text:
\begin{document}

\title{ALMOST PARACONTACT MANIFOLDS}

\author{Galia Nakova}
\address[Nakova]{University of Veliko Turnovo "St. Cyrilend St. Metodius"\\
Faculty of Education,\\
T. Tarnovski 2 str.,\\
5003 Veliko Tarnovo, Bulgaria} \email{gnakova@yahoo.com}

\author{Simeon Zamkovoy}
\address[Zamkovoy]{University of Sofia "St. Kl. Ohridski"\\
Faculty of Mathematics and Informatics,\\
Blvd. James Bourchier 5,\\
1164 Sofia, Bulgaria} \email{zamkovoy@fmi.uni-sofia.bg}

\begin{abstract}
{In this paper eleven basic classes of almost paracontact
manifolds are introduced and some examples are constructed.}

MSC: 53C15, 5350, 53C25, 53C26, 53B30
\end{abstract}

\maketitle \setcounter{tocdepth}{3} \tableofcontents

\section*{Introduction}

As is well-known, in \cite{GH} almost Hermitian manifolds are
classified with respect to the decomposition in subspaces
invariant under the action of the structural group $U(n)$.Thus we
have an adequate framework for several types of almost Hermitian
manifolds, previously defined by a number of authors in terms of
geometric properties which retain some portion of K\"{a}hler
geometry. The previous method was used in \cite{N} for Riemannian
almost product manifolds, and in \cite{GB} for almost complex
manifolds with Norden metric.

The geometry of almost contact manifolds is a natural extension in
the odd dimensional case of almost Hermitian geometry. Similarly,
the geometry of almost contact manifolds with $B$-metric can be
considered as a natural extension in the odd dimensional case of
geometry of almost complex Riemannian. A classification of almost
contact manifolds with $B$-metric with respect to the covariant
derivative of the fundamental tensor of type $(1,1)$ is made in
\cite{GMG}. The authors obtain eleven basic classes of almost
contact manifolds with $B$-metric and construct some examples.

A classification of almost paraHermitian manifolds is made in
\cite{B}.

The authors give examples of the primitive classes, which are
based on general almost paraHermitian structure on the tangent
bundles given in \cite{C}.

A classification of the almost paracontact Riemannian manifolds of
type $(n,n)$ with respect to the covariant derivative of type
$(1,1)$-tensor of the almost paracontact structure is made in
\cite{MS}. The authors consider almost paracontact Riemannian
manifolds of type $(n,n)$ with positive definite Riemannian metric
$g$, which is compatible with almost paracontact structure and it
satisfies the condition $g(\varphi \cdot,\varphi
\cdot)=g(\cdot,\cdot)-\eta (\cdot)\eta (\cdot)$.

The method used in the present paper is analogous of the one used
in \cite{GMG}. We give a classification of the almost paracontact
manifolds with respect to the covariant derivative of the
$(1,1)$-tensor of the almost paracontact structure. We consider
almost paracontact pseudo-Riemannian manifolds with indefinite
metric $g$, which it compatible with almost paracontact structure
and it satisfies the condition \eqref{con}. We obtain eleven basic
classes and construct some examples.

\section{Preliminaries}\label{sec-prelim}
A (2n+1)-dimensional smooth manifold $M^{(2n+1)}$ has an
\emph{almost paracontact structure} $(\varphi,\xi,\eta)$ if it
admits a tensor field $\varphi$ of type $(1,1)$, a vector field
$\xi$ and a 1-form $\eta$ satisfying the  following compatibility
conditions \cite{KW,Zam}:
\begin{eqnarray}
  \label{f82}
    & &
    \begin{array}{cl}
          (i)   & \varphi(\xi)=0,\quad \eta \circ \varphi=0,\quad
          \\[5pt]
          (ii)  & \eta (\xi)=1 \quad \varphi^2 = id - \eta \otimes \xi,
          \\[5pt]
          (iii) & \textrm{let $\mathbb D=Ker~\eta$ be the  horizontal distribution generated
                          by $\eta$, then}
          \\[3pt]
                & \textrm{the tensor field $\varphi$ induces an almost paracomplex structure on}
          \\[3pt]
                &  \textrm{each fibre on $\mathbb D$.}
    \end{array}
\end{eqnarray}

Recall that an almost paracomplex structure on an 2n-dimensional
manifold is a (1,1)-tensor $J$ such that $J^2=1$ and the
eigensubbundles $T^+,T^-$ corresponding to the eigenvalues $1,-1$
of $J$, respectively have equal dimension $n$. The Nijenhuis
tensor $N$ of $J$, given by
$N_{J}(X,Y)=[JX,JY]-J[JX,Y]-J[X,JY]+[X,Y],$ is the obstruction for
the integrability of the eigensubbundles $T^+,T^-$. If $N=0$ then
the almost paracomplex structure is called paracomplex or
integrable.

An immediate consequence of the definition of the almost
paracontact structure is that the endomorphism $\varphi$ has rank
$2n$, $\varphi \xi=0$ and $\eta \circ \varphi=0$ \cite{Zam}.

If a manifold $M^{(2n+1)}$ with $(\varphi,\xi,\eta)$-structure
admits a pseudo-Riemannian metric $g$ such that
\begin{equation}\label{con}
g(\varphi X,\varphi Y)=-g(X,Y)+\eta (X)\eta (Y),
\end{equation}
then we say that $M^{(2n+1)}$ has an almost paracontact metric
structure and $g$ is called \emph{compatible} metric. Any
compatible metric $g$ with a given almost paracontact structure is
necessarily of signature $(n+1,n)$ \cite{Zam}.

Setting $Y=\xi$, we have $\eta(X)=g(X,\xi).$

The fundamental 2-form
\begin{equation}\label{fund}
F(X,Y)=g(\varphi X,Y)
\end{equation}
is non-degenerate on the horizontal distribution $\mathbb D$ and
$\eta\wedge F^n\not=0$.

We have the following \cite{Zam}
\begin{defn}
If $g(X,\varphi Y)=d\eta(X,Y)$ (where
$d\eta(X,Y)=\frac12(X\eta(Y)-Y\eta(X)-\eta([X,Y])$ then $\eta$ is
a paracontact form and the almost paracontact metric manifold
$(M,\varphi,\eta,g)$ is said to be $\emph{paracontact metric
manifold}$.
\end{defn}

For a manifold $M^{(2n+1)}$ with an almost paracontact metric
structure $(\varphi,\xi,\eta,g)$ we can also construct a useful
local orthonormal basis. Let U be a coordinate neighborhood on M
and $X_1$ any unit vector field on U orthogonal to $\xi$. Then
$\varphi X_1$ is a vector field orthogonal to  both X and $\xi$,
and $|\varphi X_1|^2=-1$. Now choose a unit vector field $X_2$
orthogonal to $\xi$, $X_1$ and $\varphi X_1$. Then $\varphi X_2$
is also vector field orthogonal to $\xi$, $X_1$, $\varphi X_1$ and
$X_2$, and $|\varphi X_2|^2=-1$. Proceeding in this way we obtain
a local orthonormal basis $(X_i,\varphi X_i,\xi),i=1...n$ called a
\emph{$\varphi$-basis}.

Hence, an almost paracontact metric manifold
$(M^{2n+1},\varphi,\eta,\xi,g)$ is an odd dimensional manifold
with a structure group $\mathbb U(n,\mathbb R)\times Id$, where
$\mathbb U(n,\mathbb R)$ is the para-unitary group isomorphic to
$\mathbb {GL}(n,\mathbb R)$.

A paracontact structure for which $\xi$ is Killing vector field is
called a \emph{K-paracontact structure}.
\par
Let $\nabla$ be the Levi-Civita connection of the compatible
metric $g$. For all vectors $X,Y,Z \in
T_pM$, $p \in M$, we denote
\begin{equation}\label{4}
F(X,Y,Z)=g((\nabla_X\varphi)Y,Z).
\end{equation}
From \eqref{f82} and \eqref{con} the tensor $F$ has the following
properties:
\begin{equation}\label{5}
\begin{array}{lll}
F(X,Y,Z)=-F(X,Z,Y), \\
F(X,\varphi Y, \varphi
Z)=F(X,Y,Z)+\eta(Y)F(X,Z,\xi)-\eta(Z)F(X,Y,\xi),
\end{array}
\end{equation}
for all vectors $X,Y,Z \in T_pM$.
\par
The following 1-forms are associated with $F$:
\begin{equation}\label{6}
\theta(X)=g^{ij}F(e_i,e_j,X), \quad
\theta^*(X)=g^{ij}F(e_i,\varphi e_j,X), \quad
\omega(X)=F(\xi,\xi,X),
\end{equation}
where $X \in T_pM$, $\{e_i,\xi\}$, $(i=1,\ldots,2n)$ is a basis of
$T_pM$, and $(g^{ij})$ is the inverse matrix of $(g_{ij})$.

\section{The space of covariant derivatives of the structure $\varphi$}\label{sec-2}
\setcounter{equation}{0}

Let $V$ be a $(2n+1)$-dimensional vector space with almost
paracontact structure $(\varphi, \xi, \eta)$ and metric $g$
satisfying \eqref{con}. For an arbitrary $X \in V$ we have
$\varphi^2X=X-\eta(X)\xi \Longleftrightarrow
X=\varphi^2X+\eta(X)\xi$. Hence $V$ admits a decomposition into a
direct sum of vector subspaces
\[
V=\mathbb{D}\oplus \{\xi\},
\]
where $\mathbb{D}=Ker\eta$, $\{\xi\}=(Im\eta)\xi$. Then for an
arbitrary $X \in V$ it follows $X=hX+\eta(X)\xi$, where $X \in
\mathbb{D}, \eta(X)\xi \in \{\xi\}$. Denoting the restrictions of
$g$ and $\varphi$ on $\mathbb{D}$ with the same letters we obtain
an $2n$-dimension almost paracomplex manifold
$(\mathbb{D},\varphi,g)$.
\par
Let $\{e_1,\ldots,e_{2n}\}$ be any basis of $\mathbb{D}$. Then
$\{e_1,\ldots,e_{2n},\xi\}$ is a basis of $V$ and for an arbitrary
$X\in V$ we have $X=X^ie_i+\eta(X)\xi, \, i=1,\ldots,2n$.
\par
We define the linear operators
\begin{equation}\label{2.1}
\begin{array}{ll}
{\mathcal{A}} _{e_i} : V \longrightarrow V: \quad X\longrightarrow
{\mathcal{A}} _{e_i}X, \quad i=1,\ldots,2n; \\ \\
{\mathcal{A}} _\xi : V \longrightarrow \mathbb{D}: \quad X
\longrightarrow {\mathcal{A}} _\xi X,
\end{array}
\end{equation}
having the following properties
\begin{equation}\label{2.2}
g({\mathcal{A}} _{e_i}X,e_j)=-g({\mathcal{A}} _{e_j}X,e_i), \quad i,j=1,\ldots,2n;
\end{equation}
\begin{equation}\label{2.3}
{\mathcal{A}} _{\varphi e_i}X=-\varphi({\mathcal{A}} _{e_i}X)-g({\mathcal{A}} _\xi X,e_i)\xi;
\end{equation}
\begin{equation}\label{2.4}
\eta({\mathcal{A}} _{e_i}X)=-g({\mathcal{A}} _\xi X,\varphi e_i);
\end{equation}
\begin{equation}\label{2.5}
\eta({\mathcal{A}} _\xi X)=0.
\end{equation}
We consider the vector space ${\mathcal{F}}$ of all tensors $F$ of
type $(0,3)$ over $V$, defined by
\begin{equation}\label{2.6}
F(X,Y,Z)=Y^ig({\mathcal{A}} _{e_i}X,Z)+\eta(Y)g({\mathcal{A}} _\xi
X,\varphi Z),
\end{equation}
where ${\mathcal{A}} _{e_i} \, (i=1,\ldots,2n)$ and ${\mathcal{A}}
_\xi$ have the properties \eqref{2.2}$\div$\eqref{2.5}. It is easy
to verify that the tensors $F$ do not depend on the basis of $V$.
Using \eqref{2.6} and \eqref{2.2}$\div$\eqref{2.5} we establish
that the tensors $F \in {\mathcal{F}}$ have the properties
\eqref{5}.
\par
The compatible metric $g$ induces on ${\mathcal{F}}$ an inner
product $<,>$, defined by
\begin{equation}\label{2.7}
<F_1,F_2>=g^{iq}g^{jr}g^{ks}F_1(f_i,f_j,f_k)F_2(f_q,f_r,f_s)
\end{equation}
for $F_1, F_2 \in  {\mathcal{F}}$ and $\{f_1,\ldots,f_{2n+1}\}$ is
a basis of $V$.
\par
The standard representation of the structure group $\mathbb
U(n,\mathbb R)\times Id$ in $V$ induces a natural representation
$\lambda $ of $\mathbb U(n,\mathbb R)\times Id$ in
${\mathcal{F}}$:
\[
(\lambda (a)F)(X,Y,Z)=F(a^{-1}X,a^{-1}Y,a^{-1}Z), \quad a\in
\mathbb U(n,\mathbb R)\times Id, \quad F\in {\mathcal{F}}, \, \, X,Y,Z \in V,
\]
so that
\[
<\lambda (a)F_1,\lambda (a)F_2>=<F_1,F_2>, \quad a\in
\mathbb U(n,\mathbb R)\times Id, \quad F_1, F_2 \in {\mathcal{F}}.
\]
\par
Let $X \in V$. Then for ${\mathcal{A}} _{e_i}X \in V, \,
(i=1,\ldots,2n)$ we have
\begin{equation}\label{2.8}
{\mathcal{A}} _{e_i}X=h({\mathcal{A}}
_{e_i}hX)+\eta(X)h({\mathcal{A}} _{e_i}\xi)+\eta({\mathcal{A}}
_{e_i}hX)\xi+ \eta(X)\eta({\mathcal{A}} _{e_i}\xi)\xi .
\end{equation}
Taking into account \eqref{2.6} and \eqref{2.8} we obtain
\begin{equation}\label{2.9}
\begin{array}{ll}
F(X,Y,Z)=Y^ig(h({\mathcal{A}}
_{e_i}hX),Z)-Z^i\eta(Y)\eta({\mathcal{A}} _{e_i}hX)+Y^i\eta(Z)
\eta({\mathcal{A}} _{e_i}hX) \\ \\
+\eta(X)Y^ig(h({\mathcal{A}}
_{e_i}\xi),Z)+\eta(X)\eta(Z)Y^i\eta({\mathcal{A}} _{e_i}\xi)-
\eta(X)\eta(Y)Z^i\eta({\mathcal{A}} _{e_i}\xi).
\end{array}
\end{equation}
Analogously as \cite{GMG} we define the operators
\[
\begin{array}{l}
p_i: {\mathcal{F}} \longrightarrow {\mathcal{F}}; \quad i=1,2,3,4, \\ \\
p_1(F)(X,Y,Z)=F(hX,hY,hZ); \\ \\
p_2(F)(X,Y,Z)=-\eta(Y)F(hX,hZ,\xi)+\eta(Z)F(hX,hY,\xi); \\ \\
p_3(F)(X,Y,Z)=\eta(X)F(\xi,hY,hZ); \\ \\
p_4(F)(X,Y,Z)=\eta(X)\eta(Y)F(\xi,\xi,hZ)-\eta(X)\eta(Z)F(\xi,\xi,hY).
\end{array}
\]
\begin{lem}\label{l 2.1}
The operators $p_i \, (i=1,2,3,4)$ have the following properties
\[
\begin{array}{cl}
          (i)   & p_i\circ p_i=p_i, \quad i=1,2,3,4; \\ \\

          (ii)  & \displaystyle \sum^{4}_{i=1} p_i=id; \\ \\

          (iii) & p_i\circ p_j=0, \quad i \neq j, \quad i,j=1,2,3,4;
\end{array}
\]
and $p_i \, (i=1,2,3,4)$ commute with $\mathbb U(n,\mathbb R)\times Id$.
\end{lem}
We denote $W_i=Im p_i \quad (i=1,2,3,4)$.
\begin{pro}\label{p 2.1} ( Partial decomposition ) The decomposition
\[
{\mathcal{F}}=W_1\oplus W_2\oplus W_3\oplus W_4
\]
is orthogonal and invariant under the action of the
group $\mathbb U(n,\mathbb R)\times Id$.
\end{pro}
\par
{\bf Proof.} From well known algebraic result and Lemma~\ref{2.1} we obtain
the decomposition ${\mathcal{F}}=W_1\oplus W_2\oplus W_3\oplus W_4$ that is
$\mathbb U(n,\mathbb R)\times Id$ - invariant. By direct
computations using \eqref{2.7} we check that $W_i \bot W_j, \quad
i \neq j, \quad i,j=1,2,3,4$.
\par
From \eqref{2.9}, \eqref{2.4} by explicit calculations we have
\begin{equation}\label{2.10}
\begin{array}{ll}
p_1(F)(X,Y,Z)=F(hX,hY,hZ)=Y^ig(h({\mathcal{A}}_{e_i}hX),Z); \\  \\
p_2(F)(X,Y,Z)=-\eta(Y)F(hX,hZ,\xi)+\eta(Z)F(hX,hY,\xi)=\\
-Z^i\eta(Y)\eta({\mathcal{A}} _{e_i}hX)+Y^i\eta(Z)\eta({\mathcal{A}} _{e_i}hX)= \\ \\
\eta(Y)g({\mathcal{A}} _\xi hX,\varphi Z)-\eta(Z)g({\mathcal{A}} _\xi hX,\varphi Y);\\ \\
p_3(F)(X,Y,Z)=\eta(X)F(\xi,hY,hZ)=\eta(X)Y^ig(h({\mathcal{A}} _{e_i} \xi),Z); \\ \\
p_4(F)(X,Y,Z)=\eta(X)\eta(Y)F(\xi,\xi,hZ)-\eta(X)\eta(Z)F(\xi,\xi,hY)=\\ \\
-\eta(X)\eta(Y)Z^i\eta({\mathcal{A}} _{e_i}\xi)+
\eta(X)\eta(Z)Y^i\eta({\mathcal{A}} _{e_i}\xi),
\end{array}
\end{equation}
where $X,Y,Z \in V$ and $(i=1,\ldots,2n)$. Using \eqref{2.8},
\eqref{2.9}, \eqref{2.10} we obtain
\begin{pro}\label{p 2.2}
Let ${\mathcal{A}} _{e_i} \, (i,\ldots,2n)$ be the linear
operators, defined by \eqref{2.1} and having the properties
\eqref{2.2}, \eqref{2.3}, \eqref{2.4}, \eqref{2.5}. Then for an
arbitrary $F \in {\mathcal{F}}$ and $X,Y,Z \in V$ we have
\[
\begin{array}{clll}
          (i)   & {\mathcal{A}} _{e_i}X=h({\mathcal{A}} _{e_i}hX), \, (i=1,\ldots,2n)\Longleftrightarrow F=p_1F; \\ \\
          (ii)  & {\mathcal{A}} _{e_i}X=\eta({\mathcal{A}} _{e_i}hX)\xi =
          -g({\mathcal{A}} _\xi hX,\varphi e_i)\xi , \,
          (i=1,\ldots,2n)\Longleftrightarrow F=p_2F; \\ \\
          (iii) & {\mathcal{A}} _{e_i}X=\eta(X)h({\mathcal{A}} _{e_i}\xi ), \,
          (i=1,\ldots,2n)\Longleftrightarrow F=p_3F; \\ \\
          (iiii) & {\mathcal{A}} _{e_i}X=\eta(X)\eta({\mathcal{A}} _{e_i}\xi )\xi =
          -\eta(X)g({\mathcal{A}} _\xi \xi,\varphi e_i)\xi, \,
          (i=1,\ldots,2n)\Longleftrightarrow F=p_4F .
\end{array}
\]
\end{pro}
\section{The subspace $W_1$}\label{sec-3}
\setcounter{equation}{0}
From Proposition~\ref{p 2.2} we have
\[
W_1=\left\{F\in {\mathcal{F}}: F=p_1F\Longleftrightarrow
{\mathcal{A}} _{e_i}X=h({\mathcal{A}} _{e_i}hX) \, \, (i=1,\ldots,2n)\right\}.
\]
The condition ${\mathcal{A}} _{e_i}X=h({\mathcal{A}} _{e_i}hX) \, \,
(i=1,\ldots,2n)$ is
equivalent to
\begin{equation}\label{3.1}
\begin{array}{l}
{\mathcal{A}} _{e_i}\xi =0; \quad  \eta({\mathcal{A}} _{e_i}X)=0 \, \,(i=1,\ldots,2n).
\end{array}
\end{equation}
From equalities \eqref{2.4} and \eqref{3.1} we obtain
\begin{equation}\label{3.2}
\begin{array}{l}
{\mathcal{A}} _{\xi }X=0.
\end{array}
\end{equation}
Then the decomposition of $W_1$ over $V$ coincides with the decomposition
of $W$ over $D$, where the vector space $W$ is defined by
\[
W=\left\{F\in {\mathcal{F}}: F(X,Y,Z)=-F(X,Z,Y)=F(X,\varphi Y, \varphi Z),\, \,
X,Y,Z\in D\right\}.
\]
Taking into account \eqref{2.2}$\div$\eqref{2.5}, \eqref{2.9}, \eqref{3.1},
\eqref{3.2} we have
\begin{equation}\label{3.3}
\begin{array}{l}
W=\left\{F\in {\mathcal{F}}: F(X,Y,Z)=Y^ig({\mathcal{A}} _{e_i}X,Z):
g({\mathcal{A}} _{e_i}X,e_j)=-g({\mathcal{A}} _{e_j}X,e_i),
\right.
\end{array}
\end{equation}
\[
\left.
{\mathcal{A}} _{\varphi e_i}X=-\varphi ({\mathcal{A}} _{e_i}X), \, \,
(i=1,\ldots,2n); \, \, X,Y,Z\in D\right\}.
\]
Using \eqref{3.3} and \eqref{6} we find
\[
\theta(Z)=-Z^i{\tr}{\mathcal{A}} _{e_i}, \quad \theta ^*(Z)=\theta(\varphi Z)=
Z^i{\tr}({\mathcal{A}} _{e_i}\circ\varphi ), \quad (i=1,\ldots,2n).
\]
We define the operators
\[
\begin{array}{ll}
m_i: W \longrightarrow W, \quad i=1,2; \\
m_1(F)(X,Y,Z)=\displaystyle{\frac{1}{2}}Y^i\left\{g({\mathcal{A}} _{e_i}X,
Z)-g({\mathcal{A}} _{e_i}\varphi X,\varphi Z)\right\} \, \,
(i=1,\ldots,2n); \\ \\
m_2(F)(X,Y,Z)=\displaystyle{\frac{1}{2}}Y^i\left\{g({\mathcal{A}} _{e_i}X,
Z)+g({\mathcal{A}} _{e_i}\varphi X,\varphi Z)\right\} \, \, (i=1,\ldots,2n).
\end{array}
\]
\begin{lem}\label{l 3.1}
The operators $m_i \, (i=1,2)$ have the following properties
\[
\begin{array}{cl}
          (i)   & m_i\circ m_i=m_i, \quad i=1,2; \\ \\

          (ii)  & \displaystyle \sum^{2}_{i=1} m_i=id; \\ \\

          (iii) & m_1\circ m_2=m_2\circ m_1=0;
\end{array}
\]
and $m_i \, (i=1,2)$ commute with $\mathbb U(n,\mathbb R)$.
\end{lem}
We denote $W_{11}=Im \, m_1$, ${\mathcal{F}}_3=Im \, m_2$. Lemma~\ref{l 3.1} implies
the decomposition
$W=W_{11}\oplus {\mathcal{F}}_3$ that is $\mathbb U(n,\mathbb R)$ - invariant.
Using \eqref{2.7} we check that $W_{11} \bot {\mathcal{F}}_3$.
\begin{pro}\label{p 3.1} The decomposition
\[
W=W_{11}\oplus {\mathcal{F}}_3
\]
is orthogonal and invariant under the action of the
group $\mathbb U(n,\mathbb R)$.
\end{pro}
After direct computations using definitions of $m_1, m_2$,
\eqref{3.3} we obtain
\begin{pro}\label{p 3.2}
For an arbitrary $F \in W$ we have
\[
\begin{array}{cl}
          (i)   & F=m_1F \Longleftrightarrow {\mathcal{A}} _{e_i} \circ \varphi =
          \varphi \circ {\mathcal{A}} _{e_i}, \, \, (i=1,\ldots,2n); \\
          (ii)  & F=m_1F \Longleftrightarrow F(X,Y,Z)=-F(\varphi X,\varphi Y,Z); \\
          (iii) & F=m_2F \Longleftrightarrow {\mathcal{A}} _{e_i} \circ \varphi =-
          \varphi \circ {\mathcal{A}} _{e_i}, \, \, (i=1,\ldots,2n); \\
          (iiii)& F=m_2F \Longleftrightarrow F(X,Y,Z)=F(\varphi X,\varphi Y,Z).
\end{array}
\]
\end{pro}
From Proposition~\ref{p 3.2} the characteristic conditions of
$W_{11}, {\mathcal{F}}_3$ are
\begin{equation}\label{3.4}
\begin{array}{ll}
W_{11}=\left\{F \in W: {\mathcal{A}} _{e_i} \circ \varphi =
\varphi \circ {\mathcal{A}} _{e_i}, \, \, (i=1,\ldots,2n)\right\}\Longleftrightarrow \\ \\
\left\{F \in W: F(X,Y,Z)=-F(\varphi X,\varphi Y,Z)\right\};
\end{array}
\end{equation}
\begin{equation}\label{3.5}
\begin{array}{ll}
{\mathcal{F}}_3=\left\{F \in W: {\mathcal{A}} _{e_i} \circ \varphi =-
\varphi \circ {\mathcal{A}} _{e_i}, \, \, (i=1,\ldots,2n)\right\}\Longleftrightarrow \\ \\
\left\{F \in W: F(X,Y,Z)=F(\varphi X,\varphi Y,Z)\right\}.
\end{array}
\end{equation}
We define the operator
\[
\begin{array}{ll}
m_3: W_{11} \longrightarrow W_{11}; \\
m_3(F)(X,Y,Z)=F(X,Y,Z)-\displaystyle{\frac{1}{2(n-1)}}\left\{g(X,\varphi Y)
\theta _{F}(\varphi Z)-g(X,\varphi Z)\theta _{F}(\varphi Y)-\right.
\end{array}
\]
\[
\left.g(\varphi X,\varphi Y)
\theta _{F}(Z)+g(\varphi X,\varphi Z)\theta _{F}(Y)\right\}.
\]
\begin{lem}\label{l 3.3}
The operator $m_3$ has the following properties
\[
\begin{array}{cl}
          (i)   & m_3\circ m_3=m_3; \\ \\

          (ii)  &  <m_3F_1,F_2>=<F_1,m_3F_2>, \quad F_1, F_2 \in W_{11}; \\ \\
\end{array}
\]
and $m_3$ commutes with $\mathbb U(n,\mathbb R)$.
\end{lem}
If we denote ${\mathcal{F}}_1=Ker m_3$ and ${\mathcal{F}}_2=Im \, m_3$,
then Lemma~\ref{l 3.3} implies
\begin{pro}\label{p 3.5} The decomposition
\[
W_{11}={\mathcal{F}}_1 \oplus {\mathcal{F}}_2
\]
is orthogonal and invariant under the action of the
group $\mathbb U(n,\mathbb R)$, \\ where
\[
\begin{array}{l}
{\mathcal{F}}_1=\{F \in W:
F(X,Y,Z)=\displaystyle{\frac{1}{2(n-1)}}\left(g(X,\varphi
Y)\theta(\varphi Z)-g(X,\varphi Z)\theta(\varphi Y)\right.
\end{array}
\]
\[
\left.\left.-g(\varphi X,\varphi Y)\theta(Z)+g(\varphi X,\varphi Z)\theta(Y)
\right)\right\},
\Longleftrightarrow
\]
\[
\begin{array}{l}
{\mathcal{F}}_1=\left\{F \in W: {\mathcal{A}}_{e_i}=
\displaystyle{\frac{{\tr}{\mathcal{A}} _{e_i}}{2n}} \, id+
\displaystyle{\frac{{\tr}({\mathcal{A}} _{e_i}\circ\varphi )}{2n}}
\, \varphi , \quad (i=1,\ldots,2n)\right\},
\end{array}
\]
\[
\begin{array}{l}
{\mathcal{F}}_2=\left\{F \in W: F(X,Y,Z)=-F(\varphi X,\varphi Y,Z), \quad
\theta =0\right\}\Longleftrightarrow
\end{array}
\]
\[
\begin{array}{l}
{\mathcal{F}}_2=\left\{F \in W: {\mathcal{A}} _{e_i}\circ\varphi
=\varphi\circ {\mathcal{A}} _{e_i},
{\tr}({\mathcal{A}}
_{e_i}\circ\varphi )=0, \quad (i=1,\ldots,2n)\right\}.
\end{array}
\]
\end{pro}
Taking into account Proposition~\ref{p 3.1}, Proposition~\ref{p
3.5} we obtain
\begin{pro}\label{3.4}
The decomposition
\[
W_1={\mathcal{F}}_1 \oplus {\mathcal{F}}_2 \oplus {\mathcal{F}}_3
\]
is orthogonal and invariant under the action of the
group $\mathbb U(n,\mathbb R)\times Id$.
\end{pro}

\section{The subspace $W_2$}\label{sec-4}
\setcounter{equation}{0} From $W_2=\{F \in {\mathcal{F}}:
F=p_2F\}$, \eqref{2.10}, Proposition~\ref{p 2.2} and \eqref{2.4}
it follows
\begin{equation}\label{4.111}
\begin{array}{ll}
W_2=\left\{F \in {\mathcal{F}}: F(X,Y,Z)=-\eta(Y)g(\varphi ({\mathcal{A}} _\xi X),Z)+ \right. \\ \\
\left.
\eta(Z)g(\varphi ({\mathcal{A}} _\xi X),Y), \, {\mathcal{A}} _\xi \xi =0\right\}.
\end{array}
\end{equation}
Using \eqref{4.111} and \eqref{6} we find
\[
\theta(\xi)={\tr}({\mathcal{A}}_\xi \circ\varphi ), \quad \theta^*(\xi )=
-{\tr}{\mathcal{A}}_\xi .
\]
We define the operators
\[
\begin{array}{llllll}
q_i: W_2 \longrightarrow W_2, \quad i=1,2; \\
q_1(F)(X,Y,Z)=-\displaystyle{\frac{1}{2}}\eta(Y)\left\{g(\varphi ({\mathcal{A}} _\xi X),Z)+g({\mathcal{A}} _\xi
(\varphi X),Z)\right\}+ \\ \\
\displaystyle{\frac{1}{2}}\eta(Z)\left\{g(\varphi ({\mathcal{A}} _\xi X),Y)+g({\mathcal{A}} _\xi
(\varphi X),Y)\right\}; \\ \\
q_2(F)(X,Y,Z)=-\displaystyle{\frac{1}{2}}\eta(Y)\left\{g(\varphi ({\mathcal{A}} _\xi X),Z)-g({\mathcal{A}} _\xi
(\varphi X),Z)\right\}+ \\ \\
\displaystyle{\frac{1}{2}}\eta(Z)\left\{g(\varphi ({\mathcal{A}} _\xi X),Y)-g({\mathcal{A}} _\xi
(\varphi X),Y)\right\}.
\end{array}
\]
\begin{lem}\label{l 4.1}
The operators $q_i \, (i=1,2)$ have the following properties
\[
\begin{array}{cl}
          (i)   & q_i\circ q_i=q_i, \quad i=1,2; \\ \\

          (ii)  & \displaystyle \sum^{2}_{i=1} q_i=id; \\ \\

          (iii) & q_1\circ q_2=q_2\circ q_1=0;
\end{array}
\]
and $q_i \, (i=1,2)$ commute with $\mathbb U(n,\mathbb R)\times Id$.
\end{lem}
We denote $W^{'}=Imq_1$, $W^{''}=Imq_2$. Lemma~\ref{l 4.1} implies
the decomposition
$W_2=W^{'}\oplus W^{''}$ that is $\mathbb U(n,\mathbb R)\times Id$ - invariant.
Using \eqref{2.7} we check that $W^{'} \bot W^{''}$.
\begin{pro}\label{p 4.1} The decomposition
\[
W_2=W^{'}\oplus W^{''}
\]
is orthogonal and invariant under the action of the
group $\mathbb U(n,\mathbb R)\times Id$.
\end{pro}
After direct computations using definitions of $q_1, q_2$,
\eqref{4.111}, \eqref{2.5} we obtain
\begin{pro}\label{p 4.2}
For an arbitrary $F \in W_2$ we have
\[
\begin{array}{cl}
          (i)   & F=q_1F \Longleftrightarrow {\mathcal{A}} _\xi \circ \varphi =
          \varphi \circ {\mathcal{A}} _\xi ; \\
          (ii)  & F=q_1F \Longleftrightarrow F(X,Y,Z)=-F(\varphi X,\varphi Y,Z)-F(\varphi X,Y,\varphi Z); \\
          (iii) & F=q_2F \Longleftrightarrow {\mathcal{A}} _\xi \circ \varphi =-
          \varphi \circ {\mathcal{A}} _\xi ; \\
          (iiii)& F=q_2F \Longleftrightarrow F(X,Y,Z)=F(\varphi X,\varphi Y,Z)+F(\varphi X,Y,\varphi Z).
\end{array}
\]
\end{pro}
From Proposition~\ref{p 4.2} the characteristic conditions of
$W^{'}, W^{''}$ are
\begin{equation}\label{4.2}
\begin{array}{ll}
W^{'}=\left\{F \in W_2: {\mathcal{A}} _\xi \circ \varphi =
\varphi \circ {\mathcal{A}} _\xi \right\}\Longleftrightarrow \\ \\
\left\{F \in W_2: F(X,Y,Z)=-F(\varphi X,\varphi Y,Z)-F(\varphi X,Y,\varphi Z)\right\};
\end{array}
\end{equation}
\begin{equation}\label{4.3}
\begin{array}{ll}
W^{''}=\left\{F \in W_2: {\mathcal{A}} _\xi \circ \varphi =-
\varphi \circ {\mathcal{A}} _\xi \right\}\Longleftrightarrow \\ \\
\left\{F \in W_2: F(X,Y,Z)=F(\varphi X,\varphi Y,Z)+F(\varphi X,Y,\varphi Z)\right\}.
\end{array}
\end{equation}
We define the operators
\[
\begin{array}{llllll}
r_i: W^{'} \longrightarrow W^{'}, \quad i=1,2; \\
r_1(F)(X,Y,Z)=-\displaystyle{\frac{1}{2}}\eta(Y)\left\{g(\varphi ({\mathcal{A}} _\xi X),Z)+g(\varphi X,
{\mathcal{A}} _\xi Z)\right\}+ \\ \\
\displaystyle{\frac{1}{2}}\eta(Z)\left\{g(\varphi ({\mathcal{A}} _\xi X),Y)+g(\varphi X,
{\mathcal{A}} _\xi Y)\right\}; \\ \\
r_2(F)(X,Y,Z)=-\displaystyle{\frac{1}{2}}\eta(Y)\left\{g(\varphi ({\mathcal{A}} _\xi X),Z)-g(\varphi X,
{\mathcal{A}} _\xi Z)\right\}+ \\ \\
\displaystyle{\frac{1}{2}}\eta(Z)\left\{g(\varphi ({\mathcal{A}} _\xi X),Y)-g(\varphi X,
{\mathcal{A}} _\xi Y)\right\}.
\end{array}
\]
\begin{lem}\label{l 4.2}
The operators $r_i \, (i=1,2)$ have the following properties
\[
\begin{array}{cl}
          (i)   & r_i\circ r_i=r_i, \quad i=1,2; \\ \\

          (ii)  & \displaystyle \sum^{2}_{i=1} r_i=id; \\ \\

          (iii) & r_1\circ r_2=r_2\circ r_1=0;
\end{array}
\]
and $r_i \, (i=1,2)$ commute with $\mathbb U(n,\mathbb R)\times Id$.
\end{lem}
We denote $W^{'}_1=Imr_1$, $W^{'}_2=Imr_2$. Lemma~\ref{l 4.2}
implies the decomposition $W^{'}=W^{'}_1\oplus W^{'}_2$ that is
$\mathbb U(n,\mathbb R)\times Id$ - invariant.
Using \eqref{2.7} we check that $W^{'}_1 \bot W^{'}_2$.
\begin{pro}\label{p 4.3} The decomposition
\[
W^{'}=W^{'}_1\oplus W^{'}_2
\]
is orthogonal and invariant under the action of the group
$\mathbb U(n,\mathbb R)\times Id$.
\end{pro}
Having in mind definitions of $r_1, r_2$, \eqref{4.1},
\eqref{4.2}, \eqref{2.5} we obtain
\begin{pro}\label{p 4.4}
For an arbitrary $F \in W^{'}$ we have
\[
\begin{array}{cl}
          (i)   & F=r_1F \Longleftrightarrow g({\mathcal{A}} _\xi \, . \, , \, . )=
          g( . \, , \, {\mathcal{A}} _\xi \, . ); \\
          (ii)  & F=r_1F \Longleftrightarrow F(X,Y,Z)=-F(Y,Z,X)+F(Z,X,Y)
          -2F(\varphi X,\varphi Y,Z); \\
          (iii) & F=r_2F \Longleftrightarrow
          g({\mathcal{A}} _\xi \, . \, , \, . )=-
          g( . \, , \, {\mathcal{A}} _\xi \, . ); \\
          (iiii)& F=r_2F \Longleftrightarrow F(X,Y,Z)=-F(Y,Z,X)-F(Z,X,Y).
\end{array}
\]
\end{pro}
From Proposition~\ref{p 4.4} the characteristic conditions of
$W^{'}_1, W^{'}_2$ are
\begin{equation}\label{4.4}
\begin{array}{ll}
W^{'}_1=\left\{F \in W^{'}: g({\mathcal{A}} _\xi \, . \, , \, . )=
g( . \, , \, {\mathcal{A}} _\xi \, . )\right\} \Longleftrightarrow \\ \\
\left\{F \in W^{'}: F(X,Y,Z)=-F(Y,Z,X)+F(Z,X,Y)-2F(\varphi X,\varphi Y,Z)\right\};
\end{array}
\end{equation}
\begin{equation}\label{4.5}
\begin{array}{ll}
W^{'}_2=\left\{F \in W^{'}: g({\mathcal{A}} _\xi \, . \, , \, . )=-
g( . \, , \, {\mathcal{A}} _\xi \, . )\right\} \Longleftrightarrow \\ \\
\left\{F \in W^{'}: F(X,Y,Z)=-F(Y,Z,X)-F(Z,X,Y)\right\}.
\end{array}
\end{equation}
We define the operator
\[
\begin{array}{lll}
s: W^{'}_1 \longrightarrow W^{'}_1; \\
s(F)(X,Y,Z)=F(X,Y,Z)+\displaystyle{\frac{\theta ^{*}_{F}(\xi )}{2n}}\left\{\eta(Y)g(X,\varphi Z)-
\eta(Z)g(X,\varphi Y)\right\}.
\end{array}
\]
\begin{lem}\label{l 4.3}
The operator $s$ has the following properties
\[
\begin{array}{cl}
          (i)   & s\circ s=s; \\ \\

          (ii)  &  <sF_1,F_2>=<F_1,sF_2>, \quad F_1, F_2 \in W^{'}_1; \\ \\
\end{array}
\]
and $s$ commutes with $\mathbb U(n,\mathbb R)\times Id$.
\end{lem}
If we denote ${\mathcal{F}}_5=Kers$ and ${\mathcal{F}}_6=Ims$,
then Lemma~\ref{l 4.3} implies
\begin{pro}\label{p 4.5} The decomposition
\[
W^{'}_1={\mathcal{F}}_5 \oplus {\mathcal{F}}_6
\]
is orthogonal and invariant under the action of the
group $\mathbb U(n,\mathbb R)\times Id$, \\ where
\[
\begin{array}{llll}
{\mathcal{F}}_5=\left\{F \in W^{'}_1: F(X,Y,Z)=-\displaystyle{\frac{\theta^*(\xi)}{2n}}\{\eta(Y)g(X,\varphi Z)-
\eta(Z)g(X,\varphi Y) \}\right\}\Longleftrightarrow \\ \\
\left\{F \in W^{'}_1: {\mathcal{A}}_\xi=\displaystyle{\frac{{\tr}
{\mathcal{A}}_\xi}{2n}} \, id\right\}, \\ \\
{\mathcal{F}}_6=\left\{F \in W^{'}_1: \theta^*(\xi)=0\right\}\Longleftrightarrow
\left\{F \in W^{'}_1: {\tr}{\mathcal{A}}_\xi=0\right\}.
\end{array}
\]
\end{pro}
We define the operator
\[
\begin{array}{lll}
t: W^{'}_2 \longrightarrow W^{'}_2; \\
t(F)(X,Y,Z)=F(X,Y,Z)-\displaystyle{\frac{\theta _{F}(\xi
)}{2n}}\left\{\eta(Y)g(\varphi X,\varphi Z)- \eta(Z)g(\varphi
X,\varphi Y)\right\}.
\end{array}
\]
\begin{lem}\label{l 4.4}
The operator $t$ has the following properties
\[
\begin{array}{cl}
          (i)   & t\circ t=t; \\ \\

          (ii)  &  <tF_1,F_2>=<F_1,tF_2>, \quad F_1, F_2 \in W^{'}_2; \\ \\
\end{array}
\]
and $t$ commutes with $\mathbb U(n,\mathbb R)\times Id$.
\end{lem}
If we denote ${\mathcal{F}}_4=Kert$ and ${\mathcal{F}}_7=Imt$,
then Lemma~\ref{l 4.4} implies
\begin{pro}\label{p 4.6} The decomposition
\[
W^{'}_2={\mathcal{F}}_4 \oplus {\mathcal{F}}_7
\]
is orthogonal and invariant under the action of the
group $\mathbb U(n,\mathbb R)\times Id$, \\ where
\[
\begin{array}{llll}
{\mathcal{F}}_4=\left\{F \in W^{'}_2: F(X,Y,Z)=\displaystyle{\frac{\theta (\xi)}{2n}}\{\eta(Y)
g(\varphi X,\varphi Z)-\eta(Z)g(\varphi X,\varphi Y) \}\right\}\Longleftrightarrow \\ \\
\left\{F \in W^{'}_2: {\mathcal{A}}_\xi \circ \varphi =\displaystyle{\frac{{\tr}
({\mathcal{A}}_\xi \circ \varphi )}{2n}} \, id\right\}, \\ \\
{\mathcal{F}}_7=\left\{F \in W^{'}_2: \theta (\xi)=0\right\}\Longleftrightarrow
\left\{F \in W^{'}_2: {\tr}({\mathcal{A}}_\xi \circ \varphi )=0\right\}.
\end{array}
\]
\end{pro}
Now we consider the subspace $W^{''}$ of $W_2$. We define the operators
\[
\begin{array}{llllll}
l_i: W^{''} \longrightarrow W^{''}, \quad i=1,2; \\
l_1(F)(X,Y,Z)=-\displaystyle{\frac{1}{2}}\eta(Y)\left\{g(\varphi ({\mathcal{A}} _\xi X),Z)-g(\varphi X,
{\mathcal{A}} _\xi Z)\right\}+ \\ \\
\displaystyle{\frac{1}{2}}\eta(Z)\left\{g(\varphi ({\mathcal{A}} _\xi X),Y)-g(\varphi X,
{\mathcal{A}} _\xi Y)\right\}; \\ \\
l_2(F)(X,Y,Z)=-\displaystyle{\frac{1}{2}}\eta(Y)\left\{g(\varphi ({\mathcal{A}} _\xi X),Z)+g(\varphi X,
{\mathcal{A}} _\xi Z)\right\}+ \\ \\
\displaystyle{\frac{1}{2}}\eta(Z)\left\{g(\varphi ({\mathcal{A}} _\xi X),Y)+g(\varphi X,
{\mathcal{A}} _\xi Y)\right\}.
\end{array}
\]
\begin{lem}\label{l 4.5}
The operators $l_i \, (i=1,2)$ have the following properties
\[
\begin{array}{cl}
          (i)   & l_i\circ l_i=l_i, \quad i=1,2; \\ \\

          (ii)  & \displaystyle \sum^{2}_{i=1} l_i=id; \\ \\

          (iii) & l_1\circ l_2=l_2\circ l_1=0;
\end{array}
\]
and $l_i \, (i=1,2)$ commute with $\mathbb U(n,\mathbb R)\times Id$.
\end{lem}
We denote ${\mathcal{F}}_9=Iml_1$, ${\mathcal{F}}_8=Iml_2$.
Lemma~\ref{l 4.5} implies the decomposition \\
$W^{''}={\mathcal{F}}_8 \oplus {\mathcal{F}}_9$ that is
$\mathbb U(n,\mathbb R)\times Id$ - invariant.
Using \eqref{2.7} we check that ${\mathcal{F}}_8 \bot {\mathcal{F}}_9$.
\begin{pro}\label{p 4.7} The decomposition
\[
W^{''}={\mathcal{F}}_8 \oplus {\mathcal{F}}_9
\]
is orthogonal and invariant under the action of the
group $\mathbb U(n,\mathbb R)\times Id$.
\end{pro}
Taking into account definitions of $l_1, l_2$, \eqref{4.1},
\eqref{4.3}, \eqref{2.5} we obtain
\begin{pro}\label{p 4.8}
For an arbitrary $F \in W^{''}$ we have
\[
\begin{array}{cl}
          (i)   & F=l_1F \Longleftrightarrow g({\mathcal{A}} _\xi \, . \, , \, . )=
          g( . \, , \, {\mathcal{A}} _\xi \, . ); \\
          (ii)  & F=l_1F \Longleftrightarrow F(X,Y,Z)=-F(Y,Z,X)-F(Z,X,Y); \\
          (iii) & F=l_2F \Longleftrightarrow
          g({\mathcal{A}} _\xi \, . \, , \, . )=-
          g( . \, , \, {\mathcal{A}} _\xi \, . ); \\
          (iiii)& F=l_2F \Longleftrightarrow F(X,Y,Z)=-F(Y,Z,X)+F(Z,X,Y)+2F(\varphi X,\varphi Y,Z).
\end{array}
\]
\end{pro}
From Proposition~\ref{p 4.8} the characteristic conditions of
${\mathcal{F}}_8, {\mathcal{F}}_9$ are
\begin{equation}\label{4.6}
\begin{array}{ll}
{\mathcal{F}}_8=\left\{F \in W^{''}: g({\mathcal{A}} _\xi \, . \, , \, . )=-
g( . \, , \, {\mathcal{A}} _\xi \, . )\right\} \Longleftrightarrow \\ \\
\left\{F \in W^{''}: F(X,Y,Z)=-F(Y,Z,X)+F(Z,X,Y)+2F(\varphi X,\varphi Y,Z)\right\};
\end{array}
\end{equation}
\begin{equation}\label{4.7}
\begin{array}{ll}
{\mathcal{F}}_9=\left\{F \in W^{''}: g({\mathcal{A}} _\xi \, . \, , \, . )=
g( . \, , \, {\mathcal{A}} _\xi \, . )\right\} \Longleftrightarrow \\ \\
\left\{F \in W^{''}: F(X,Y,Z)=-F(Y,Z,X)-F(Z,X,Y)\right\}.
\end{array}
\end{equation}

Finally, we denote ${\mathcal{F}}_{10}=W_3$ and
${\mathcal{F}}_{11}=W_4$. Taking into account Proposition~\ref{p
2.1}, Proposition~\ref{p 3.5}, Proposition~\ref{p 4.1},
Proposition~\ref{p 4.3}, Proposition~\ref{p 4.5},
Proposition~\ref{p 4.6}, Proposition~\ref{p 4.7} we obtain
\begin{thm}\label{4.1}
The decomposition
\[
{\mathcal{F}}={\mathcal{F}}_1 \oplus \ldots \oplus {\mathcal{F}}_{11}
\]
is orthogonal and invariant under the action of the
group $\mathbb U(n,\mathbb R)\times Id$.
\end{thm}
Next we summarize the characterization conditions for the factors ${\mathcal{F}}_i
(i=1,\ldots,11)$.
\par
Let $X,Y,Z \in V$. Then
\[
\begin{array}{llll}
{\mathcal{F}}_1: F(X,Y,Z)=\displaystyle{\frac{1}{2n}}\{g(X,\varphi Y)\theta(\varphi
Z)-g(X,\varphi Z)\theta(\varphi Y)- \\ \\
\left. g(\varphi X,\varphi Y)\theta(hZ)+g(\varphi X,\varphi Z)\theta(hY)\right\}, \\ \\
{\mathcal{F}}_2: F(\varphi X,\varphi Y,Z)=-F(X,Y,Z); \quad \theta =0, \\ \\
{\mathcal{F}}_3: F(\varphi X,\varphi Y,Z)=F(X,Y,Z), \\ \\
{\mathcal{F}}_4: F(X,Y,Z)=\displaystyle{\frac{\theta(\xi)}{2n}}\{\eta(Y)g(\varphi X,\varphi Z)-
\eta(Z)g(\varphi X,\varphi Y) \}, \\ \\
{\mathcal{F}}_5: F(X,Y,Z)=-\displaystyle{\frac{\theta^*(\xi)}{2n}}\{\eta(Y)g(X,\varphi Z)-
\eta(Z)g(X,\varphi Y) \}, \\ \\
{\mathcal{F}}_6: F(X,Y,Z)=-F(\varphi X,\varphi Y,Z)-F(\varphi X,Y,\varphi Z)=\\ \\
-F(Y,Z,X)+F(Z,X,Y)-2F(\varphi X,\varphi Y,Z); \quad \theta^*(\xi)=0, \\ \\
{\mathcal{F}}_7: F(X,Y,Z)=-F(\varphi X,\varphi Y,Z)-F(\varphi X,Y,\varphi Z)=\\ \\
-F(Y,Z,X)-F(Z,X,Y); \quad \theta(\xi)=0, \\ \\
\end{array}
\]
\[
\begin{array}{llll}
{\mathcal{F}}_8: F(X,Y,Z)=F(\varphi X,\varphi Y,Z)+F(\varphi X,Y,\varphi Z)=\\ \\
-F(Y,Z,X)+F(Z,X,Y)+2F(\varphi X,\varphi Y,Z), \\ \\
{\mathcal{F}}_9: F(X,Y,Z)=F(\varphi X,\varphi Y,Z)+F(\varphi X,Y,\varphi Z)=\\ \\
-F(Y,Z,X)-F(Z,X,Y), \\ \\
{\mathcal{F}}_{10}: F(X,Y,Z)=\eta(X)F(\xi,\varphi Y,\varphi Z), \\ \\
{\mathcal{F}}_{11}: F(X,Y,Z)=\eta(X)\{\eta(Y)\omega (Z)-\eta(Z)\omega (Y)\}.
\end{array}
\]
\section{Basic classes of almost paracontact manifolds and some examples}\label{sec-5}
\setcounter{equation}{0}
\par
Let $(M^{2n+1},\varphi,\xi,\eta,g)$ be an almost paracontact
manifold. The tensor $F$, defined by \eqref{4} we can write in the
form \eqref{2.6}, where the linear operators ${\mathcal{A}} _{e_i}
\, (i,\ldots,2n)$ and ${\mathcal{A}} _\xi$ are defined by
\[
{\mathcal{A}} _{e_i}X=(\nabla _X\varphi )e_i, \quad (i,\ldots,2n); \quad
{\mathcal{A}} _\xi X=\nabla _X\xi .
\]
We verify immediately that so defined operators ${\mathcal{A}}
_{e_i} \, (i,\ldots,2n)$ and ${\mathcal{A}} _\xi$ have the
properties \eqref{2.2}$\div$\eqref{2.5}. Using the decomposition
of the space ${\mathcal{F}}$ over $V=T_pM, \, p \in M$, we define
the corresponding subclasses of the class of almost paracontact
manifolds with respect to the covariant derivative of the
structure tensor field $\varphi$.
\par
An almost paracontact manifold is said to be in the class
${\mathcal{F}}_i \, (i=1,\ldots,11)$ if the tensor $F(X,Y,Z)=g(\nabla _X\varphi )Y,Z)$
belongs to the class ${\mathcal{F}}_i$ over $V=T_pM$ for each $p \in M$.
\par
In a similar way we define the classes ${\mathcal{F}}_i \oplus {\mathcal{F}}_j$.
It is clear that $2^{11}$ classes of almost paracontact manifolds are possible.
\par
The class ${\mathcal{F}}_0$ of almost paracontact manifolds is
defined by the condition \\ $F(X,Y,Z)=0$. This special class
belongs to everyone of the defined classes.
\begin{example}\label{5.1}
Let $(M^5,\varphi,\xi,\eta,g)$ be an almost paracontact metric
manifold. We consider a $\varphi$-basis $\{e_1,e_2,\varphi
e_1,\varphi e_2,\xi \}$ of $T_pM, \, p \in M$ such that
\[
g(e_i,e_i)=-g(\varphi e_i,\varphi e_i)=1, \quad i=1,2.
\]
\end{example}
We denote the matrixes of the the operators ${\mathcal{A}}_{e_i}$
and ${\mathcal{A}}_{\varphi e_i}$ $(i=1,2)$ with respect to the
basis $\{e_1,e_2,\varphi e_1,\varphi e_2,\xi \}$ by
${\mathcal{A}}_i$ $(i=1,2)$ and ${\mathcal{A}}_j$ $(j=3,4)$
respectively. We define
\[
{\mathcal{A}}_1=\left(
\begin{array}{rrrrr}
0 & 0 & 0 & 0 & 0 \cr a & b & c & d & 0 \cr 0 & 0 & 0 & 0 & 0 \cr
-c & -d & -a & -b & 0 \cr 0 & 0 & 0 & 0 & 0 \cr
\end{array}
\right), \quad {\mathcal{A}}_2=\left(
\begin{array}{rrrrr}
-a & -b & -c & -d & 0 \cr 0 & 0 & 0 & 0 & 0 \cr
c & d & a & b & 0
\cr 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr
\end{array}
\right),
\]
\[
{\mathcal{A}}_3=\left(
\begin{array}{rrrrr}
0 & 0 & 0 & 0 & 0 \cr c & d & a & b & 0 \cr
0 & 0 & 0 & 0 & 0 \cr
-a & -b & -c & -d & 0 \cr 0 & 0 & 0 & 0 & 0 \cr
\end{array}
\right), \quad {\mathcal{A}}_4=\left(
\begin{array}{rrrrr}
-c & -d & -a & -b & 0 \cr
0 & 0 & 0 & 0 & 0 \cr a & b & c & d & 0 \cr
0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr
\end{array}
\right),
\]
where $a, b, c, d$ are functions over $M$.
\par
From the definitions of the matrixes ${\mathcal{A}_j}$
$(j=1,2,3,4)$ we have $\eta({\mathcal{A}}_{e_i}X)=$
\\ $\eta({\mathcal{A}}_{\varphi e_i}X)=0$ $(i=1,2)$. From
\eqref{2.4} it follows ${\mathcal{A}}_\xi X=0$. Using \eqref{2.6}
we compute
\[
F(X,Y,Z)=\left(aX^1+bX^2+cX^3+dX^4\right)\left(
Y^1Z^2-Y^2Z^1+Y^3Z^4-Y^4Z^3\right)+
\]
\[
\left(cX^1+dX^2+aX^3+bX^4\right)\left(Y^1Z^4-Y^2Z^3+Y^3Z^2-Y^4Z^1\right),
\]
where $X=X^ie_i+X^{i+2} \varphi e_i+\eta(X)\xi, \quad
Y=Y^ie_i+Y^{i+2} \varphi e_i+\eta(Y)\xi$, \\
$Z=Z^ie_i+Z^{i+2} \varphi e_i+\eta(Z)\xi \quad i=1,2$.
We verify that
\[
F(X,Y,Z)=F(\varphi X,\varphi Y,Z),
\]
which is the characterization condition of the class ${\mathcal{F}}_3$.
\begin{example}\label{5.2}
Let $(M^5,\varphi,\xi,\eta,g)$ be an almost paracontact metric manifold.
We consider a $\varphi$-basis $\{e_1,e_2,\varphi e_1,\varphi e_2,\xi \}$ of
$T_pM, \, p \in M$ such that
\[
g(e_i,e_i)=-g(\varphi e_i,\varphi e_i)=1, \quad i=1,2.
\]
\end{example}
We denote the matrixes of the operators ${\mathcal{A}}_{e_i}$,
${\mathcal{A}}_{\varphi e_i}$ $(i=1,2)$ and ${\mathcal{A}}_\xi$
with respect to the basis $\{e_1,e_2,\varphi e_1,\varphi e_2,\xi
\}$ by ${\mathcal{A}}_i$ $(i=1,2)$, ${\mathcal{A}}_j$ $(j=3,4)$
and ${\mathcal{A}}$
respectively. We define
\[
{\mathcal{A}}_1=\left(
\begin{array}{rrrrr}
0 & 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & 0 \cr
-d & -e &-a &-b & 0 \cr
\end{array}
\right), \quad
{\mathcal{A}}_2=\left(
\begin{array}{rrrrr}
0 & 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & 0 \cr
-e & -f & -b & -c & 0 \cr
\end{array}
\right),
\]
\[
{\mathcal{A}}_3=\left(
\begin{array}{rrrrr}
0 & 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & 0 \cr
-a & -b & -d & -e & 0 \cr
\end{array}
\right), \quad
{\mathcal{A}}_4=\left(
\begin{array}{rrrrr}
0 & 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & 0 \cr
-b & -c  & -e & -f & 0 \cr
\end{array}
\right),
\]
\[
{\mathcal{A}}=\left(
\begin{array}{rrrrr}
a & b & d & e & 0 \cr
b & c & e & f & 0 \cr
-d & -e & -a & -b & 0 \cr
-e & -f & -b & -c & 0 \cr
0 & 0 & 0 & 0 & 0 \cr
\end{array}
\right),
\]
where $a,b,c,d,e,f$ are functions over $M$.
Using \eqref{2.6} we compute
\[
F(X,Y,Z)=\eta(Y)\left\{a(X^1Z^3+X^3Z^1)+b(X^1Z^4+X^2Z^3+X^3Z^2+X^4Z^1)+\right.
\]
\[
c(X^2Z^4+X^4Z^2)+d(X^1Z^1+X^3Z^3)+e(X^1Z^2+X^2Z^1+X^3Z^4+X^4Z^3)+
\]
\[
\left.f(X^2Z^2+X^4Z^4)\right\}-\eta(Z)\left\{a(X^1Y^3+X^3Y^1)+
\right.
\]
\[
b(X^1Y^4+X^2Y^3+X^3Y^2+X^4Y^1)+c(X^2Y^4+X^4Y^2)+d(X^1Y^1+X^3Y^3)+
\]
\[
\left.e(X^1Y^2+X^2Y^1+X^3Y^4+X^4Y^3)+f(X^2Z^2+X^4Z^4)\right\},
\]
where $X=X^ie_i+X^{i+2} \varphi e_i+\eta(X)\xi, \quad
Y=Y^ie_i+Y^{i+2} \varphi e_i+\eta(Y)\xi$, \\
$Z=Z^ie_i+Z^{i+2} \varphi e_i+\eta(Z)\xi \quad i=1,2$.
We verify that
\[
F(X,Y,Z)=F(\varphi X,\varphi Y,Z)+F(\varphi X,Y,\varphi Z)=
-F(Y,Z,X)-F(Z,X,Y),
\]
which is the characterization condition of the class ${\mathcal{F}}_9$.

\begin{example}\label{5.3}
Let $(M^5,\varphi,\xi,\eta,g)$ be an almost paracontact metric
manifold. We consider a $\varphi$-basis $\{e_1,e_2,\varphi
e_1,\varphi e_2,\xi \}$ of $T_pM, \, p \in M$ such that
\[
g(e_i,e_i)=-g(\varphi e_i,\varphi e_i)=1, \quad i=1,2.
\]
\end{example}
We denote the matrixes of the the operators ${\mathcal{A}}_{e_i}$
and ${\mathcal{A}}_{\varphi e_i}$ $(i=1,2)$ with respect to the
basis $\{e_1,e_2,\varphi e_1,\varphi e_2,\xi \}$ by
${\mathcal{A}}_i$ $(i=1,2)$ and ${\mathcal{A}}_j$ $(j=3,4)$
respectively. We define
\[
{\mathcal{A}}_1=\left(
\begin{array}{rrrrr}
0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & a \cr 0 & 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & b \cr 0 & 0 & 0 & 0 & 0 \cr
\end{array}
\right), \quad {\mathcal{A}}_2=\left(
\begin{array}{rrrrr}
0 & 0 & 0 & 0 & -a \cr 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & -b
\cr 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr
\end{array}
\right),
\]
\[
{\mathcal{A}}_3=\left(
\begin{array}{rrrrr}
0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & -b \cr 0 & 0 & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & -a \cr 0 & 0 & 0 & 0 & 0 \cr
\end{array}
\right), \quad {\mathcal{A}}_4=\left(
\begin{array}{rrrrr}
0 & 0 & 0 & 0 & b \cr 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & a \cr
0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr
\end{array}
\right),
\]
where $a, b$ are functions over $M$.
\par
From the definitions of the matrixes ${\mathcal{A}_j}$ $(j=1,2,3,4)$
we have $\eta({\mathcal{A}}_{e_i}X)=$
\\ $\eta({\mathcal{A}}_{\varphi e_i}X)=0$ $(i=1,2)$. From
\eqref{2.4} it follows ${\mathcal{A}}_\xi X=0$. Using \eqref{2.6}
we compute
\[
F(X,Y,Z)=\eta(X)\left\{Y^1(aZ^2-bZ^4)+Y^2(bZ^3-aZ^1)+ \right.
\]
\[
\left.Y^3(aZ^4-bZ^2)+Y^4(bZ^1-aZ^3)\right\},
\]
where $X=X^ie_i+X^{i+2} \varphi e_i+\eta(X)\xi, \quad
Y=Y^ie_i+Y^{i+2} \varphi e_i+\eta(Y)\xi$, \\
$Z=Z^ie_i+Z^{i+2} \varphi e_i+\eta(Z)\xi \quad i=1,2$. We verify
that
\[
F(X,Y,Z)=\eta(X)F(\xi,\varphi Y,\varphi Z),
\]
which is the characterization condition of the class
${\mathcal{F}}_{10}$.
\begin{rem}\label{4.1}
Taking into account the characterization of the classes
${\mathcal{F}}_i \, (i=1,\ldots,11)$ by the linear operators
${\mathcal{A}} _{e_i} \, (i,\ldots,2n)$ and ${\mathcal{A}} _\xi$,
we can construct examples for the rest of the classes too. Using
the matrixes of the operators with respect to a $\varphi $-basis
we obtain also that an almost paracontact manifold of dimension
$3$ can not belong to the classes ${\mathcal{F}}_1,
{\mathcal{F}}_2, {\mathcal{F}}_3, {\mathcal{F}}_6$.
\end{rem}

\textbf{Acknowledgement} Simeon Zamkovoy acknowledges support from
the European Operational programm HRD through contract
BGO051PO001/07/3.3-02/53 with the Bulgarian Ministry of Education.
He also was partially supported by Contract 154/2008 with the
University of Sofia "St. Kl. Ohridski".

Galia Nakova was partially supported by Scientific researches fund
of " St. Cyril and St. Methodius " University of Veliko Tarnovo
under contract RD-491-08 from 27.06.2008.

\bibliographystyle{hamsplain}


TITLE: I want to know why $\omega \neq \omega+1$.
QUESTION [13 upvotes]: In Kunen's book, Set Theory,chapter I.7, he said: $1+\omega=\omega \neq \omega+1$.
I want to know why $\omega \neq \omega+1$.

REPLY [4 votes]: $\omega + 1$ has a limit point (i.e. $\omega$ — using the von Neumann definition $\omega + 1 = \omega \cup \lbrace\omega\rbrace$) in the order topology while $\omega$ is discrete in the order topology.


TITLE: Weak convergence of functionals in dual space given by a sum of functionals
QUESTION [0 upvotes]: Suppose that $X_1$, $X_2$ are Hilbert spaces and $Y=X_1\times X_2$. Let $y\in Y$ be $y=(x_1, x_2)$ and define $F\in Y^*$ by
$$\langle F,y\rangle_{Y^*\times Y}=\langle f,x_1\rangle_{X^*_1\times X_1}+\langle g,x_2\rangle_{X^*_2\times X_2},$$
where $f\in X_1^*$ and $g\in X_2^*$are given. Assume we have a sequence $(F_n)\subset Y^*$ which is equivalent that we have sequences $(f_n), (g_n)$ in $X^*_1$ and $X^*_2$, respectively. Is the following statement true:
$$F_n\to F\,\,\text{weakly in}\,\,Y^* \Leftrightarrow f_n\to f\,\,\text{weakly in}\,\,X^*_1\,\,\text{and}\,\, g_n\to g\,\,\text{weakly in}\,\,X^*_2\,\,\,?$$
"$\Leftarrow$"
$$|\langle F_n-F,y\rangle|\le|\langle f_n-f,x_1\rangle|+|\langle g_n-f,x_2\rangle|\to 0$$
since weak and weak* topologies coincide. What about "$\Rightarrow$"?

REPLY [0 votes]: Assume that $F_n\to F$ weakly in $Y^*$. Then, for all $y$ we have that
$$|\langle F_n-F,y\rangle|=|\langle f_n-f,x_1\rangle+\langle g_n-g,x_2\rangle|\to0.$$
If this is true for all $y$, then in particular for $y=(x_1,0)$ and $y=(0,x_2)$. Such choice gives however the desired convergances.


TITLE: Why am I not shocked when I touch a tree, and the tree touches a power line?
QUESTION [0 upvotes]: Why am I not shocked when I touch a tree, and the tree touches a power line?

REPLY [3 votes]: Wood is a poor conductor
Oven-Dry Wood has a resistivity of $1.00 \times 10^{14}$ to $1.00 \times 10^{16} \Omega m$
Damp Wood has a resistivity of $1.00 \times 10^3$ to $1.00 \times 10^4 \Omega m$
Copper has a resistivity of $1.68 \times 10^{−8} \Omega m$


TITLE: Excitation of mercury atom: Confused about statement in textbook
QUESTION [1 upvotes]: A physics textbook I am currently reading states that

"if an electron with kinetic energy equal to 1.12 aJ collides with a single mercury atom in its ground state, the atom may be excited and the electron will lose kinetic energy. If the same atom is hit by a photon with an energy of 1.12 aJ, the photon will not be absorbed, and the atom will not be excited by the photon."

I am a bit confused about this. Why is it that the electron may excite the atom, but the photon may not, when they both carry exactly the same amount of energy?
if anyone can explain the statement to me, then I would greatly appreciate it!

REPLY [3 votes]: In photon absorption excitation, the photon needs to carry an energy that is exactly the difference between the atom's (Mercury here) ground state and a higher energy state (or an energy that is exactly the difference between any lower energy and a higher energy state).
This is different to collisional excitation where an electron collides with an atom, such that the atom will absorb some fraction of the electron's initial kinetic energy. This fraction of kinetic energy changes the atom's energy (causing excitation) and some kinetic energy is still left over in the scattered electron.
So in collisional excitation, the incoming electron will have an energy greater than that required to excite the atom. On the other hand, for photon absorption, the incident photon needs to have an exact energy, equal to the difference between two energy levels.


TITLE: Proof that a sum of the first period of powers of integer roots of rationals is irrational
QUESTION [1 upvotes]: I'm looking for a simple proof or a reference to any proof that
For $j \in ℤ$, $0<j<m$, when each $k^j \notin ℚ$ and each $d_j \in ℚ$, $d_j \neq 0$, then  $\sum_{j=1}^{m-1} d_j k^j \notin ℚ$
My searches have turned up a few papers that appear to make use of this property, but none that prove it, refer to a source, or even give it a name.
EDIT:
I'm working with $k$ restricted to positive integer roots of rational numbers:  $k=r^{1/m}=(a/b)^{1/m}$ with $r$ and $m$ chosen so that $m$ is the least positive integer for which $k^m \in ℚ$ (so when $k \in ℚ$, $m=1$).
The set of positive integer roots of rational numbers (set $$) is not closed under addition (see the $\sqrt{2}-1$ case in comments)
Given $l$, $h$ members of $$, $\left(l\pm h\right) = l \left(1\pm{h\over l}\right) = l \left(1+k\right) \notin $ occurs when $\left(1+k\right) \notin $.
The sum before the edit is a rearrangement of the binomial expansion of $\left(1+k\right)^n$ where a different choice of $n$ changes the values of $d_j$, and excluding the rational $j=0$ term.  If the sum is irrational, then $\left(1+k\right)^n \notin ℚ$ for any $n$, $\left(1+k\right) \notin $, and $\left(l\pm h\right) \notin $ occurs when $\pm {h\over l} = k \notin ℚ$.

REPLY [1 votes]: The result that you want can be found in a fair number of places. The one I know is Lang's Algebra. Here is a link.  It starts at page $297$, in the Galois theory chapter.


TITLE: A variational optimal control problem
QUESTION [2 upvotes]: Steer the system $\dot{x}(t)=u(t)$ from $x(0)$ to $x(1)$ and minimize $$\int_0^1 x^2+u^2 \, \mathrm{d}t$$
The problem i'm having is to find the boundary values because i feel enough information is not given. i got a diff eq $\ddot{\lambda }(t)=\lambda$ but forming hamiltonian and doing other calculations, but how to find say $\lambda_1(0),\lambda_2(0)$ ? i'm confused about it.

REPLY [4 votes]: A straightforward calculus of variations approach using a test function $\eta$ with $\eta(0) = \eta(1) = 0$ yields
$$0 = \left.\frac{d}{d\epsilon}\right|_{\epsilon=0}\int_0^1(x+\epsilon\eta)^2 + (\dot{x}+\epsilon\dot{\eta})^2dt = \int_0^12x\eta+2\dot{x}\dot{\eta}dt = \int_0^12x\eta-2\ddot{x}\eta dt$$
where the last equality uses integration by parts and the fact that $\eta(0)=\eta(1) = 0$. Therefore, $\int_0^12\eta(x-\ddot{x})dt = 0$ for any test function $\eta$, and so $\ddot{x} = x$ on the interval [0,1]. Both $e^t$ and $e^{-t}$ are solutions to this differential equation, so the general solution is given by $x(t) = ae^{t}+be^{-t}$ for some constants $a, b$. Our boundary values tell us that $x(0) = a+b$ and that $x(1) = ae+be^{-1}$, so solving this linear system for $a$ and $b$ yields  $a = \frac{ex(1)-x(0)}{e^2-1}, b= \frac{e^2x(0)-ex(1)}{e^2-1}$ and therefore
$$ x(t) = \frac{x(0)\cdot(e^{2-t}-e^t) + x(1)\cdot(e^{1+t}-e^{1-t})}{e^2-1} $$
is the specific solution for the given boundary values.


TITLE: Physically, what is the Fabri-Picasso theorem really trying to say?
QUESTION [3 upvotes]: The Fabri-Picasso (FP) theorem states that if a symmetry is spontaneously broken the corresponding conserved charge operator $Q$ does not exist in the Hilbert space. The state $Q|0\rangle$ will have infinite norm!
Assuming situations where FP theorem holds, what conclusion should one draw from this theorem? Will it be correct to say that the charge conservation fails? But if that is true it will be in contradiction with Weinberg's statement here

The conservation of a current is usually a symptom of some symmetry of the underlying theory and holds whether or not the symmetry is spontaneously broken.

I am thoroughly confused. What is FP theorem really trying to tell (when it holds, of course)?

REPLY [3 votes]: These two statements are not contradicting. Charge is conserved regardless of the symmetry being broken or being not broken.
The usual wisdom goes like this: assume you have a conserved current $j^\mu(x)$ with
$$ \partial_\mu j^\mu = 0 \ . $$
Now pick a constant-time ball $\Sigma$ and define
$$ Q = \int_\Sigma j^0(x) d^3 x  \ . $$
This exists if the charge density decays fast enough. This is a conserved charge since:
$$ \frac{d Q}{d t} = \int_\Sigma \partial_0 j^0(x) d^3 x = - \int_\Sigma \nabla \cdot j(x)  d^3 x = - \int_{\partial \Sigma} j(x) \cdot d \sigma $$
Where in the last equality we used that the currents decay at infinity.
Now this is classical reasoning. On the quantum level, the current becomes an operator, and all the statements about the fall-off of functions at infinity are statements about the states we wish to apply these operators on. In particular, it suffices to look at the vacuum, since we usually only consider states which differ from the vacuum in a finite region, so once the vacuum is fine, all excited states will be fine as well.
But in the case of spontaneous symmetry breaking, it is just not true that
$$ j^0(x)|0\rangle $$
decays at infinity, hence the integral defining the charge does not exist. Note however that while $Q$ doesn't make sense in itself, one may still give meaning to commutators of $Q$ with field operators $\phi(x)$:
$$ [ Q, \phi(x) ] $$
And in particular, these still implement the symmetry.


TITLE: center of quotients of $C^*$ algebra
QUESTION [0 upvotes]: Does there exist a non-unital $C^*$ algebra  $A$ such that $\mathcal{Z}(A/I)\neq 0$,where $ \mathcal{Z}(A/I)$ denotes the center of the $A/I$.

REPLY [0 votes]: Of course. There are non-unital commutative C$^*$-algebras, like $C_0(\mathbb R)$. Any quotient of it will be commutative and nonzero if the ideal is non-trivial.