A natural symplectic form for every field theory
Olaf M\"uller
\noindent In this article, a natural weakly symplectic form on a natural configuration space associated to the fiber bundle of a given field theory is constructed. Although the form is in general not strongly symplectic, in the case of a vector bundle it gives rise to a Poisson bracket on a dense subspace of the differentiable $L^2$-functions on this space. If one has a local field theory whose Lagrangian's kinetic term is a nondegenerate bilinear form on the total space, then the Poisson bracket produces the usual commutation relations.
math-ph0108016.tex


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\begin{document}

\title{A natural symplectic form for every field theory}

\author{Olaf M\"uller}
\maketitle
\begin{center}Max-Planck-Institute for Mathematics in the Sciences\\Inselstrasse 22-26\\D-04103 Leipzig, Europe\end{center}
\date{}

\begin{abstract}
\noindent In this article, a natural weakly symplectic form on a natural configuration space associated to the fiber bundle of a given field theory is constructed. Although the form is in general not strongly symplectic, in the case of a vector bundle it gives rise to a Poisson bracket on a dense subspace of the differentiable $L^2$-functions on this space. If one has a local field theory whose Lagrangian's kinetic term is a nondegenerate bilinear form on the total space, then the Poisson bracket produces the usual commutation relations. 
\end{abstract}
%

\bigskip


\noindent Classical field theories are described by sections of fiber bundles as states of a physical system and functions on these sections as physical observables. In the spirit of geometric quantization (cf. ~\cite{wo} for a good overview), to get commutator relations for the quantum analogues of these observables, one needs a Poisson bracket for such functionals, i.e. a symplectic structure on the space of sections.

\bigskip

\noindent There are attempts to get such a structure for particular cases of field theories one of which can be found in ~\cite{wo}. Chernoff and Marsden ( ~\cite{cm}) use the natural symplectic form on the {\em tangent} space of a space of sections which is not enough for our purposes because we want to have a bracket for observables on the space itself. Kijowski (~\cite{k}) restricts the set of possible observables to Poincar\'e generators, field strength and its time-derivative.     

\noindent Most general approaches deal with functions on the total space of a related jet bundle instead of functions on the space of sections (cf. e.g. ~\cite{kt}, ~\cite{b}, ~\cite{gim}, ~\cite{fr}). 

\bigskip

\noindent Here we want to construct a weakly symplectic structure on the infinite-dimensional manifold $ \Gamma (\pi'')$ of smooth sections with compact support of a fiber bundle with bundle projection $ \pi''  : E \rightarrow \Sigma$ over a spacelike submanifold of spacetime (resp. of the world sheet in the case of string theory). Compact support means here that we fix a section of $\pi''$  playing the role of the zero section and consider only sections differing from it in a compact subset of $\Sigma$. 


\bigskip
\noindent The definition of a symplectic form for sections of a trivial bundle with a symplectic form on its fiber that is used in this construction can already be found in ~\cite{dkn} (p.185), but there, for the question of closedness, the reader is referred to ~\cite{n} which does not contain any proof of closedness.
\bigskip

\noindent For fruitful discussions and advices, I want to thank J\"urgen Jost, Peter Albers, Stefan Fredenhagen, Michael Holicki, Frank Klinker, Marc Nardmann, Tobias Preusser, Markus Roth, and Arnold Wassmer. Special thanks I would like to give to H\^ong V\^an L\^e for her great and friendly help and the correction of some bad mistakes.

\bigskip



\section{The construction of the form}


All the following takes place in the category $\mathcal{C}$ whose objects are fiber bundles whose projections are Pseudo-Riemannian submersions with a distinguished codimension-1 submanifold $\Sigma$ in the basis $M$. The morphisms are the obvious ones. 

\bigskip

\noindent First, from an original bundle with bundle projection $\pi $ we go over to the bundle $  d \pi  = \pi ' : T E =: E' \rightarrow TM =: M' $. We denote the normal vector field on $\Sigma$ by $\nu$. Then we go over to the bundle $\pi'' = \nu ^* \pi ' : TE \rightarrow \Sigma$


\noindent We want to construct a weakly symplectic structure $\Omega$ on $ \Gamma (\pi'')$ as a configuration set of a physical system.

\bigskip


Now the total space of $\pi'$ and of $\pi''$, namely $TE$, is equipped with a symplectic structure via the nondegenerate bilinear form on $E$ as bundle isomorphism between $TE$ and $T^* E$, namely just the pull-back of the canonical form on the latter. The sections of $\pi $ are a submanifold in the sections of $ \pi '$. A special feature of the symplectic structure on $E '$ is


\begin{Lemma}
\label{vert}

The above symplectic form $\omega$ on $TE$ is "vertically nondegenerate" in the following sense: 

\noindent To each vertical tangent vector $v$ at a point $ e \in TE$ there is another vertical tangent vector $w$ such that 
$ \omega \vert_e (v,w) \neq 0  $.  

\end{Lemma}

\noindent {\em Proof.} We will prove this fact in $T^*E$, then it is transferred to $TE$ by the isomorphism. As $\nu$ is an embedding, $d \nu$ is nondegenerate at every point, so the vertically nondegenerate vector fields on $TE$ w.r.t. $\pi'$ are exactly the vertically nondegenerate vector fields on $TE$ w.r.t. the pull-back bundle $\pi''$. The usual prescription $\theta \vert_{(q_1, ... q_n, p_1, ... p_n)} = \sum_i p_i dq_i $ for the natural symplectic 1-form does not depend of the used coordinates $q_1 ... q_n$ on $E$. So to distinguish horizontal from vertical parts, we take charts coming from trivializations. Then the corresponding coordinates on $ T^* E$ are $ x_1...x_k, y_1...y_m, \phi _1... \phi_k, \xi_1 ... \xi_m$, where latin indices stand for position coordinates, greek ones for the corresponding momenta, $x_i$, $\phi_i$ being horizontal and $y_i$, $\xi _i$ being vertical parts. Then the corresponding 2-form $\omega =  d \theta$ is of the direct sum form

$$ \omega \vert_{( x_1...x_k, y_1...y_m, \phi _1... \phi_k, \xi_1 ... \xi_m)} = \sum_i d \phi_i \wedge dx_i  + \sum_j d \xi_j \wedge dy_j ,$$ 

\noindent from which we can read off vertical nondegeneracy as we know nondegeneracy  \qed


\bigskip

\noindent Let us consider a vector bundle with bundle projection $\tau : A \rightarrow N$. For our construction, a crucial tool is the identification of a tangent vector $V$ resp. the value of a vector field $V$ on $\Gamma ( \tau )$ at a fixed section $\gamma$ with a vector field {\em along} $\gamma$, i.e. a section of $\gamma^*T^v A$:

$$\hat{} \  : T \vert_{\gamma} \Gamma(\tau)   \rightarrow \Gamma (\gamma^*T^v A),$$

$$\hat{V} \vert_{\gamma} (p) := \mathcal{L}_V ev_p$$

or equivalently,

$$ \hat{V} \vert_{\gamma} : p \mapsto \partial_t (\gamma_t(p)),$$

\noindent where $p \in N$, $\gamma_t$ a curve representing $V( \gamma)$. This means, we just fix a point $p \in N$ and note the direction in which it is moved infinitesimally by the family of maps $\gamma_t$. If we start with a tangent vector $V$ at $\Gamma_0 (\tau)$ tangent to the submanifold $\Gamma_0 (\tau)$, then  $\hat{V} \vert_{\gamma}$ has also compact support because $ \bigcup_{t \in [-1;1]} \gamma_t$ has compact support.

\bigskip

\noindent Conversely, a vertical vector field $V$ on $A$ maps to a vector field $ \tilde{V}$ on $\Gamma (\tau)$ by 

$$ \tilde{} : {\rm Vect} (A)  \rightarrow {\rm Vect} (  \Gamma (\tau))$$

$$ \tilde{V}: \gamma \mapsto \partial_t \vert_{t=0} (Fl_t^V \o \gamma ) ,$$

where $Fl_t^V $ is the flow of $V$.


\begin{Lemma}

\label{correspondence}

\begin{enumerate}

\item[(i)]
Both maps $\ \tilde{} \ $ and $\ \hat{}\ \vert_{\gamma}  $ are smooth for all $\gamma$; 

moreover, $\  \hat{} \ $ depends smoothly on $ \gamma  $.

\item[(ii)]
$ \ \hat{} \  \vert_{\gamma} \o \ \tilde{} \   = \gamma^*  \qquad \forall \gamma \in \Gamma(\theta) $, i.e. the composition is the pull-back along $\gamma$,
therefore $T \vert_{\gamma} \Gamma (\tau) = \Gamma ( \gamma ^* T^v A)$ and  $T \vert_{\gamma} \Gamma_0 (\tau) = \Gamma_0 ( \gamma ^* T^v A)$.

\item[(iii)]
The Lie bracket is preserved by $\ \tilde{\ } \ $: $ [\tilde{V}, \tilde{W}] = \widetilde{[V,W]} .$

\end{enumerate}


\end{Lemma}

\noindent {\em Proof.} The infinite-dimensional manifold of vector fields on a manifold is in particular a Fr\^echet space, and the flow of a vector field depends smoothly on it, so both maps are smooth. Looking deeply at the definitions, one sees that they are inverse to each other in the sense of (ii). For a proof of the third statement we define

$$ F: {\rm Diff}(A) \rightarrow {\rm Diff}(\Gamma(\tau)) ,$$

$$ f \mapsto ( \gamma \mapsto f \o \gamma) .$$

\noindent As this map is a differentiable Lie group homomorphism, its differential  
$dF \vert_{\1} : T_{\1} {\rm Diff}(A) \rightarrow T_{\1}{\rm Diff}(\Gamma(\tau)) $ preserves the Lie bracket in the sense that 

\noindent $  [dF (V), dF(W)] = dF [V,W] $. Now the source resp. target of $dF$ is as Lie algebra isomorphic to ${\rm Vect}(A)$ resp. ${\rm Vect}(\Gamma(\theta))$ via $\Phi$ resp. $\Psi$, and $ \Psi \o dF \o \Phi^{-1} = \  \tilde{} \ $ by the chain rule     \qed

\bigskip

\noindent The construction of he symplectic form is an easy consequence of a general theorem for fiber bundles with a vertically nondegenerate symplectic structure in the total space: 
Let $\tau : A \rightarrow N $ be the bundle projection, $\alpha$ the symplectic structure in the total space, then one can define a form $\hat{\alpha} $ on $\Gamma(\tau)$ acting on each two vector fields $X,Y \in {\rm Vect} (\Gamma (\tau))$ as follows:




$$ \hat{\alpha} \vert_{\gamma} (X,Y) := \int_N (\alpha \o \gamma )( \hat{X} \vert_{\gamma}  , \hat{Y}\vert_{\gamma} ). $$


\begin{Theorem}
\label{hoeppe}
$\hat{\alpha}$ as above is a weakly symplectic form.

\end{Theorem}


\noindent {\em Proof.} Obviously, $\hat{\alpha}$ is an antisymmetric two-tensor; we have to show that it is  closed and nondegenerate. We calculate  $ d \hat{\alpha} (X,Y,Z)$ at a  section $ \gamma $ using

\bigskip

$ d \Omega (X,Y,Z) = X ( \Omega (Y,Z)) - Y ( \Omega (Z,X)) + Z ( \Omega (X,Y))$

$  - \Omega ([X,Y],Z) + \Omega ([Y,Z],X) - \Omega ([Z,X],Y) . $

\bigskip

\noindent Now the crucial point is that because of tensoriality of $ d \hat{\alpha}$ we can choose    $X = \tilde{x}, Y = \tilde{y}, Z = \tilde{z}  $ for some vertical vector fields $x,y,z$ on $A$ . Lemma \ref{correspondence} then implies $ \widehat{[X,Y]} = \gamma^* ( [ x , y ]) $, and because of $ \hat{V} \vert_{\gamma} =  \gamma^* v$ for $V=X,Y,Z$ and $v=x,y,z$ we have


$$ X ( \hat{\alpha} (Y,Z)) = X \int_N (\alpha \o  \gamma  ) ( \ \hat{Y} \vert_{\gamma}  , \ \hat{Z} \vert_{\gamma}  )        = X \int_N (\alpha \o \gamma ) ( y , z) =  \int_N  x ( \alpha ( y, z)) .$$ 

\noindent Using these equations respectively their cyclic permutations we get the result from the closedness of $\alpha$.

\bigskip

\noindent Up to this point, we could have used {\em any} linear function on $ C^{\infty} (N)$ instead of the integral. Now to show nondegeneracy, take $X$ to be an arbitrary tangent vector at a section $\gamma$, $\hat{X}$ the corresponding vector field along $\gamma$. Then, because of the vertical nondegeneracy of $\omega$, there is a point $n \in N$ and a vector field $\hat{Y}$ such that $ \omega \vert_{n} ( \hat{X}, \hat{Y}) \neq 0$. If we scale $\hat{Y}$ with a function $f$ on $M$, then $\hat{\alpha} (\hat{X}, f \hat{Y} ) = \int_N f \cdot  (\omega \o \gamma) (\hat{X} , \hat{Y})$, and the statement follows from the nondegeneracy of the scalar product of $L_2 (N)$   $\qed$ 


\bigskip


\noindent Finally, we define 

$$\Omega := \hat{\omega}$$


\noindent This is well-defined for differentials of sections of $\pi'' $ of compact support. We should keep in mind that everything we needed to define $\Omega$ is a volume form on $\Sigma$ and a nondegenerate bilinear form on $E$.

\bigskip
Obviously, $\Omega$ has some nice features:

\begin{itemize}
\item{It is natural w.r.t. pull-back of bundles in $\mathcal{C}$} 
\item{It is "covariant" under Whitney sum of vector bundles in $\mathcal{C}$} 
\item{It is "covariant" under tensor product of bundles in $\mathcal{C}$}
\end{itemize}



\section{The construction of Poisson brackets}

Up to now, we know only that $\Omega$ is a weakly symplectic form. But for sake of quantization, what we would like to have is a symplectic form in the sense that the insertion into $\Omega$ is an isomorphism between the space of smooth sections and its dual. As we shall see below, this is not the case, but to define a Poisson bracket on the space of classical observables it will be enough that the insertion map is a dense embedding such that we get a Poisson bracket on a dense subset of the differentiable functions. The strategy to single out a dense subset of functions with symplectic gradient appears e.g. in ~\cite{hv}. From now on, we restrict ourselves to the case of vector bundles instead of general fiber bundles.

\bigskip

\begin{Theorem}

There is a dense subset in all differentiable probability distributions on $\Gamma_0(\pi'')$ that admits Hamiltonian vector fields and therefore Poisson brackets under which it is closed.

\end{Theorem}

\bigskip

\noindent {\em Proof.} We first choose a scalar product $h$ on $TE$ compatible with the symplectic structure $\omega$, i.e. $h(v,w) = \omega(jv,w) $ for an isomorphism tensor field $j$ on $TE$. 


\noindent Then this scalar product induces a scalar product $H$ on $ \Gamma (\pi '') $ in the same way as $\omega$ induces $\Omega$. One defines the subspace $L$ where this scalar product is finite at every $\gamma$; its completion is a Hilbert space denoted by $W$.

\noindent The isomorphism $j$ in the tangent space of the total space induces an isomorphism  $J$ of $T \Gamma_0 (\pi'')$ by 
$J(V) \vert_{ \gamma } := ( \widetilde{ j \o \hat{V} \vert_{\gamma} } )\vert_{\gamma}$ 
which intermediates between $H$ and $\Omega$: $\Omega = H(J \cdot , \cdot)$.
 

\noindent Then, we define a norm on the continously differentiable functions on $\Gamma_0 (\pi'') $: 

\noindent $\parallel f \parallel _{C^1} := sup_{\gamma \in \Gamma_0 (\pi'')} f + sup_{\gamma \in \Gamma_0 (\pi'')} \parallel df \parallel_{op}$, where the operator norm is taken w.r.t. $H$, i.e.
$ \parallel A \parallel_{op} :=  sup \{ \frac {\vert A v \vert}{\sqrt{H(v, v)}} , v \in T_{\gamma} \Gamma_0 (\pi'') \}$

\noindent Now, once defined any measure on $\Gamma_0 (\pi '')$, the subspace 

\noindent $\tilde {C} ^1 := \{ f \in C^1 ( \Gamma _0 (\pi'') ) : \parallel f \parallel_{C^1} < \infty , diam (supp f) < \infty \}$

\noindent is dense in $ C^1 ( \Gamma _0 (\pi'') ) \cap L^2 ( \Gamma_0(\pi''))$.


\noindent Define $\lambda :  W \rightarrow W^*$ by $v \mapsto \omega (v, \cdot )$ .

\noindent Obviously, this is an isomorphism because it is just a composition of the isomorphism $J$ and the usual insertion isomorphism for the Hilbert space 
$W$ w.r.t. the scalar product $H$.


\bigskip

\noindent Now we want to show that the functions on $\Gamma_0 (\pi'')$ possessing a Hamiltonian vector field (a symplectic gradient) are dense in the differentiable probability distributions. 


\bigskip

\noindent Given any $f \in \tilde C^1 ( \Gamma _0 (\pi'') )$, we choose an "almost Hamiltonian vector field" $ X_f ^{\epsilon} $ with 
$\parallel df - \Omega (X_f ^{\epsilon} , \cdot ) \parallel _{op} < \epsilon $ everywhere whose existence is guaranteed by the density of $L$ in $L^*$. Then we define a function $f _{\epsilon} $ by: 

\noindent $f _{\epsilon} (p_0) = f (p_0)$ for a fixed $p_0$, for simplicity chosen out of the support of $f$, and

\noindent $ f_{\epsilon}  (p) := \int _c \Omega (x_f^{\epsilon} , \dot{c} )$, where $c$ is any curve from $p_0$ to $p$. Of course this definition requires integrability of the covector field which follows from the fact that each closed curve in $\Gamma _0 (\pi) $ bounds an immersed disk and that on the corresponding 2-dimensional disk equipped with the pulled-back symplectic 2-form we can apply Stokes' Theorem. It holds

\noindent $X_{f^{\epsilon}} = X_f^{\epsilon}$ and therefore

\noindent $ \parallel df_{\epsilon} (p) - df(p) \parallel_{op} < \epsilon $,

\noindent $\vert f_{\epsilon} (p) - f(p) \vert < diam (supp f) \cdot \epsilon$ 

\noindent which in turn implies 

\noindent $\parallel f - f_{\epsilon} \parallel_{C^1} \rightarrow 0$ with $ \epsilon \rightarrow 0$ everywhere, so we see that the functions possessing a symplectic gradient are indeed dense in all differentiable functions.

\bigskip

\noindent Moreover, the functions with symplectic gradient build a closed algebra under the Poisson bracket 
$\{ f , g \} := \Omega ( X_f , X_g )$ because of the well-known property

\noindent $ X _{ \{ f , g \} } = [ X_f , X_g ] $ where the right hand side denotes the Lie bracket (cf. e.g. ~\cite{wo}).

\bigskip

\noindent So, in summary, we have a well-defined Poisson bracket on a dense subspace of $  C^1 ( \Gamma _0 (\pi'') ) \cap L^2 ( \Gamma_0(\pi''))$    \qed



\section{Application to field theories}

When applied to a local field theory whose kinetic term is a nondegenerate bilinear form on $E$, the Poisson bracket above satisfies the usual commutation relations in the sense that one can consider the observables $\Phi _f ^{\mu}, \Pi _g ^{\nu}$ defined by

$$  \Phi _f ^{\mu} (\gamma) = \int_{\Sigma} f \cdot \gamma ^{\mu},$$

$$ \Pi _g ^{\nu} (\gamma) = \int_{\Sigma} g \cdot \dot{\gamma} ^{\nu},$$

\noindent $f,g \in C_0^{\infty}(\Sigma)$. Then the above Poisson bracket between them is easily computed to be  

\noindent $\{ \Phi _f^{\mu}, \Pi _g^{\nu} \} = \ \delta^{\mu \nu} \int_M f \cdot g .$ 

\noindent The sharp position and momentum observables correspond to $f,g$ being delta-distributions and do not possess a gradient.

\noindent Moreover, equipped with the above symplectic form $\omega$, the theory allows immediately Geometric Quantization because $\omega$ is not only closed but also exact because the outer derivative of forms on $\Gamma_0 (\pi'')$ defined by $\hat{}$ corresponds to the outer derivative of forms on $E$ : $d \hat{\alpha} = \hat{d \alpha}$, as we can see from the proof of Theorem \ref{hoeppe}.

\noindent If we consider the special case of the (bosonic degrees of freedom of) electrodynamics, the comparison with the symplectic form obtained in ~\cite{wo} (ch. 7, p.134) is instructive. There (with the definition $F_{ab} (\Phi) = 2 (\nabla _a \Phi_b - \nabla _b \Phi_a )$ and with $\sigma$ being the normal vector field on $\Sigma$ representing the time multiplied by the volume form) the symplectic form reads 

$$\omega ( \Phi, \Psi ) = \frac{1}{2} \int_{\Sigma} (\Phi^b F_{ab} (\Psi)  -\Psi^b  F_{ab} (\Phi) ) d \sigma^a $$

which is degenerate. The above construction leads to 

$$\omega ( \Phi, \Psi ) = \frac{1}{2} \int_{\Sigma} (\Phi^b \nabla _a \Psi_b  -\Psi ^b  \nabla _a \Phi _b ) d \sigma^a $$

\noindent which is nondegenerate and is just the part of the first form which is tensorial in the normal vector field $\sigma$.

\bigskip



\begin{thebibliography}{99}

\bibitem{wo}
N.M.J. Woodhouse: Geometric Quantization. Oxford Mathematical Monographs 1991

\bibitem{cm} 
Paul R. Chernoff, Jerrold E. Marsden : Properties of Infinite Dimensional Hamiltonian Systems. Lecture Notes in Mathematics, no. 425, Springer-Verlag, Berlin, New York 1974

\bibitem{k}
Jerzy Kijowski : A Finite-dimensional Canonical Formalism in the Classical Field Theory, Comm.Math.Phys., 30 (1973), pp. 99-128

\bibitem{kt}
J.Kijowski, W.Tulczyjew : A symplectic framework for field theories, Lecture Notes in Physics, no. 107, Springer-Verlag, New York 1979


\bibitem{b}
G.Barnich: Brackets in the jet-bundle approach to field theory, hep-th/9709164


\bibitem{gim}
Mark J. Gotay, James Isenberg, Jerrold E. Marsden: Momentum maps and Classical relativistic fields, physics/9801019


\bibitem{fr}
Michael Forger, Hartmann R\"omer: A Poisson bracket on multisymplectic phase space, math-ph/0009037


\bibitem{dkn}
B.A.Dubrovin, I.M.Krichever, S.P.Novikov: Integrable Systems I, in: Dynamical Systems IV, Encyclopedia of Mathematical Sciences, no. 4, Springer-Verlag, Berlin, New York 1990


\bibitem{n}
S.P.Novikov: The Hamiltonian formalism and a many-valued analogue of Morse theory, Russian Math. Surveys 37:5 (1982), pp. 1-56


\bibitem{hv}
H\^ong V\^an L\^e : Geometry of a class of soliton solutions of nonlinear Schr\"odinger equations, preprint, Leipzig 2001



\end{thebibliography}

\end{document}
math-ph0108016.extracted_bib
\begin{thebibliography}{99}

\bibitem{wo}
N.M.J. Woodhouse: Geometric Quantization. Oxford Mathematical Monographs 1991

\bibitem{cm} 
Paul R. Chernoff, Jerrold E. Marsden : Properties of Infinite Dimensional Hamiltonian Systems. Lecture Notes in Mathematics, no. 425, Springer-Verlag, Berlin, New York 1974

\bibitem{k}
Jerzy Kijowski : A Finite-dimensional Canonical Formalism in the Classical Field Theory, Comm.Math.Phys., 30 (1973), pp. 99-128

\bibitem{kt}
J.Kijowski, W.Tulczyjew : A symplectic framework for field theories, Lecture Notes in Physics, no. 107, Springer-Verlag, New York 1979


\bibitem{b}
G.Barnich: Brackets in the jet-bundle approach to field theory, hep-th/9709164


\bibitem{gim}
Mark J. Gotay, James Isenberg, Jerrold E. Marsden: Momentum maps and Classical relativistic fields, physics/9801019


\bibitem{fr}
Michael Forger, Hartmann R\"omer: A Poisson bracket on multisymplectic phase space, math-ph/0009037


\bibitem{dkn}
B.A.Dubrovin, I.M.Krichever, S.P.Novikov: Integrable Systems I, in: Dynamical Systems IV, Encyclopedia of Mathematical Sciences, no. 4, Springer-Verlag, Berlin, New York 1990


\bibitem{n}
S.P.Novikov: The Hamiltonian formalism and a many-valued analogue of Morse theory, Russian Math. Surveys 37:5 (1982), pp. 1-56


\bibitem{hv}
H\^ong V\^an L\^e : Geometry of a class of soliton solutions of nonlinear Schr\"odinger equations, preprint, Leipzig 2001



\end{thebibliography}