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\begin{center}
{\bf THE JACOBI ALGEBRAS CLASSIFICATION

ON HOMOGENEOUS MANIFOLDS}
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\begin{center}
{V. S. KALNITSKY}
\end{center}


%\begin{abstract}{
\section{Introduction}
The problem of classification of the geodesic flows symmetries
polynomial on velocity (the Jacobi fields) on the tungent bundle
of a homogeneous manifold has the qualitative solution. The
question is reduced to the straightforward calculation, with the
construction of the manifold and explicit description of the
algebra being possible. The idea of such procedure itself is not
new, it is equations described earlier by the author which are.
The key notion is the tensor equation monodromy group. The list of
possible Jacobi algebras coincides with a number of conjugate
classes of the monodromy representations in the group of affine
symmetries of the universal cover space. Remarkably that for the
flat connections this conclusion follows right from the algebraic
structure of the Jacobi algebra. Here the calculation is fulfilled
for some important cases.

This article is the part of the technical report (St.Petersburg
Government and Ministry of Education RF \#PD03-1.1-27).

%\end{abstract}

\medskip


\section{Classification theorem}

Let $M^n$ be connected homogeneous $C^\infty$-manifold provided by the symmetrical
affine connection
 $\nabla$, $\Gamma$\,--- a transitive pseudogroup of local affine transformations.
 All objects we will face in this work belong $C^\infty$-category.
The elements of the group connecting the points $p,q$ will be denoted
$\varphi_{pq}$. The theorem cited below belongs to~K.~Nomizu~(\cite{Nom}). We will
apply it for the case of the infinite set of equivariant tensor operators introduced
by the author and use the additional algebraic structure (graded Lie algebra) which
is invariant as well.

\noindent {\bf Theorem} (\cite{Nom}). {\it If a tensor equation is
given on a simply connected manifold, such that the dimension of
the germs space is constant over it, every germ of local solution
is expandable over whole manifold.}

The author has proved in~(\cite{Kalnitsky:Kal1}) that every
generalized Jacobi equation $$ \delta^2A+R\circ A =0$$ has finite
dimensional germ space in each point. Here $\delta$ and $\circ$
are the symmetrised covariant derivation and contraction, $R$ is
curvature tensor of the connection. In view of homogeneity and the
Jacobi equation invariance the dimension function is constant. It
is enough to consider the universal cover space $(\widetilde
M,\pi,M)$. The symmetrical affine connection with the same
transitive pseudogroup $\Gamma$ is induced on the cover space.
Each local solution is expandable over the cover. Hence, the
Jacobi algebra $\frak Y(\widetilde M,\widetilde\nabla)$ consists
of globally defined tensor fields. But this algebra is wider than
that on base. Not every field on the cover space is the lift of
one from the base. In order to describe such lifts we will
introduce the notion of the {\it shift} operator. Let us consider
the germ space in arbitrary point $p\in\widetilde M$. Every germ
is expandable, hence each germ defines some germ over another
point $q$. Such correspondence defines the germ spaces
isomorphism. More over this operation saves the Lie bracket. Hence
this operation gives the isomorphism $T_{pq}$ of the graded Lie
algebras $\frak Y(U_p)$ and $\frak Y(U_q)$ over some neighborhoods
of points $p,q$, which will be called the shift operator.  From
another side, any element $\varphi_{qp}\in\Gamma$ defines an
isomorphism $\varphi_{qp}^*$ of these algebras and we can consider
an automorphism of the space $\frak Y(U_p)$
\[
T_{pq}^{\varphi_{qp}}=\varphi^*_{qp}\circ T_{pq}:\frak Y(U_p)\to\frak Y(U_p).
\]
For any pair of points projecting in one on the base correspond to
some loop on base with this origin. If $\pi$ is the cover map and
$p,q\in\pi^{-1}(x)$, than the local diffeomorphism
$\pi_q^{-1}\circ\pi_p$
 of the neighborhoods is the connection automorphism. Hence, the operator
$T_{pq}^{\pi_p^{-1}\circ\pi_q}$ is well defined. It gives the
monomorphism of the fundamental group to the affine group, with
the image acting free and totally disconnected on the cover space
\[
R:\pi_1(M)\to\frak A(\widetilde M).
\]
Using this monomorphism one can construct the monodromy
representation
\[
\left\{ \rho([\gamma])=\left.T_{pR([\gamma])(p)}^{R^{-1}([\gamma])|_p}\in{\rm
Aut}\frak Y(\widetilde M) \right| [\gamma]\in\pi_1(M)\right\}
\]
Indeed, it is a homomorphism
\[
\rho([\beta]\circ[\alpha])=\rho([\beta])\circ\rho([\alpha]).
\]
In fact, if the loop is covered by different paths $\widehat{pr}$ and $\widehat{qm}$
on $\widetilde M$, than each germ extends along them uniformly. It means that the
shift operators are conjugate with respect to the change of coordinates determined
by the cover
\[
T_{pr}=(\pi_r^{-1}\circ\pi_m)\circ T_{qm}\circ(\pi_q^{-1}\circ\pi_p).
\]
By the definition we can write
\[
\rho([\alpha])=\left(\pi_p^{-1}\circ\pi_q\right)^*\circ
T_{pq},\quad\rho([\beta])=\left(\pi_p^{-1}\circ\pi_r\right)^*\circ T_{pr},
\]
\[
\rho([\beta])\circ\rho([\alpha])=(\pi_p^{-1}\circ\pi_r)\circ
T_{pr}\circ(\pi_p^{-1}\circ\pi_q)\circ T_{pq}=
\]
\[
(\pi_p^{-1}\circ\pi_r)\circ \left( (\pi_r^{-1}\circ\pi_m)\circ
T_{qm}\circ(\pi_q^{-1}\circ\pi_p) \right) \circ(\pi_p^{-1}\circ\pi_q)\circ T_{pq}=
\]
\[
(\pi_p^{-1}\circ\pi_m)\circ T_{qm}\circ T_{pq}=(\pi_p^{-1}\circ\pi_m)\circ T_{pm}=
\rho([\beta]\circ[\alpha]).
\]
Finally, we can formulate the {\it classification theorem}.

{\bf Theorem.} {\it Any subgroup $\pi$ of $\frak A(\widetilde M)$
acting free and totally disconnected on homogeneous simply
connected manifold $\widetilde M$ determines the set of
automorphisms $\rho(a)$, $a\in\pi$,
\[
\rho(a)=T_{pa(p)}^{a^{-1}}
\]
of the Jacobi algebra $\frak Y(\widetilde M)$. The factor space
$\widetilde M/\pi$ is homogeneous manifold, has the fundamental
group $\pi$ and the Jacobi algebra
\[
\frak Y(M)=\bigcap_{a\in\pi}{\rm Inv}\rho(a).
\]
}

\noindent The explicit description is now based on extension of the PDE equations
system solution. Any simplification leads to the excessiveness of the list, i.e. the
question on realizability must be investigated in addition. An arbitrariness of the
map construction in the theorem is double\,--- the choice of the universal cover
$\pi$ and that of point $p\in\tilde M$.  All cover spaces are diffeomorphic. The
diffeomorphism itself is affine map. Hence, both maps $R, R'$ corresponding to the
different covers are conjugate by this affine map. As to the choice of the point
$p'$ differ from the point $p$, in this case the operators  $\rho$ and $\rho'$ are
conjugate with respect to the shift operator  $T_{pp'}$. In fact, for the operator
 $R$ is affine the following diagram is commutative for any $a\in\pi_1$
\[
\begin{CD}
U_p @>T_{pp'}>>U_{p'}\\
@VVR(a)V @VVR(a)V\\
U_{R(a)(p)}@>T_{R(a)(p)R(a)(p')}>>U_{R(a)(p')}
\end{CD}
\]
By the definition
\[
\rho'(a)=T_{p'R(a)p'}^{R^{-1}(a)}=(T_{pp'}\circ T_{p'p})\circ
T_{p'R(a)p'}^{R^{-1}(a)}\circ(T_{pp'}\circ T_{p'p})=
\]
\[
T_{pp'}\circ (T_{p'p}\circ (R^{-1}(a))^*)\circ (T_{p'R(a)p'}\circ T_{pp'})\circ
T_{p'p}=
\]
\[
T_{pp'}\circ ( (R^{-1}(a))^*\circ T_{R(a)p'R(a)p})\circ T_{pR(a)p'} \circ T_{pp'}=
\]
\[
T_{pp'}\circ ( (R^{-1}(a))^*\circ T_{pR(a)p})\circ T_{pp'}= T_{pp'}\rho(a)T_{p'p}.
\]
So, for any pair $a\in\frak A(\widetilde M)$ and $q\in\widetilde M$ the conjugation
of the representation  $\rho$ by the operator $T_{pq}^{(a^{-1}Ra)^{-1}}$ corresponds
to some change of coordinates on the manifold. Hence, the algebra is still the same.
It means that the set  $\frak A\times\widetilde M$ acts on the set of monodromy
representations ${\rm Rep}[\pi\to{\rm Aut}\frak Y]$. To each realizable algebra
corresponds some conjugacy class with respect to this action. Finally, {\it the
Jacobi algebras classification is reduced to the description of the realizable
classes
\[ {\rm Rep}[\pi\to{\rm Aut}\frak Y]/\frak A\times\widetilde M
\] }
Thus for the explicit classification we need a) to know the group
$\frak A(\widetilde M)$; b) to list all subgroup acting free and
totally discontinuous; c) to know the algebra $\frak Y(\widetilde
M)$; d) to describe all shift operators; e) to find all realizable
classes. For the homogeneous spaces the items a)--b) are studied
rather well  (see, for inst.~\cite{loos}).

{\it Remark.} Let us consider now any loop $\gamma$ with the
origin $x\in M$. Any germ is extendable along the cover path with
endpoints $p$ and $q$, $p,q\in\pi^{-1}(x)$. The projection of
$T_{pq}$ by $\pi$ on a neighborhood of point $x$ determines an
automorphism $\varphi_{[\gamma]}:\frak Y(U_x)\to\frak Y(U_x)$
depending on the homotopy class of the loop only. This operator is
the automorphism of the graded Lie algebra.  In the case of the
algebra $\frak Y(U_x)\cong\frak Y(\Bbb R^1)$  all such
automorphisms have the form  $\varphi^*$, where $\varphi$ is some
affine transformation of $\Bbb R^1$~((\cite{Kalnitsky:Kal1},
1995)). The hypothesis it is true for $\Bbb R^n$ was proved by the
author in \cite{Kalnitsky:Kal2} (2002). Namely, any automorphism
of the graded Lie algebra $\frak Y(\Bbb R^n)$, commuting with the
symmetrized covariant derivative on its domain of definition, is
generated by an affine transformation of $\Bbb R^n$. So a field
$X$ is determined on $M$ along the loop  $\gamma$ if its cover
field  $\tilde X\in{\bf Inv}\varphi_{[\gamma]}$. The strategy of
the {\it flat} Jacobi algebras classification arises from the
above conclusions. One needs to describe the maximal trivial
subrepresentation of the fundamental group representation in the
affine transformation group.  The simplest examples show that not
all algebras from the list appeared are realizable. More over,
implicit description of $\frak Y(\Bbb R^n)$ is hard combinatorial
task. The key fact in the proof of the last theorem is the
perfectness of the affine fields Lie algebra.  For the non-flat
connection one may hardly expect similar result. In searching of
generalizations the new concept of spray algebra was introduced.
New interesting interconnections and tensor structures were
detected. The last appeared in \cite{Kalnitsky:Kal3}.

\section{Explicit descriptions and realizability}

As it follows from the classification theorem the explicit
description is based on the explicit expression of the operator
$T_{pq}^{\varphi_{qp}}$. For the {\it flat} case such description
is  possible completely.

\noindent{\bf Complete connection.} If the manifold $\widetilde M$
is the complete, flat, simply connected one, than it is
diffeomorphic to  $\Bbb R^n$ with canonical connection.  In this
case $\frak A(\widetilde M)\cong {\rm Aff}(n)$  and the shift
operators  $T_{pq}^{\varphi_{qp}}$ are the tensor field shifts
with respect to the affine transformation $\varphi_{qp}(\bar
x)+\vec{pq}$. So, action $\rho(a)$, $a\in{\rm Aff}(n)$, coincides
with the map $a$. For map $a$ acts free such transformation has
not fix points. The description of the Jacobi algebras is reduced
to the enumeration of all $\pi < {\rm Aff}(n)$ acting free and
totally discontinuous
 on $\Bbb R^n$, up to the group ${\rm
Aff}(n)$ conjugation.

\noindent{\bf Theorem.} {\it  The classification of the Jacobi algebras is reduced
to the enumeration of complete flat not simply connected manifolds generated by the
actions of some subgroup of the group ${\rm Aff}(n)$ on $\Bbb R^n$ and the
description of the maximal invariant subalgebra with respect to this action on the
algebra of polynomial symmetries.}

\bigskip

\noindent{\bf Case $\Bbb R^2$ with
$\mathbf{\boldsymbol{\pi}_1\boldsymbol{=}\boldsymbol{\Bbb Z}}$.}
In the case of $\Bbb Z$ action it is enough to define an
automorphism $\rho(1)\in {\bf Aff}(2)$ acting free on the plane.
Any such automorphism can be reduced by some affine transformation
of coordinates to one of the map listed below
\[
1)\quad \left(\begin{array}{cc}
1&0\\
0&\lambda
\end{array}\right)
\left(\begin{array}{c}
x\\
y
\end{array}\right)
+ \left(\begin{array}{c}
1\\
0
\end{array}\right);
\quad 2)\quad \left(\begin{array}{cc}
1&0\\
1&1
\end{array}\right)
\left(\begin{array}{c}
x\\
y
\end{array}\right)
+ \left(\begin{array}{c}
1\\
0
\end{array}\right).
\]

Case 1). Let $\lambda=1$. Polynomial fields are invariant with respect to the
parallel shift iff their coefficients are independent of one variable $x$. Thus, the
Jacobi algebra is a subalgebra of $\frak A(\Bbb R^2)$ consisting of fields depending
on one variable. In this case the Jacobi equation is simply second partial
derivation with respect to the rest variable. It means that the algebra consists of
the tensor fields with linear one variable components
\[
\frak Y\cong\Bbb R^4\oplus\Bbb R^8 \oplus \Bbb R^{12} \oplus \dots.
\]

Let $\lambda=-1$. Among the polynomial fields $C^{i}_{j_1\dots j_s}(y)$ the only
fields depending on $y$ in degree $p=0,1$, such that $(-1)^{i+j_1+\dots+j_s+p}=1$,
will survive after $Ox$-symmetry. It is the complete algebra description.

$\lambda>0$. It is enough to note that this map is conjugate to an affine shift by
some diffeomorphism of the plane. This conjugating diffeomorphism has the form
\[
\varphi(x,y)=(x,ye^{-\mu x}), \mu=\ln(\lambda).
\]
A polynomial field shifted by this diffeomorphism must be independent of $x$ in new
coordinates. New components of the field of type $(1,k)$ have the form
\[
x^ry^se^{t\mu x}L(\mu),\quad r,s\in\Bbb N,t\in\Bbb Z,
\]
where $L(\mu)$ is some linear combination of the initial polynomial field
coefficients. All linear expressions with  $r,t\ne0$, which have to be zero, form a
homogeneous linear system. The $k$-th components of the Jacobi algebra is the
solution space of this system. Straightforward calculation shows that the space is
independent of spectral parameter and each component is three-dimensional. The
explicit form of the invariant polynomial fields is
\[
A^1_{1\dots1}=a,\quad A^1_{\bar\imath}=0;
\]
\[
\quad A^2_{1\dots1}=by, \quad A^2_{21\dots1}=c, \quad A^2_{\bar\imath}=0,
\]
with arbitrary constants $a,b,c\in\Bbb R$. The Lie bracket of these fields is
trivial. Thus, the Jacobi algebra is the commutative infinite dimensional Lie
algebra
\[
\frak Y\cong\Bbb R^2\oplus\Bbb R^3 \oplus \Bbb R^3 \oplus \dots
\]



$\lambda<0$. The algebra corresponding this spectrum is the intersection of the
algebras $\lambda=-1$ and $\lambda>0$.

Case 2). Polynomial field transferred with such map has to coincide with initial one
in new coordinates $(x,y)$. The appeared conditions on tensor components are some
functional equations system. Straightforward calculations show that the component
$A^2_{1\dots1}$ has the maximal degree. But it is this component which has degree
not exceeded 1. The last follows directly from the generalized Jacobi
equation~\cite{Kalnitsky:Kal1}. More over this component does not depend on variable
$y$ as a sequence of properties of the appeared tensor equations. The general form
of the tensor field is
\[
A^2_{1\dots1}=ax+b, \quad A^1_{1\dots1}=a, \quad A^2_{21\dots1}=c.
\]
Particularly, the affine algebra of such connection is two-dimensional, commutative
and consists of the fields $(0,1)$ and $(1,x)$. The structure of the Jacobi algebra
\[ \frak Y\cong\Bbb R^2\oplus\Bbb
R^3 \oplus \Bbb R^3 \oplus \dots
\]
is not trivial. The fields corresponding to the coefficients  $a$ and $c$ do not
commute. The last remark is that matrix with negative determinant generates the
M\"obius strip.

{\bf Theorem.} {\it There exist exactly three type of the Jacobi algebras of the
complete flat affine connection on cylinder.

1) The typical algebra is infinite dimensional commutative Lie algebra
\[
\frak Y_1\cong\Bbb R^2\oplus\Bbb R^3 \oplus \Bbb R^3 \oplus \dots.
\]

2) The set of connections of co-dimension 1 in the space of complete connections on
cylinder has non-commutative infinite dimensional Jacobi algebra
\[ \frak Y_2\cong\Bbb R^2\oplus\Bbb
R^3 \oplus \Bbb R^3 \oplus \dots.
\]

3) The Jacobi algebra on the Euclidean cylinder consists of tensor fields depending
on one variable and linear with respect to it
\[
\frak Y_3\cong\Bbb R^4\oplus\Bbb R^8 \oplus \Bbb R^{12} \oplus \dots.
\]

There exist exactly two type of algebras on the M\"obius strip. Both of them are
infinite dimensional and commutative. One of them are typical.}

In this investigation we do not touch the question of the symmetries completeness.
For the obtained explicit classification however the answer is clear. {\it For any
flat complete connection there exists a group of the geodesic flow automorphisms of
arbitrary much dimension!} This follows directly from the commutativity and the
Palais theorem~\cite{Palais}.

From the above description it is easy to obtain the classification for the torus.
Indeed, the torus can be generate as factor space only by pair of the commutative
independent shifts of cover space. Easy to show that such pair are (up to the
conjugation) a) the multiple spectrum map and shift along $Oy$ axis; b) the pair of
shifts.


{\bf Theorem.}{\it There exist exactly two types of the complete plane affine
connections on torus

1) The typical connection has the algebra
\[ \frak Y_2\cong\Bbb R^2\oplus\Bbb
R^3 \oplus \Bbb R^3 \oplus \dots.
\]

2) To the Euclidean tori corresponds the algebra of fields with constant components
\[\frak Y\cong\Bbb R^2\oplus\Bbb
R^4 \oplus \Bbb R^6 \oplus \dots.
\]
}


In higher dimensions there are more types of the Jordan forms and the spectral
parameters domain is multi-dimensional. The dependance of the solution space on
spectrum is not known. In the next subsection an instance of such dependance is
described for an uncomplete connection. One can assume that this picture is not
trivial for complete ones too.

\medskip

\noindent{\bf Non-complete connection.} In this case  $\frak Y(\widetilde
M)\cong\frak Y(\Bbb R^n)$ and $\frak A<{\rm Aff}(n)$. The shift operator $T_{pq}$
 coincide with that of Euclidean plane if the points $p,q$ belongs to one normal map
 (in which the Christoffel symbols are zero identically). If the points are far from
 each other, any "good enough"
 path can be split by the set of points, each pair of which belongs to one map.
 The shift operator in this case is the composition of Euclidean shifts and operators
 generated by the gluing cocycles
of normal maps.  Hence, the shift operator belongs to the operator group generated
by affine maps of Euclidean plane and the gluing cocycles. In the flat case the
latter are affine maps (for hyperbolic spaces,--- hyperbolic plane isometries and so
on). Henceforth, one should describe all homomorphisms from $\pi$ to ${\rm Aff}(n)$
in order to enumerate all possible Jacobi algebras. The question of realizability
will be still open. This is exactly the conclusion of the classification theorem of
the author~\cite{Kalnitsky:Kal2}.

\medskip

\noindent{\bf Case
$\mathbf{\boldsymbol{\pi}_1\boldsymbol{=}\boldsymbol{\Bbb Z}}$ in
$\Bbb R^n$.} The case of $\rho(1)\in {\bf Aff}(n)$ without fix
points can be regarded as previous one, as we can just remove
invariant set from the cover space with saving the fundamental
group. The last gives an uncomplete space with the same Jacobi
algebra. Let us consider now the elements of ${\bf Aff}(n)$ having
fix point. The action of an automorphism $A$  on the space of the
type $(1,k)$ fields can be regarded as coordinates changing. In
such form the description of the algebra coincides with that of
the normal form of the field in sense of Birkhof~(\cite{Bib}) with
the linear part
\[ \left(\begin{array}{cc}
A&0\\
0&A
\end{array}\right).
\]
The classification of the last is completely determined by the operator $A$ spectrum
and existence of "resonance" arrays. The explicit description for the prime real
spectrum can be found in~(\cite{Bru}). The perfect account of the temporary state of
the theory one can read in~{\cite{Bas}}. As an example let us describe the
classification on two dimensional cylinder for the prime positive spectrum.


Let coordinates of a point on the plane correspond to the spectrum of $A$. All
points lying on the curve $y=x^{p/q}, x\ne1$, where $p,q$ are coprime natural
numbers, correspond to isomorphic non-commutative Jacobi algebras. The lines $x=1$
and $y=1$ do as well. The others points correspond to one-dimensional algebra
consisting of the geodesic flow.

In conclusion, let us note that all algebras of this two dimensional classification
are realizable.



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\bibitem{Kalnitsky:Kal1} Kalnitsky V.S., {\it  Algebra of generalized Jacobi fields}
//~J. Math. Sci. (New-York) 91 (1998), no. 6, pp. 3476--3491.

\bibitem{Kalnitsky:Kal2} Kalnitsky V.S., The Jacobi algebra of flat manifold //{\it
J. Math. Sci.}, 2003 (in transl.)

\bibitem{Kalnitsky:Kal3} Kalnitsky V.S., Spray Algebra, {\it Proc. Math. Inst. Nat. Acad. Sci.
Ukr.,} v. 50, Part III, pp. 1356--1360, 2004.

\bibitem{Palais} Palais, R. {\it A global formulation of the Lie theory of transformation
groups.} //~Mem. Amer. Math. Soc. \#22, 1957

\bibitem{loos} Loos O. Symmetric spaces. Benjamin, NY-Amsterdam, 1969.

\bibitem{Bib} Bibikov Yu.N. Local theory of nonlinear analytic ordinary differential
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\bibitem{Bru} Bruno A.D. {\it Analitic form of the differential equation} //~Proc.
of Moscow Math. Soc. (in Russian) 1971. v.~25.

\bibitem{Bas} Basov V.V., {\it The normal form method in the local qualitative theory of differential equation}
 (in Russian) St.P. State Univ. Press, 2002.


\bibitem{Kalnitsky:Kal2Vestn} Kalnitsky V.S., {\it Automorphisms of a geodesic vector
field} //~Vestn. St. Petersb. Univ. Math., Alerton Press, N.-Y., v.~28 (1995) ({\bf
2}), 19--20.

\end{thebibliography}




\end{document}
