_{j_r}$ and merely
$(\Sigma2)$ on $Q_{j_1} \bigcap \; \ldots \; \bigcap
\widehat{Q_{j_s}} \bigcap \ldots \; \bigcap Q_{j_{n+2}}$, we
obtain the requested compatibility of solutions.

Continuing, we can extend the $\psi^{(i_a)}$'s on the
$\Bigl(^{n+1}_{\;\;n}\Bigr)=\frac{1}{2}(n+1)(n+2)$ sets

  $$Q_{j_3} \bigcap \; \ldots \; \bigcap Q_{j_{n+2}}\;\;,\;\ldots\;,$$

  $$Q_{j_1} \bigcap \; \ldots \; \bigcap \widehat{Q_{j_s}} \bigcap
   \;\; \ldots \;\; \bigcap \widehat{Q_{j_k}} \bigcap \ldots \;
   \bigcap Q_{j_{n+2}}\;,\;\;\ldots$$

  $$\;\ldots\;\;\;,\;Q_{j_1} \bigcap \; \ldots \; \bigcap
  Q_{j_{n}}$$

where $1 \leq s < k \leq n+2$, requesting the additional equations

\[
    (\Sigma3) \;\; \left\{
    \begin{array}{lll}
    \Phi_{(j_{r})}\; =\;\underset{[\B_i] \nsubseteq Q_{j_s} \; \bigcup \; Q_{j_k}}
    {\overset{n+2}{\underset{i_a = 1}{\sum}}}\{
    \Theta^{(i_1)}_{(j_{r})}\;\psi^{(i_1)}+ \ldots + \Theta^{(i_a)}_{(j_{r})}
           \;\psi^{(i_a)}+\ldots +
           \Theta^{(i_N)}_{(j_{r})}\;\psi^{(i_N)}\}\\
    \hspace{230pt}1 \leq r \leq n+2\;,\;\;\;\;\;r \neq s, k
    \end{array}
    \right.
\]

Continuing inductively, we can determine the $\psi^{(i_a)}$'s on
$Q_{j_1}$, . . . , $Q_{j_{n+2}}$, and thus on the whole $Q_{j_1}
\bigcup\; \ldots \; \bigcup Q_{j_{n+2}}$.

\noindent Hence, the final set of equations we request to hold so
that to extend the solutions as previously described is


\[
    (\Sigma1) \;\; \left\{
    \begin{array}{lll}
    \Phi_{(j_{r})}\; =\;\overset{n+2}{\underset{i_a = 1}{\sum}}\{
    \Theta^{(i_1)}_{(j_{r})}\;\psi^{(i_1)}+ \ldots + \Theta^{(i_a)}_{(j_{r})}
           \;\psi^{(i_a)}+\ldots +
           \Theta^{(i_N)}_{(j_{r})}\;\psi^{(i_N)}\}\\
    \hspace{250pt} 1 \leq r \leq n+2
    \end{array}
    \right.
\]
on $Q_{j_1} \bigcap\; \ldots \; \bigcap Q_{j_{n+2}}$
\[
    (\Sigma2) \;\; \left\{
    \begin{array}{lll}
    \Phi_{(j_{r})}\; =\;\underset{[\B_i] \nsubseteq Q_{j_s}}
    {\overset{n+2}{\underset{i_a = 1}{\sum}}}\{
    \Theta^{(i_1)}_{(j_{r})}\;\psi^{(i_1)}+ \ldots + \Theta^{(i_a)}_{(j_{r})}
           \;\psi^{(i_a)}+\ldots +
           \Theta^{(i_N)}_{(j_{r})}\;\psi^{(i_N)}\}\\
    \hspace{240pt} 1 \leq r \neq s \leq n+2
    \end{array}
    \right.
\]
on $Q_{j_1} \bigcap \; \ldots \; \bigcap \widehat{Q_{j_s}}
        \bigcap \ldots \; \bigcap Q_{j_{n+2}}$
\[
    (\Sigma3) \;\; \left\{
    \begin{array}{lll}
    \Phi_{(j_{r})}\; =\;\underset{[\B_i] \nsubseteq Q_{j_s} \; \bigcup \; Q_{j_k}}
    {\overset{n+2}{\underset{i_a = 1}{\sum}}\{
    \Theta^{(i_1)}_{(j_{r})}}\;\psi^{(i_1)}+ \ldots + \Theta^{(i_a)}_{(j_{r})}
           \;\psi^{(i_a)}+\ldots +
           \Theta^{(i_N)}_{(j_{r})}\;\psi^{(i_N)}\}\\
    \hspace{205pt}1 \leq r \leq n+2\;,\;\;\;\;\;r \neq s, k
    \end{array}
    \right.
\]
on $Q_{j_1} \bigcap \; \ldots \; \bigcap \widehat{Q_{j_s}} \bigcap
   \;\; \ldots \;\; \bigcap \widehat{Q_{j_k}} \bigcap \ldots \;
   \bigcap Q_{j_{n+2}}$

   $$\hspace{80pt} \vdots$$
   $$\hspace{80pt} \vdots$$

\[
    (\Sigma(n+2)) \;\; \left\{
    \begin{array}{lll}
    \Phi_{(j_{r})}\; =\;\underset{[\B_i] \nsubseteq Q_{j_1} \bigcap \; \ldots \;
     \bigcap \widehat{Q_{j_r}}
        \bigcap \ldots \; \bigcap Q_{j_{n+2}}}
    {\overset{n+2}{\underset{i_a = 1}{\sum}}\{
    \Theta^{(i_1)}_{(j_{r})}\;\psi^{(i_1)}+ \ldots +} \Theta^{(i_a)}_{(j_{r})}
           \;\psi^{(i_a)}+\ldots +
           \Theta^{(i_N)}_{(j_{r})}\;\psi^{(i_N)}\}\\
    \hspace{245pt} r = 1, \ldots , n+2
    \end{array}
    \right.
\]
on $Q_{j_1}$, . . . , $Q_{j_{n+2}}$.

\noindent The total number of all the equations $(\Sigma1)$,
$(\Sigma2)$, . . . , $(\Sigma(n+2))$ is $M = \Bigl(^{n+2}_{n+2}
\Bigr) + \Bigl(^{n+2}_{n+1} \Bigr) + \ldots +
\Bigl(^{n+2}_{\;\;1} \Bigr)$, that is

   $$M\;=\;2n \;+\; 5 \;+\; \overset{n}{\underset{t = 2}{\sum}}\;
   \frac{(n+3-t)\cdots(n+2)}{t!}$$
or
   $$M\;=\;2(\overset{m}{\underset{k = 1}{\sum}}n_k)
   \;+\; 5 \;+\; \overset{\overset{m}{\underset{k = 1}{\sum}}n_k}
   {\underset{t = 2}{\sum}}\;
   \frac{(\overset{m}{\underset{k = 1}{\sum}}n_k + 3-t)\cdots
   (\overset{m}{\underset{k = 1}{\sum}}n_k + 2)}{t!}$$
Thus, setting
          $$N\;\equiv\;M\;+\;1$$


we obtain more $\psi^{(i_a)}$'s than equations, thus, there do
exist solutions as desired so that the curvature condition given
by \emph{eq.} \eqref{Locally Einstein} to hold on the union
$Q_{j_1}\bigcup\; \ldots \; \bigcup Q_{j_{n+2}}$.

\noindent We claim that in fact we have specified the metric
$\gE$ through the $\psi^{(i_a)}$'s on the whole $\mE$. If
$\bigcup Q_j$, $\bigcup Q_{j'}$ are $2$ unions of $Q_j$'s
overlapping on at least $1$ $Q_{j_0}$, then the metrics
$\gE|_{\;\bigcup Q_j}$, $\gE|_{\;\bigcup Q_{j'}}$ coincide on
$\bigcup Q_{j'} \bigcap \bigcup Q_{j'} \subseteq Q_{j_0}$,
provided that we have requested the equation

  $$\Phi_{(j_{0})}\; =\;\underset{[\B_i] \subseteq Q_{j_0}}
    {\overset{n+2}{\underset{i_a = 1}{\sum}}}\{
    \Theta^{(i_1)}_{(j_{0})}\;\psi^{(i_1)}+ \ldots + \Theta^{(i_a)}_{(j_{0})}
    \;\psi^{(i_a)}+\ldots + \Theta^{(i_N)}_{(j_{0})}\;\psi^{(i_N)}\}$$

to hold on both systems solved on $\bigcup Q_{j}$ and $\bigcup
Q_{j'}$.


\medskip \noindent \textbf{Step 7: The isometry \& $C^{\infty}_{\alpha}$
 embedding conditions.}

Having shown that $\gE$ given by \eqref{Global metric} does exist
so that relation \eqref{Globally Einstein} to hold on $\mE$, the
isometry condition holds trivially by restriction of the product
metric on the submanifolds $\mM_k$:

     $$\gE|_{\;\mM_k}\; = \; g_{\mM_k} \circ (dpr_{\mM_k} \times dpr_{\mM_k})
      \; \equiv g_{\mM_k}
     \;\; \in\; \Gamma(T^*\mM_k \bigotimes T^*\mM_k)$$

if any $\xi$, $\eta$ $\in T \mM_k$ are applied. The Riemannian
connection $\nE$ of $\mE$ restricted on (the open in the induced
subspace topology) $\mM_k$'s reduce to the same $\nabla^{\mM_k}$
on $\mM_k$
                    $$\nE|_{\;\mM_k}\;=\;\nabla^{\mM_k}$$

which is the Riemannian connection $\nabla^{\mM_k}$ of $\mM_k$
with respect to $\gM$. The trivial $C^{\infty} _{\alpha}
$-analyticity of the inclusion maps $i_{\mM_k}\;:\;\mM_k
\hookrightarrow \mE$ clearly completes the proof.
\end{proof}



\section{Homotopy of Analytic Einstein Global
 Embedding Manifolds.}

\noindent In this section we study Homotopically the possible
topologies that the embedding space $\mE$ can have, in complete
analogy with the respective of \cite{Katz 1}.

The key-property is not the geometry of $\mE$ presented in
\emph{Th.} \ref{Katz 3}, but the product structure of $\mE$.


\begin{proposition}\label{Homotopy} (\textbf{Homotopy of the embedding
 Einstein manifold})

\noindent The Homotopy of the embedding Einstein space
$(\mE;\;\nE, \gE)$ of $\mM_1$, . . . , $\mM_m$ splits in every
dimension as
\begin{align}
          \pi_k(\mE)\; \cong \; \pi_k(\mM_1) \bigoplus \; \cdots \; \bigoplus
           \pi_k(\mM_m) \bigoplus \pi_k(\mF)
\end{align}

for all $k \in \N$ and any analytic manifold $\mF$, $dim(\mF)
\geq 1$.
\end{proposition}

\begin{proof}
The proof is a triviality. We simply recall that the bulk $\mE$
was defined as the product
             $$\mE = \mM_1 \times \; \ldots \; \times \mM_m \times \mF$$

and classical Homotopy theory (e.g. \cite{Bre}, \cite{Hatcher},
\cite{Steen}) implies the requested splitting relation for the
Homotopy of $\mE$.
\end{proof}

\begin{corollary}
If we choose in the case of the embedding codimension $1$, the
fibre $\mF$ to be homotopically equivalent to $\R$ (or the
relatively compact $(0,1)$) or the circle $S^1$:
                      $$\mF \simeq \R\;\;\;\;\; or
                      \;\;\;\;\; \mF \simeq S^1$$

we obtain the following respective Homotopies for $\mE$

        $$\pi_k(\mE)\; \cong \; \pi_k(\mM) \;,\;\;\forall\;k \in \N$$
or
\[
    \pi_k(\mE)\; \cong  \left\{
    \begin{array}{lll}
     \pi_1(\mM_1) \bigoplus \; \cdots \; \bigoplus
           \pi_1(\mM_m) \bigoplus \Z, & for\;\;k = 1\\
    \;\;\; \\
     \pi_k(\mM_1) \bigoplus \; \cdots \; \bigoplus
           \pi_k(\mM_m), &  \forall\;k \in \N \smallsetminus \{1\} .
    \end{array}
    \right.
\]

\end{corollary}

\begin{remark}
In the proof of \ref{Katz 3}, $\mE$ was defined as a product
manifold for technical reasons, providing a bulk containing the
given $\mM$.

Notwithstanding, the direct sum decomposition of its Homotopy
implies the \emph{existence} of "\emph{simple}" topologies for
$\mE$, with respect to those that the $\mM_k$'s may admit.
\end{remark}


\noindent {\bf Acknowledgement.} \textit{The Author is profoundly
indebted to professor Lorenz Magaard at University of Honolulu,
Hawaii for his interest in the present work, P. Krikelis \& A.
Kartsaklis at University of Athens for helpful comments \&
technical support and Ed. Anderson for discussions on the physical
aspects of the embedding results}.

\begin{thebibliography}{45}
\bibitem{Bre} Glen E. Bredon, \emph{Topology and Geometry},
Springer, 1997

\bibitem{Camp} J. E. Campbell, \emph{A Course of Differential
Geometry}, Oxford, Clarendon Press, 1923

\bibitem{DC} Do Carmo, M.P., \emph{Riemannian Geometry},
Birkha\"{u}ser, 1992

\bibitem{Eis} L. Pf. Eisenhart, \emph{Riemannian Geometry},
Princeton University Press, 1964

\bibitem{OG-SH} O. Gron and S. Hervik, \emph{Einstein's
 General Theory of Relativity} (e-book),
http://www.fys.uio.no/~sigbjorh/gravity.pdf, 2004

\bibitem{Hatcher} A. Hatcher, \emph{Algebraic Topology},
  http://www.math.cornell.edu/~hatcher, (e-book), 2000

\bibitem{Hawk} S. W. Hawking \& G. F. R. Ellis \emph{The Large scale structure
 of spacetime}, Cambridge University Press, 1973

\bibitem{Maa} L. Magaard, \emph{Zur Einbettung Riemannscher Raume in Einstein - Raume
 und Konfor - Euclidische Raume}, PhD thesis, (Kiel), 1963

\bibitem{Mun} J. R. Munkres, \emph{Topology, a first course},
Prentice-Hall, Inc., Englewood Cliffs, New Jersey

\bibitem{Nar} R. Narasimhan, \emph{Analysis on Real and Complex
Manifolds}, Masson and Cie, Paris, North Holland, 1968

\bibitem{Pen} R. Penrose, \emph{Techniques of Differential Topology in
Relativity}, SIAM Publ. Philadelphia, 1972

\bibitem{Sa-Wu} R. K. Sachs and H. Wu, \emph{General Relativity for
Mathematicians}, Springer-Verlang, 1977

\bibitem{Steen} N. Steenrod, \emph{The topology of fibre
Bundles}, Princeton University Press, 1972

\bibitem{Stef} H. Stefani, \emph{General Relativity}, 2nd Edition, Cambridge
University Press, 1990

\bibitem{W} P. S. Wesson, \emph{Space-Time-Matter, Modern Kaluza-Klein
Theory}, World Scientific, 2000

\noindent (\textit{\underline{Articles}})

\bibitem{Anders-Lidsey} E. Anderson, J. E. Lidsey, \emph{Embeddings in
Non�Vacuum Spacetimes}, Class.Quant.Grav. 18 (2001) 4831-4844,
ArXiv: gr-qc/0106090

\bibitem{D-R2} F. Dahia, C. Romero, \emph{The embedding of spacetime in $5$
dimensions: An extension of Campbell-Magaard theorem},
J.Math.Phys. 43 (2002) 5804-5814, ArXiv: gr-qc/0109076

\bibitem{D-R3} F. Dahia, C. Romero, \emph{On the embedding of branes in
five-dimensional spaces}, Class.Quant.Grav. (2004) 21, No 4,
927-933, ArXiv: gr-qc/0308056

\bibitem{Davidson-1}A. Davidson, \emph{$\Lambda = 0$ cosmology of a
brane-like universe}, Class. Quantum Grav. 16 (1999) 653 � 659

\bibitem{Deffayet} C. Deffayet, \emph{Cosmology of
Brane-Induced Gravity Models,} International Journal of
Theoretical Physics, Vol. 41, No. 11, November 2002
http://www.kluweronline.com/article

\bibitem{Gogberashvili-1} M. Gogberashvili, \emph{Brane-Universe in Six
Dimensions with Two Times}, Phys.Lett. B484 (2000) 124-128,
ArXiv: hep-ph/0001109

\bibitem{Gogberashvili-2} M. Gogberashvili, \emph{Embedding of the Brane
into Six Dimensions}, J.Math.Phys. 43 (2002) 4886 - 4888, ArXiv:
gr-qc/0202061

\bibitem{Gogberashvili-3} M. Gogberashvili, \emph{Brane Gravity from Bulk
Vector Field}, Phys.Lett. B553 (2003) 284-288, ArXiv:
hep-th/0203090

\bibitem{Grauert} H. Grauert, \emph{On Levi's problem and the imbedding of real
analytic manifolds}, Annals of Mathematics, 68 (47-82)

\bibitem{Gra-Wi} E. Gravanis, S. Willison, \emph{Israel conditions for the
Gauss-Bonnet theory and the Friedmann equation on the brane
universe}, ArXiv: hep-th/0209076

\bibitem{Katz 1} N. I. Katzourakis \emph{"The Global Embedding Problem of Semi-
Riemannian into Einstein Manifolds"}, ArXiv: math-ph/0407067

\bibitem{Katz 2} N. I. Katzourakis \emph{"On the Global Embedding of Spacetime
into Singular $\mE^{\Sigma}$ Einstein Manifolds: Wormholes"},
ArXiv: math.DG/0503129

\bibitem{Langlois1} D. Langlois, \emph{Brane cosmology: an introduction},
Prog.Theor.Phys.Suppl. 148 (2003) 181-212, ArXiv: hep-th/0209261

\bibitem{Langlois2} D. Langlois, \emph{Gravitation and Cosmology in Brane-Worlds},
(Invited plenary talk at the GR17 conference, Dublin, Ireland,
July 2004), ArXiv: gr-qc/0410129

\bibitem{L-R-T-R} J. E. Lidsey, C. Romero, R. Tavakol, S. Rippl,
\emph{On Applications of Campbell's Embedding Theorem},
 Class.Quant.Grav. 14 (1997) 865-879, ArXiv: gr-qc/9907040

\bibitem{Lidsey} J. E. Lidsey, \emph{The Embedding of Superstring Backgrounds
 in Einstein Gravity}, Phys.Lett. B417 (1998) 33, ArXiv: hep-th/9907095

\bibitem{Maartens2} R. Maartens, \emph{Geometry and Dynamics of the
Brane-World}, Reference Frames and Gravitomagnetism, ed. J
Pascual-Sanchez et al. (World Sci., 2001), p. 93-119, ArXiv:
gr-qc/0101059

\bibitem{Maartens1} R. Maartens, \emph{Brane-World Gravity}, Living Rev.
Relativity, 7, (2004), 7, http://www.livingreviews.org/lrr-2004-7

\bibitem{Pavisic} M. Pavisic, \emph{Resource Letter on Geometrical Results
for Embeddings and Branes}, arXiv: gr-qc/0010045

\bibitem{Perez} A. Perez-Lorenzana, \emph{An Introduction to the Brane World},
ArXiv: hep-ph/0406279

\bibitem{Quevedo} F. Quevedo, \emph{Lectures on String/Brane Cosmology},
 arXiv: hep-th/0210292

\bibitem{RS-1} L. Randall, R. Sundrum, \emph{A Large Mass Hierarchy from a
Small Extra Dimension}, Phys.Rev.Lett. 83 (1999) 3370-3373,
ArXiv: hep-ph/9905221

\bibitem{RS-2} L. Randall, R. Sundrum, \emph{An Alternative to
Compactification}, Phys.Rev.Lett. 83 (1999) 4690-4693, ArXiv:
hep-th/9906064

\bibitem{Seahra-1} S. S. Seahra, \emph{Physics in Higher-Dimensional Manifolds},
$\hspace{100pt}$ http://etd.uwaterloo.ca/etd/ssseahra2003.pdf

\bibitem{S-W} S. S. Seahra, P. S. Wesson, \emph{Application of the Campbell-
Magaard theorem to higher-dimensional physics}, Class.Quant.Grav.
20 (2003) 1321-1340, ArXiv: gr-qc/0302015

\bibitem{Shi-Ko} T. Shiromizu, K. Koyama, \emph{Low energy effective theory for
two branes system - Covariant curvature formulation}, Phys.Rev.
D67 (2003) 084022, ArXiv: hep-th/0210066

\bibitem{Ste-Pa} D. A. Steer, M. F. Parry, \emph{Brane Cosmology,
Varying Speed of Light and Inflation in Models With One or More
Extra Dimensions}, Int.J.Th.Phys., Vol. 41, No. 11, Nov. 2002,
http://www.kluweronline.com/article

\bibitem{Wands}  D. Wands, \emph{Brane-world cosmology},

http://www.tech.port.ac.uk/staffweb/wandsd/powerpoint/sussex.pdf

\bibitem{Whit 2} H. Whitney, \emph{Differentiable Manifolds},
Annals of Mathematics, 37 (645-680), 1936
\end{thebibliography}
\end{document}

% ------------------------------------------------------------------------

math0411630.extracted_bib
\begin{thebibliography}{45}
\bibitem{Bre} Glen E. Bredon, \emph{Topology and Geometry},
Springer, 1997

\bibitem{Camp} J. E. Campbell, \emph{A Course of Differential
Geometry}, Oxford, Clarendon Press, 1923

\bibitem{DC} Do Carmo, M.P., \emph{Riemannian Geometry},
Birkha\"{u}ser, 1992

\bibitem{Eis} L. Pf. Eisenhart, \emph{Riemannian Geometry},
Princeton University Press, 1964

\bibitem{OG-SH} O. Gron and S. Hervik, \emph{Einstein's
 General Theory of Relativity} (e-book),
http://www.fys.uio.no/~sigbjorh/gravity.pdf, 2004

\bibitem{Hatcher} A. Hatcher, \emph{Algebraic Topology},
  http://www.math.cornell.edu/~hatcher, (e-book), 2000

\bibitem{Hawk} S. W. Hawking \& G. F. R. Ellis \emph{The Large scale structure
 of spacetime}, Cambridge University Press, 1973

\bibitem{Maa} L. Magaard, \emph{Zur Einbettung Riemannscher Raume in Einstein - Raume
 und Konfor - Euclidische Raume}, PhD thesis, (Kiel), 1963

\bibitem{Mun} J. R. Munkres, \emph{Topology, a first course},
Prentice-Hall, Inc., Englewood Cliffs, New Jersey

\bibitem{Nar} R. Narasimhan, \emph{Analysis on Real and Complex
Manifolds}, Masson and Cie, Paris, North Holland, 1968

\bibitem{Pen} R. Penrose, \emph{Techniques of Differential Topology in
Relativity}, SIAM Publ. Philadelphia, 1972

\bibitem{Sa-Wu} R. K. Sachs and H. Wu, \emph{General Relativity for
Mathematicians}, Springer-Verlang, 1977

\bibitem{Steen} N. Steenrod, \emph{The topology of fibre
Bundles}, Princeton University Press, 1972

\bibitem{Stef} H. Stefani, \emph{General Relativity}, 2nd Edition, Cambridge
University Press, 1990

\bibitem{W} P. S. Wesson, \emph{Space-Time-Matter, Modern Kaluza-Klein
Theory}, World Scientific, 2000

\noindent (\textit{\underline{Articles}})

\bibitem{Anders-Lidsey} E. Anderson, J. E. Lidsey, \emph{Embeddings in
Non�Vacuum Spacetimes}, Class.Quant.Grav. 18 (2001) 4831-4844,
ArXiv: gr-qc/0106090

\bibitem{D-R2} F. Dahia, C. Romero, \emph{The embedding of spacetime in $5$
dimensions: An extension of Campbell-Magaard theorem},
J.Math.Phys. 43 (2002) 5804-5814, ArXiv: gr-qc/0109076

\bibitem{D-R3} F. Dahia, C. Romero, \emph{On the embedding of branes in
five-dimensional spaces}, Class.Quant.Grav. (2004) 21, No 4,
927-933, ArXiv: gr-qc/0308056

\bibitem{Davidson-1}A. Davidson, \emph{$\Lambda = 0$ cosmology of a
brane-like universe}, Class. Quantum Grav. 16 (1999) 653 � 659

\bibitem{Deffayet} C. Deffayet, \emph{Cosmology of
Brane-Induced Gravity Models,} International Journal of
Theoretical Physics, Vol. 41, No. 11, November 2002
http://www.kluweronline.com/article

\bibitem{Gogberashvili-1} M. Gogberashvili, \emph{Brane-Universe in Six
Dimensions with Two Times}, Phys.Lett. B484 (2000) 124-128,
ArXiv: hep-ph/0001109

\bibitem{Gogberashvili-2} M. Gogberashvili, \emph{Embedding of the Brane
into Six Dimensions}, J.Math.Phys. 43 (2002) 4886 - 4888, ArXiv:
gr-qc/0202061

\bibitem{Gogberashvili-3} M. Gogberashvili, \emph{Brane Gravity from Bulk
Vector Field}, Phys.Lett. B553 (2003) 284-288, ArXiv:
hep-th/0203090

\bibitem{Grauert} H. Grauert, \emph{On Levi's problem and the imbedding of real
analytic manifolds}, Annals of Mathematics, 68 (47-82)

\bibitem{Gra-Wi} E. Gravanis, S. Willison, \emph{Israel conditions for the
Gauss-Bonnet theory and the Friedmann equation on the brane
universe}, ArXiv: hep-th/0209076

\bibitem{Katz 1} N. I. Katzourakis \emph{"The Global Embedding Problem of Semi-
Riemannian into Einstein Manifolds"}, ArXiv: math-ph/0407067

\bibitem{Katz 2} N. I. Katzourakis \emph{"On the Global Embedding of Spacetime
into Singular $\mE^{\Sigma}$ Einstein Manifolds: Wormholes"},
ArXiv: math.DG/0503129

\bibitem{Langlois1} D. Langlois, \emph{Brane cosmology: an introduction},
Prog.Theor.Phys.Suppl. 148 (2003) 181-212, ArXiv: hep-th/0209261

\bibitem{Langlois2} D. Langlois, \emph{Gravitation and Cosmology in Brane-Worlds},
(Invited plenary talk at the GR17 conference, Dublin, Ireland,
July 2004), ArXiv: gr-qc/0410129

\bibitem{L-R-T-R} J. E. Lidsey, C. Romero, R. Tavakol, S. Rippl,
\emph{On Applications of Campbell's Embedding Theorem},
 Class.Quant.Grav. 14 (1997) 865-879, ArXiv: gr-qc/9907040

\bibitem{Lidsey} J. E. Lidsey, \emph{The Embedding of Superstring Backgrounds
 in Einstein Gravity}, Phys.Lett. B417 (1998) 33, ArXiv: hep-th/9907095

\bibitem{Maartens2} R. Maartens, \emph{Geometry and Dynamics of the
Brane-World}, Reference Frames and Gravitomagnetism, ed. J
Pascual-Sanchez et al. (World Sci., 2001), p. 93-119, ArXiv:
gr-qc/0101059

\bibitem{Maartens1} R. Maartens, \emph{Brane-World Gravity}, Living Rev.
Relativity, 7, (2004), 7, http://www.livingreviews.org/lrr-2004-7

\bibitem{Pavisic} M. Pavisic, \emph{Resource Letter on Geometrical Results
for Embeddings and Branes}, arXiv: gr-qc/0010045

\bibitem{Perez} A. Perez-Lorenzana, \emph{An Introduction to the Brane World},
ArXiv: hep-ph/0406279

\bibitem{Quevedo} F. Quevedo, \emph{Lectures on String/Brane Cosmology},
 arXiv: hep-th/0210292

\bibitem{RS-1} L. Randall, R. Sundrum, \emph{A Large Mass Hierarchy from a
Small Extra Dimension}, Phys.Rev.Lett. 83 (1999) 3370-3373,
ArXiv: hep-ph/9905221

\bibitem{RS-2} L. Randall, R. Sundrum, \emph{An Alternative to
Compactification}, Phys.Rev.Lett. 83 (1999) 4690-4693, ArXiv:
hep-th/9906064

\bibitem{Seahra-1} S. S. Seahra, \emph{Physics in Higher-Dimensional Manifolds},
$\hspace{100pt}$ http://etd.uwaterloo.ca/etd/ssseahra2003.pdf

\bibitem{S-W} S. S. Seahra, P. S. Wesson, \emph{Application of the Campbell-
Magaard theorem to higher-dimensional physics}, Class.Quant.Grav.
20 (2003) 1321-1340, ArXiv: gr-qc/0302015

\bibitem{Shi-Ko} T. Shiromizu, K. Koyama, \emph{Low energy effective theory for
two branes system - Covariant curvature formulation}, Phys.Rev.
D67 (2003) 084022, ArXiv: hep-th/0210066

\bibitem{Ste-Pa} D. A. Steer, M. F. Parry, \emph{Brane Cosmology,
Varying Speed of Light and Inflation in Models With One or More
Extra Dimensions}, Int.J.Th.Phys., Vol. 41, No. 11, Nov. 2002,
http://www.kluweronline.com/article

\bibitem{Wands}  D. Wands, \emph{Brane-world cosmology},

http://www.tech.port.ac.uk/staffweb/wandsd/powerpoint/sussex.pdf

\bibitem{Whit 2} H. Whitney, \emph{Differentiable Manifolds},
Annals of Mathematics, 37 (645-680), 1936
\end{thebibliography}