\section{Introduction}
The evolution of the velocity field $X$ of an ideal fluid in an arbitrary Riemannian three manifold $(M,g)$ is modeled by the Euler equations:
\begin{equation*}
\begin{cases}
\partial_t X + \nabla_X X &= -\nabla p \\
\operatorname{div}X=0
\end{cases}
\end{equation*}
where the differential operators are taken with respect to the metric $g$ and the scalar function $p \in C^{\infty}(M)$ is the pressure. Finding stationary solutions for a fixed metric can be challenging, and it is common in the literature to study stationary solutions to the Euler equations by allowing the metric to vary. 


In the analysis of stationary fluids, Arnold's celebrated structure theorem inaugurated the modern field of topological hydrodynamics \cite{AKh}. For closed manifolds, it provides an almost complete description of the rigid behavior of the flow if the Bernoulli function, which depends on the pressure, is non constant and analytic (or $C^2$ Morse-Bott). Except on an analytic stratified subset of positive codimension, the manifold is trivially fibered by invariant tori where the flow is conjugate to a linear field. However, Arnold's theorem is an \textit{a posteriori} conclusion: it gives the structure of such solutions but says nothing about their existence. When the Bernoulli function is constant and Arnold's theorem does not apply, the solutions are Beltrami fields: vector fields parallel to their curl. The existence of non-vanishing Beltrami fields for some metric has been extensively studied. For instance, it was proved in \cite{EG} that solutions of these type exist in every homotopy class of vector fields (each solution for some particular metric) of any three manifold. This result was recently extended to every odd dimensional manifold in \cite{C1}. Beltrami fields are much more flexible solutions, and can be used to prove the existence of steady Euler flows with some specific dynamical properties \cite{EP,CMPP}.
 
 To the authors knowledge, the only known examples of flows that satisfy the hypotheses of the structure theorem are in the round sphere and the flat torus \cite{KKP}. A natural question is to ask in which other manifolds such solutions exist. Motivated by this problem, it is left as an open question in \cite{P} to study which manifolds admit Euler flows of this type for some metric. We will adress it in two different contexts: when the steady flow is non-vanishing, and when it can vanish but the Bernoulli function is of Morse-Bott type.
 
 
 With the extra assumption that the vector field $X$ is non-vanishing, it was shown in \cite{CV} that it forces the manifold to be of a certain topological type: it has to be the union of Seifert manifolds glued along their torus boundaries, or equivalently a graph manifold. A more general case was observed before in \cite{EG0}, and which follows from previous works \cite{FM}: a non-singular flow with a stratified integral can only exist in a graph manifold. We will refer to a non-vanishing solution to the Euler equations for some metric with non constant analytic (or Morse-Bott) Bernoulli function as an Arnold fluid. 

In the first part of this note, we show that every possible three manifold, which we know to be of graph type, admits an Arnold fluid with analytic Bernoulli function.
\begin{theorem}\label{main}
Any closed, oriented graph three manifold $M$ admits a non-vanishing steady solution to the Euler equations for some metric and non constant analytic Bernoulli function.
\end{theorem}
By the discussion above, the statement is in fact an if and only if. We prove this using the standard construction of Seifert manifolds. However, adapting the arguments in the proof of the theorem above, one can use another decomposition of graph manifolds developped by Fomenko et alli to obtain a richer source of examples of Arnold fluids. In particular, one can produce an Arnold fluid realizing as Bernoulli function, up to diffeomorphism, any possible Morse-Bott function. In the language developped in \cite{F1}, the topological characterization of a graph three manifold and the foliation by level sets of a Bott integral is given by a molecule with gluing matrices. We will show that any such configuration with or without critical Klein bottles can be realized by an Arnold fluid. A concrete way to reformulate the result is the following.

\begin{theorem}\label{ArnTop}
Given an admissible Morse-Bott function $B$ in a graph manifold $M$, there exist an Arnold fluid with Bernoulli function $B$ (modulo diffeomorphism of $M$).
\end{theorem}
By admissible we mean a Morse-Bott function which can topologically be the integral of a fluid. This theorem shows that when we allow the metric to vary, the apparent difficulty to construct fluids with such a rigid behavior is overcomed, and one can realize flows in all allowed Morse-Bott topological configurations. An immediate corollary, explained in the Appendix, is that the invariants developped for Bott integrable systems lead to a topological classification of Arnold fluids with Morse-Bott Bernoulli function. This classification can be compared with the classification of vorticity functions of Morse type studied in \cite{IK} for the 2D Euler equation in surfaces.


 The ressemblance between Arnold's structure theorem and the Arnold-Liouville theorem in the theory of integrable systems already suggest that there might be some connection between these two worlds. By taking a careful look at the Arnold fluids constructed in Theorem \ref{main2}, we can relate them to Bott integrable systems by means of the symplectization of an appropiate stable Hamiltonian structure. This yields an alternative proof that any molecule with gluing matrices can be realized by an integrable system, with the additional property that the isoenergy hypersurface is of stable Hamiltonian type. Without this extra property, this was originally proved in \cite{BFM}.

\begin{theorem}\label{maincor}
The constructed steady Euler flows together with their vorticity in Theorem \ref{ArnTop} can be realized as the isoenergy hypersurface of a Hamiltonian system with a Bott integral (the Bernoulli function) in a symplectic manifold with boundary. Additionally, the Hamiltonian vector field is up to rescaling a Reeb field of a stable Hamiltonian structure.
\end{theorem}
This materializes the intuition that Arnold's structure theorem and the classical Arnold-Liouville theorem are closely related. However, the construction is \emph{ad hoc} and we can find examples of Arnold fluids that cannot reasonably be though as energy levels set of an integrable system. 
 
 
 In the last part of this work, we will drop the assumption that the solution is non-vanishing. In this general case, no obstruction on the topology of $M$ is known. Hence, we analyze the topology of Euler flows with Morse-Bott Bernoulli function that can have stagnation points. We show that the Bernoulli function cannot have a non-degenerate critical point, and use it to prove that the topology of the ambient manifold can prevent the existence of Bott integrable fluids.
 
 \begin{theorem}\label{thm:sing}
Let $M$ be a three manifold that is not of graph type. Then $M$ does not admit a solution to the Euler equations (for any metric) with a Morse-Bott Bernoulli function.
\end{theorem}  
For example, the class of hyperbolic manifolds satisfies the hypotheses of this theorem. This is the first example of a topological obstruction to the existence of an integrable fluid, and answers the question raised in \cite{P} in the Morse-Bott case.
 
 \textbf{Plan of the paper:} In Section \ref{sec:pre} we recall the necessary definitions and results concerning the Euler equations in Riemannian manifolds and the construction of Seifert manifolds. In Section \ref{sec:arn}, we use the topological decomposition of Seifert manifolds to prove Theorem \ref{main}. Finally, in Section \ref{sec:fom}, we combine the techniques introduced in the previous section with the theory of Bott integrable systems developped by Fomenko et alli. We obtain both Theorem \ref{ArnTop} and \ref{maincor}, with the simplifying assumption that the Morse-Bott function has only simple atoms. In Section 5, we analyze Euler flows with a Morse-Bott Bernoulli function that can have stagnation points and obtain Theorem \ref{thm:sing}. In the Appendix, we cover the construction of Section $4$ for general $3$-atoms and discuss the topological classification of the moduli of Arnold fluids with Morse-Bott Bernoulli function.

\textbf{Acknowledgements:} The author is grateful to Eva Miranda, C\'edric Oms and Daniel Peralta-Salas for useful conversations. The author thanks also Alexey Bolsinov for helpful correspondence concerning the topological classification of Bott integrable systems.


\section{Preliminaries}\label{sec:pre}

In this section, we introduce the basic background in the Euler equations and Seifert manifolds, required to prove Theorem \ref{main}. We will eventually introduce other preliminaries when required through the other sections.

\subsection{Steady Euler equations and Arnold structure theorem}
When the solution does not depend on time, we can rewrite the Euler equations in terms of the dual one form $\alpha=g(X,\cdot)$ and the so called Bernoulli function. The equations are then 
\begin{equation*}
\begin{cases}
\iota_X d\alpha=-dB \\
d\iota_X\mu=0
\end{cases},
\end{equation*}
where the Bernoulli function is $B=p+\frac{1}{2}g(X,X)$ and $\mu$ is the induced Riemannian volume. A key property of this function is that it is an integral of the field $X$, since the first equation implies that $\iota_X dB=0$. We aim to study vector fields that are solutions to such equations for some metric.
\begin{defi}
Let $M$ be an odd dimensional manifold with a volume form $\mu$. A volume preserving vector field $X$ is \textbf{Eulerisable} if there is a metric $g$ on $M$ for which $X$ satisfies the Euler equations for some Bernoulli function $B:M \rightarrow \mathbb{R}$.
\end{defi}
We might refer to such vector fields as Euler flows too. In \cite{PRT} they are characterized, and the following lemma is introduced.
\begin{lemma}\label{eul}
Let $X$ be a non-vanishing volume preserving vector field in $M$. Then there exists a one form $\alpha$ such that $\alpha(X)>0$ and $\iota_Xd\alpha=-dB$ for some function $B\in C^\infty(M)$ if and only if $X$ satisfies the Euler equations for some metric $g$ satisfying $g(X,\cdot)=\alpha$.
\end{lemma}
When the Bernoulli function is assumed to be analytic, Arnold \cite{Arn} introduced a dichotomy depending on whether $B$ is constant or not. When the Bernoulli function is non constant, the flow has a rigid structure that is reminiscent of integrable systems in the context of symplectic geometry. When the function is constant, the vector field is Beltrami: it is everywhere parallel to its curl. The following statement is a version of Arnold's theorem in the context of closed manifolds. It holds also when the Bernoulli function is assumed to be $C^2$ Morse-Bott.
\begin{theorem}[Arnold's structure theorem]
Let $M$ be a closed three manifold and $X$ a flow satisfying the Euler equations for some non constant analytic Bernoulli function. Denote by $C$ the union of critical level sets of $B$, which is an analytic set of codimension at least $1$. Then $M\setminus C$ consists of finitely many domains $M_i$ and each domain is trivially fibered by invariant tori of $X$, where the flow is conjugated to the linear flow.
\end{theorem}
In \cite{CV} the authors study the case of non-vanishing solutions in the hypotheses of the structure theorem. It is proved that for an analytic Bernoulli function $B$, one can always find some other metric in $M$ such that $X$ is a solution to the Euler equations with constant Bernoulli function. In particular, it is the Reeb field of some stable Hamiltonian structure. As a corollary of the methods in the proof, topological obstructions to the existence of solutions with non constant analytic Bernoulli function are obtained. The manifold has to be a union of Seifert manifolds glued along their torus boundary components, i.e. a graph manifold. This result was obtained in \cite{EG0} for a more general context: it holds for the existence of a non-vanishing vector field with a stratified integral. It follows from a previous work in \cite{FM}. 
\begin{theorem}[{\cite[Corollary 2.10]{CV}}, {\cite[Theorem 5.1]{EG0}}]\label{top}
If a three manifold admits steady solution to the Euler equations for some metric and non constant analytic (or in general stratified) Bernoulli function, then the manifold is of graph type.
\end{theorem}
 To simplify the notation, we might refer to a vector field which is a non-vanishing steady solution to the Euler equations for some metric and non constant analytic Bernoulli function as an \textbf{Arnold fluid}. This is reminiscent of the nomenclature introduced in \cite{KKP}, where one speaks of a divergence free vector field which is \emph{Arnold integrable} when it is almost everywhere fibered by invariant tori.

\subsection{Seifert and graph manifolds}\label{Se}
Let us recall the definition and construction of Seifert manifolds, introduced and classified by Seifert \cite{Se}.



\begin{defi}
A \textbf{Seifert fiber space} is a three manifold together with a decomposition as a disjoint union of circles. 
\end{defi}
Equivalently, a Seifert fiber space admits a circle bundle over a two dimensional orbifold. Denote by $\pi:M\rightarrow B$ the bundle map over the compact base $B$. When the manifold $M$ is oriented, Seifert fiberings are classified up to bundle equivalence by the following invariants (up to some operations that yield isomorphic fiberings):
$$ M= \{ g ; (\alpha_1,\beta_1),...,(\alpha_m,\beta_m)\}.$$
The only thing we will need in this work is the way to reconstruct the manifold $M$ when given a collection of invariants. The integer $g$ denotes the genus of the base space $B$, and $(\alpha_i,\beta_i)$ are pairs of relatively prime positive integers $0<\beta_i< \alpha_i$. The integer $g$ can be negative, and then $B$ is connected sum of $g$ copies of $ \mathbb{RP}^2$. The integer $m$ represents the amount of orbifold singularities. Given such a collection of invariants, there is a precise construction to obtain the manifold $M$ that we describe following \cite{JN}. 

The base space $B$ is either an orientable surface $\Sigma_g$ of genus $g$, or decomposes as $\Sigma_{g'}\# \mathbb{RP}^2$ or $\Sigma_{g'}\# \mathbb{RP}^2 \# \mathbb{RP}^2$ for some $g'$.  To simplify, denote in any case by $\Sigma_g$ the orientable surface with genus $g$ or $g'$ respectively in each case. Remove $m$ open disks of $\Sigma_g$, that we denote by $D_i, i=1,...,m$. If $B$ has  some non-orientable part, one or two additional open disks $\tilde D_1, \tilde D_2$ are removed depending on the amount of $\mathbb{RP}^2$ components that have to be added to $\Sigma_g$ to obtain $B$. We will denote by $\Sigma_0$ the resulting surface with boundary.

Over $\Sigma_0$, we take the trivial $S^1$ bundle and denote it by $M_0= \Sigma_0 \times S^1$. At each component of the boundary, which is of the form $\partial D_i \times S^1$, we glue a solid torus $V_i= \bar D^2 \times S^1$ by means of a Dehn surgery with coefficients $(\alpha_i, \beta_i)$. 

If $B$ was orientable, this concludes the construction: we obtain a closed three manifold. If $B$ was non orientable, we denote by $M_1$ the resulting three manifold with boundary and we need to fill one or two holes $\partial \tilde D_1 \times S^1$ and $\partial \tilde D_2 \times S^1$ if there were respectively one or two copies of $\mathbb{RP}^2$ attached to the orientable part of $B$. To do so, consider the only orientable $S^1$-bundle over the M\"oebius band: namely, the twisted $I$-bundle over the Klein bottle fibered meridianally. Denote two copies of such a space as $M_2$ and $M_3$. The torus boundary of both $M_2$ and $M_3$ can be framed longitudinally by a fiber and meridianally by a section to the bundle.  The boundary components $\partial \tilde D_1 \times S^1$ and $\partial \tilde D_2 \times S^1$ in $M_1$ can be framed also longitudinally by a fiber and meridianally by a section to the bundle. Then glue $M_2$ and $M_3$ respectively to the boundary components of $M_1$ according to the identity between first homology groups represented by the given framings. This concludes the construction of $M$ in the most general case.


For a Seifert manifold with boundary, we take the base surface $B$ with some boundary circles. The boundary of $M$ is then a collection of tori. 

\begin{defi}
A \textbf{graph manifold} is a manifold obtained by gluing Seifert spaces with boundary along their torus boundary components.
\end{defi}
These manifolds were introduced and classified by Waldhausen \cite{W1,W2}. Observe that this is a larger class of manifolds, since the gluing of the boundary tori might not match the fibering in each piece. Hence the resulting total space might not admit a foliation by circles.

\section{Arnold fluids in Seifert and graph manifolds}
\label{sec:arn}
In this section we will prove Theorem \ref{main}. To do so, we first show that any Seifert manifold $M$ admits an Arnold fluid. In view of Lemma \ref{eul}, we only need to prove that there is vector field $X$ in $M$ and a one form $\alpha$ such that
\begin{itemize}
\item $X$ is volume preserving,
\item $\alpha(X)>0$,
\item $\iota_Xd\alpha=-dB$ for some analytical function $B$.
\end{itemize}
The strategy of the proof is to break the manifold $M$ into pieces according to the construction detailed above, construct a vector field and a one form satisfying some conditions in each piece and glue them together in an appropriate way. Concretely, we will show that in a neighborhood of the gluing locus, one can interpolate between the Arnold fluids in each piece to obtain a globally defined steady Euler flow. Once we have a globally defined pair $(X,\alpha)$, we show it is a steady Euler flow for some analytic Bernoulli function and hence an Arnold fluid. Finally, we adapt the interpolation to glue Seifert manifolds along their boundary to deduce the general case of graph manifolds.

 
\subsection{Building pieces}\label{ssec:blocks}
Let $M$ be a three dimensional manifold which is a Seifert fibered space. Hence $M$ is an $S^1$-fibration over an orientable surface $\Sigma$. To cover the most general case, we shall assume that the base space decomposes as $\Sigma_g \# \mathbb{RP}^2 \# \mathbb{RP}^2$. As in Subsection \ref{Se}, $\Sigma_0$ is an orientable surface of genus $g$ and with some removed open disks $D_i, i=1,...,m$ and $\tilde D_i, i=1,2$. It is a surface with boundary $M_0= \Sigma_0 \times S^1$. We think of $\Sigma_0$ as being embedded in $\mathbb{R}^3$ so that there is a naturally defined height $h$ function, whose minimum is equal to $1$ and is reached at the boundary components of $\Sigma_0$. This height function lifts trivially  to $M_0$. See Figure \ref{fig:emb} for an example with genus $2$, two singular fibers and one $\mathbb{RP}^2$ component.
\begin{figure}[!ht] 

\begin{center}
\begin{tikzpicture}
     \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[scale=0.16]{embedding.png}};
     
     \node at (4.5,5){$\Sigma_0$};
     \node at (0,0)[scale=0.8]{$\partial D_1$};
     \node at (2,0)[scale=0.8]{$\partial D_2$};
     \node at (4,0)[scale=0.8]{$\partial \tilde D_1$};
     \draw [->] (6,0) -- (6,6);
     \node at (6.5,3) {$h$};
     \node at (6.35,0.43) {$1$};
     \node at (6,0.43) {$-$};
\end{tikzpicture}
\caption{Height function in $\Sigma_0$}
\label{fig:emb}
\end{center}
\end{figure}


In $M_0$ denote by $\theta$ the coordinate in the $S^1$ component. Consider the vector field $ X= \pp{}{\theta}$, 
and as connection form $\alpha= hd\theta$. We have $\alpha(X)>0$ and $\iota_X d\alpha=-dh$.

For each solid torus $V_i \cong \bar D^2 \times S^1$, we take coordinates $(r,\varphi',\theta')$. Consider in $V_i$ the vector field $Y= \pp{}{\theta'}$ and as one form $\beta= v(r) d\theta'$, where $v'>0$ in $(0,1]$, close to $r=1$ the function is $r$ and close to $r=0$ the function is $\varepsilon + r^2$. We have $\beta(Y)>0$  and $\iota_Yd\beta= -v'(r)dr$. Hence taking as Bernoulli function $B=\int v'(r)dr$ such that $B(0)=0$, we have that $\iota_Y d\beta=-dB$.

The remaining blocks $M_1$ and $M_2$ are twisted $I$-bundles over a Klein bottle. This space $M_1\cong M_2\cong K^2\tilde \times I$ can be seen as the mapping torus of
\begin{align*}
\phi: S^1\times [-1,1] &\longrightarrow S^1\times [-1,1] \\
 		(\theta,r)  &\longmapsto (-\theta, -r).
\end{align*} 
Hence we have $K^2 \tilde \times I= \frac{(S^1\times [-1,1])\times [0,1]}{ ((p,0)\sim (\phi(p),1))}$. Such manifold is foliated by tori parallel to the boundary, together with a Klein bottle at the core. If we denote by $r$ the coordinate in $[-1,1]$, the function $r^2$ is well defined in the mapping torus total space. Furthermore, we have a natural defined vector field $Z=\pp{}{\theta'}$, where $\theta'$ is in $[0,1]$, induced by the flow of the mapping torus and that is $2$-periodic. Construct a function $v(r^2)$ such that close to $r=0$ it is $\varepsilon + r^2$ and close to $r=1$ it is $r^2$. By taking $\gamma=v(r)d\theta'$, we have $\gamma(Z)>0$ and $\iota_Y d\gamma =-dG$ where $G=\int v'(r)dr$ with $G(0)=0$. 

\subsection{Interpolation Lemma and gluing}

In order to reconstruct the whole manifold, we need first to paste the sets $V_i= \overline{D_i}\times S^1$ to the boundary components $\partial D_i \times S^1$ of $M_0$ by means of a Dehn twist. Denote $U_i \cong [1,2) \times S^1 \times S^1$ the boundary components at $D_i$ of $M_0$ and fix one of them that we denote by $U$ with coordinates $(t, \varphi, \theta)$. We can assume the first component $t$ corresponds to the restriction of $h$ to $U$, and the coordinate $\theta$ is on the $S^1$ fibre over $\Sigma_0$. Denote by $\lambda= \{1\} \times \{\varphi\} \times S^1$ and by $\mu= \{1\} \times S^1 \times \{ \theta \}$ the longitude and the meridian of 
the boundary component $\{1\} \times T^2$ of $U$. We will glue a solid torus $V= D^2 \times S^1$ to the boundary of $U$. Again take $\lambda_1= \{ p\in \partial {\bar D^2} \} \times S^1$ and $\mu_1= \partial {\bar D^2} \times \{\theta'\}$ the longitude and the meridian of $V$. The Dehn surgery is described by gluing in a way that 
\begin{align*}
\varphi:  \partial V_i &\longrightarrow \partial U_i \times S^1 \\
\mu_1 &\longrightarrow p\mu + q\lambda \\
\lambda_1 &\longrightarrow m \mu + n\lambda
\end{align*}
Coordinate wise we have that $\varphi'=p\varphi+q\theta$ and $\theta'= m \varphi+ n\theta$. We can assume that the surgery is such that the radial coordinate $r$ in $V$ is sent to $t$ in the neighborhood of the boundaries.  The vector field $Y=\pp{}{\theta'}$ which generates the longitude, is sent to $Y|_{\partial U}=  \frac{q}{qm-pn} \pp{}{\varphi}+\frac{p}{pn-qm} \pp{}{\theta}$ in the surgered target coordinates, the boundary of $U$. This can be easily deduced by the fact the coordinate wise we have 
\begin{equation*}
\begin{cases}
\varphi&= \frac{n}{pn-qm} \varphi' + \frac{q}{qm-pn} \theta' \\
\theta&= \frac{m}{qm-pn} \varphi' + \frac{p}{pn-qm}\theta'
\end{cases}.
\end{equation*} 
Analogously, the one form $\beta$ is sent to $\beta|_{\partial  U}= t(m d\varphi + nd\theta)$. This follows from the fact that near the boundary of $B$, the one form is $\beta= r d\theta$. It satisfies $\beta(Y)=t(\frac{qm}{qm-pn}+\frac{pn}{pn-qm})>0$.


Once we have our building blocks of an Euler flow, we need to glue the flows in a smooth way: both the vector field and the one form, and such that the critical set of the Bernoulli function is controlled. More precisely, we can do this interpolation by keeping the Bernoulli function regular.

\begin{lemma}[Interpolation Lemma]\label{int}
Suppose we are given a vector field $Y=A\pp{}{\theta}+B\pp{}{\varphi}$ in the torus $T^2$ and a one form $\gamma=Cd\theta + D d\varphi$, for some constants $A,B,C,D$, such that $\gamma(Y)=AC+DB=1>0$. Denote $t$ the coordinate in $[1,2]$. Then there is a volume perserving vector field $X$ and a one form $\alpha$ in $T^2\times [1,2]$ such that
\begin{itemize}
\item $X|_{\{t=1\}}=\pp{}{\theta}$, $\alpha|_{\{t=1\}}=d\theta$,
\item $X|_{\{t=2\}}=Y$, $\alpha|_{\{t=2\}}=\gamma$,
\item $\alpha(X)>0$ everywhere,
\item $\iota_X d(t\alpha)=-dh$ where $h(t)$ is a function without critical points and equal to $t$ at the boundary.
\end{itemize}
\end{lemma}
\begin{proof}
Break the interval $[1,2]$ into seven disjoint intervals $I_1,...,I_7$, for instance $I_i=[1+\frac{i-1}{7},1+\frac{i}{7}]$. Denote by $H_i(t)$ a cutoff function with support in $I_i$ such that $H_i'\geq 0$, with $H_i=0$ at $1+\frac{i-1}{7}$ and $H_i=1$ at $1+\frac{i}{7}$.

Since we have $AC+DB>0$, we might assume that $A$ and $C$ are of the same sign. Otherwise, the constants $D$ and $B$ are of the same sign and an analogous interpolation is done.

\begin{enumerate}
\item In the first interval, take $X=\pp{}{\theta}$ and $\alpha=d\theta+ H_1(t)d\varphi$. We have $\alpha(X)=1>0$.

\item In the second interval, take $X=\pp{}{\theta} + H_2(t)\pp{}{\varphi} $ and $\alpha=d\theta+ d\varphi$. We have $\alpha(X)=1+H_2>0$.

\item In the third interval, take $X= (1-H_3(t))\pp{}{\theta}+ \pp{}{\varphi}$ and $\alpha=d\theta+d\varphi$. We have $\alpha(X)=(1-H_3)+1>0$.

\item In the fourth interval, take $X=\pp{}{\varphi}$ and $\alpha=(1+H_4(t)(C-1))d\theta + d\varphi$. We have $\alpha(X)=1>0$.
\item In the fifth interval, take $X=H_5(t)A\pp{}{\theta} + (1-H_5(t))\pp{}{\varphi}$ and $\alpha= Cd\theta + d\varphi$. We have $\alpha(X)= ACH_5+(1-H_5)>0 $ since $AC>0$.
\item In the sixth interval, take $X=A\pp{}{\theta}$ and $\alpha=Cd\theta + (1+H_6(t)(D-1))d\varphi$. We have $\alpha(X)=AC>0$.
\item In the last interval, take $X=A\pp{}{\theta}+ H_7(t) B\pp{}{\varphi}$ and $\alpha= Cd\theta + Dd\varphi$. We have $\alpha(X)=AC+H_7BD>0$. It is positive because if $BD$ is positive, then $AC+H_7 BD> AC>0$ and if $BD$ is negative then $AC+H_7BD>AC+BD>0$.
\end{enumerate}
This a priori arbitrary way of interpolating by steps is done to achieve both that $h$ has no critical points and that it suits an application in the last part of this section. The key point in achieving that $h$ has no critical points is to avoid decreasing a component in $\alpha$ which is non-vanishing when evaluated at $X$.


We constructed $(X,\alpha)$ satisfying the first three conditions. To check the fourth condition, we will do it interval by interval. If we compute $\iota_Xd(t\alpha)$ we obtain:
\begin{equation*}
\iota_Xd(t\alpha)=
\begin{cases}
- dt \qquad &t\in I_1 \\
-[1+ H_2 ]dt \qquad &t\in I_2\\
-[ 2-H_3 ]dt \qquad &t\in I_3\\
-dt \qquad &t\in I_4\\
-[ H_5AC+(1-H_5) ] dt \qquad &t\in I_5\\
- AC dt \qquad &t\in I_6\\
- [ AC+H_7BD ]dt\qquad &t\in I_7
\end{cases}
\end{equation*}
It is clear from the case by case description that the function $h$, which is the indefinite integral with respect to $t$ of the $dt$-term, has no critical points in $[1,2]$. At the boundary we have that $\iota_Xd(t\alpha)=-dt$.

The vector field preserves the volume form $\mu=dt\wedge d\varphi \wedge d\theta$, since at any point the vector field is of the form $X=A(t) \pp{}{\theta} + B(t) \pp{}{\varphi}$ which implies that $\mathcal{L}_X \mu=d\iota_X\mu=0$.
\end{proof}
We can now apply this lemma in $U= [1,2] \times T^2$ to interpolate between $(Y,\beta)$ and $(X,\alpha)$. By Lemma \ref{int}, we can do so in such a way that the function $h$ extends in the building blocks as $B$ and has no critical points in $U$. Hence we constructed a pair $(\tilde {X},\tilde {\alpha})$ in $M_1$ with $\tilde {\alpha}(\tilde {X})>0$ and $\iota_{\tilde {X}}d\tilde {\alpha}=-dB'$ for some function $B'\in C^\infty(M)$.


Finally, it only remains to glue the $M_1$ and $M_2$ to $W_1= \partial \tilde D_1 \times S^1$ and $W_2= \partial \tilde D_2 \times S^1$. Take for example $M_1$, and $M_2$ is glued analogously. The vector field $Z$ is tangent to the leaves of the foliation (by torus and core Klein bottle) in $M_1$, and on the torus leaves $Z$ is linear and periodic. Hence at the surgered boundary, the vector field $Z$ and the one form $\gamma$ are sent to $Z=C_1\pp{}{\theta}+C_2\pp{}{\varphi}$ and to $t\gamma=t(D_1d\theta+D_2\varphi)$ for some constants $C_1,C_2,D_1,D_2$ with $\gamma(Z)=C_1D_1+C_2D_2=1$.


We are under the hypotheses of Lemma \ref{int} in a neighborhood of the surgery locus, so we can obtain a vector field and one form $(X', \alpha')$ in $W_1$ that can be extended in $M_0$ as $(\tilde {X},\tilde {\alpha})$ and in $M_1$ as $(Z, \gamma)$. Doing this to both pieces $M_1$ and $M_2$, we constructed a globally defined function $B\in C^\infty(M)$ such that $\alpha'(X')>0$, the equation $\iota_{X'} \alpha'=-dB$ satisfied and $B$ coincides with $\tilde B$ in $M_1 \setminus U_1 \cup U_2$ and with $G$ in $M_1$ and $M_2$. 

\subsection{Volume preservation and critical set}\label{ss:volcrit}

To prove that $X'$ is volume preserving, we will prove that it preserves some volume in each part of the manifold. 

\begin{enumerate}
\item Denote by $A$ a neighborhoods of the boundary circles of $M_0$ where we applied the interpolation lemma. Then in $M_0\setminus A$ we have that $X'=X=\pp{}{\theta}$ and $X$ clearly preserves $\mu= \mu_{\Sigma_0} \wedge d\theta$, where $\mu_{\Sigma_0}$ is any area form of $\Sigma_0$. This follows from the fact that $\iota_X\mu= \mu_{\Sigma_0}$, which is a closed form.
\item In any interpolation area, it is by construction volume preserving. The vector field is of the form $X'= A(t)\pp{}{\theta}+B(t)\pp{}{\varphi}$. Taking as volume $\mu_U= dt \wedge d\varphi \wedge d\theta$, we have that 
$$ \iota_{\tilde {X}}\mu_U= A(t) dt \wedge d\varphi - B(t)dt\wedge d\theta, $$
 which is again closed.
 \item In every solid torus $V_i$, we have $Y=\pp{}{\theta'}$ which preserves the volume $rdr\wedge d\varphi'\wedge d\theta'$. 
 \item In the mapping tori $M_1$ and $M_2$, the vector field $X'=\pp{}{\theta'}$ preserves the volume form $\mu= \mu_S \wedge d\theta' $, where $\mu_S=d\varphi \wedge dt$ is a volume of $S^1\times [-1,1]$. This follows from the fact that diffeomorphism $\varphi$ that we used for the mapping torus preserves the area form $d\varphi \wedge dt$
\end{enumerate}
It is now standard to construct a globally defined volume $\mu$ on $M$ preserved by $X'$. We just proved that $X'$ is volume preserving and admits a one form $\alpha'$ such that $\alpha'(X')>0$ and $\iota_{X'} \alpha'=-dB$ for some function $B\in C^\infty(M)$. By applying Lemma \ref{eul}, we deduce that $X'$ satisfies the Euler equations for some metric with Bernoulli function $B$.



\paragraph{\textbf{Analiticity}}
The function $B$ is a priori only smooth. However, we have the following theorem of equivalence between smooth and analytic functions. We state a particular case which is enough for our purposes.

\begin{theorem}[{\cite[Theorem 7.1]{Sh}}] \label{An}
Let $f$ be a smooth function on a manifold $M$. Suppose that at every point we have locally that:
\begin{enumerate}
\item $f$ is regular,
\item $f$ is the sum of a constant and a power of a regular function,
\item $f$ is $\pm x_1^2 \pm ... \pm x_k^2 + \operatorname{const}$ for suitable coordinates $(x_1,...,x_n)$.
\end{enumerate}
Then $f$ is equivalent to an analytic function.
\end{theorem}


We only need to prove that $B$ satisfies the conditions of this theorem, then $B$ is analytic taking some suitable charts. In $M_0\setminus A$, we have that $B=h$. We initially defined $h$ in $\Sigma_0$ where it is a Morse function, hence $h$ is a Morse-Bott function in the considered three dimensional space. In the solid tori $V_i$ attached via Dehn surgery, we have that $B=r^2$ and the only singularity is of Morse-Bott type (the critical core circle). In the pieces $M_1$ and $M_2$, the Bernoulli function has a Klein bottle as singular set, since at the core of the mapping torus we have $B=r^2$. The singularity is again of Morse-Bott type. Finally, on the regions $U_i$ and $W_i$, we know that $B$ is regular. 


Hence, the only singular points of $B$ admit an expression of type (2) or (3). We just proved that every Seifert manifold with orientable base admits a steady Euler flow with a Morse-Bott Bernoulli function, and by Theorem \ref{An} equivalent to a non constant analytic Bernoulli function.

\begin{theorem}\label{main2}
Every Seifert manifold admits a non-vanishing steady solution to the Euler equations (for some metric) with non constant analytic Bernoulli function. Analogously for a smooth Morse-Bott function.
\end{theorem}


To prove the general case of graph manifolds, it only remains to glue Seifert manifolds with boundary and obtain globally defined Arnold fluids.

\subsection{Proof of Theorem \ref{main}}
Let $M_1,M_2$ be two Seifert manifolds with boundary. We shall assume that there is only a single torus component in the boundary. The manifold $M_1$ is glued to $M_2$ by a diffeomorphism 

$$\varphi: \partial M_1 \longrightarrow \partial M_2,$$
 between both torus boundaries. By Theorem \ref{main2}, we can construct steady Euler flows with non constant analytic Bernoulli function in both manifold $M_1,M_2$. The idea is to introduce a singular torus to the Bernoulli function and then apply the interpolation lemma. It follows from the construction in the previous section that we can assume that in the neighborhood of the boundaries $\partial M_i=U_i=T^2 \times [1,2]$ we have that $X_i=\pp{}{\theta_i}$ and $\alpha=t_i d\theta_i$ where $(\theta_i,\varphi_i,t_i)$ are coordinates in $U_i$. Once we glue the boundaries, we can construct a coordinate $t$ in $U=T^2 \times [-1,1]$ such that 
\begin{align}\label{par}
&t^2=t_1 \text{ for } t\in [-1,-1+\varepsilon] \\
  &t^2=t_2 \text{ for } t\in [1-\varepsilon,1]\label{par2}
\end{align} 
  
We can assume that $T^2 \times \{-1\}$ is the boundary of $M_1$ and $T^2\times \{+1\}$ is the boundary of $M_2$, and we thickened the gluing torus. Consider the coordinates $(\theta_2,\varphi_2,t)$ of $U$, obtained by extending $\theta_2,\varphi_2$ from the boundary of $M_2$ to $U$. 
 
 
The gluing diffeomorphism, which is in the mapping class of a two torus, can be assumed to be a Dehn twist (up to isotopy). This implies that the vector field $X_1$ and the one form $\alpha_1$, which are defined $\{-1\}\times T^2$ are of the form $X_1=C_1\pp{}{\theta_2} + C_2\pp{}{\varphi_2}$ and $\alpha_1=t^2( D_1d\theta_2 + D_2d\varphi_2)$ for some constants $C_1,C_2,D_1,D_2$. In the other hand, we have $X_2=\pp{}{\theta_2}$ and $\alpha_2=t^2d\theta_2$ defined in $T^2 \times \{1\}$. Using Lemma \ref{int} in $[-1,-1+\varepsilon]$, we obtain a vector field $X'$ and a one form $\beta$ satisfying $\iota_{X'}\beta=-dH$ for some function $H$ without critical points that extends as $t^2$ in the boundary of $[-1,-1+\varepsilon]$. Furthermore, in $\{-1\}\times T^2$, the pair is equal to $(X_1,\alpha_1)$ and in $\{-1+\varepsilon\}$ is equal to $(\pp{}{\theta},t^2d\theta)$. Finally, the vector field $\pp{}{\theta}$ and the one form $t^2d\theta$ extend trivially to $[-1+\varepsilon, 1]$ and hence to all $M_1 \cup_{\varphi}M_2$ by conditions \eqref{par} and \eqref{par2}. 

We obtained a global volume preserving vector field $X'$ and a one form $\beta$ such that $\beta(X)>0$ and $\iota_X \beta=-dB$. The only new critical level set of the Bernoulli function is given by $\{0\} \times T^2$, a non degenerate critical torus, since there $\alpha=t^2d\theta$.

This concludes the proof of the Theorem \ref{main}, since the real analiticity of $B$ follows from Theorem \ref{An}.
\section{Fomenko's theory for Arnold fluids}\label{sec:fom}

In this section, we use the theory of Bott integrable systems studied by Fomenko et alli to realize Arnold fluids with any possible Morse-Bott Bernoulli function. The connection between these steady flows and the theory of Morse-Bott integrable systems was already observed in \cite{EG0}. We will show that any topological configuration (in the sense of a graph manifold and a given admissible Morse-Bott function) can be realized by an Arnold fluid. To simplify the discussion, we will treat in this section the case where the Morse-Bott integral contains a single connected critical submanifold in each connected component of the critical level set. In the language of atoms introduced in \cite{BF}, this means that we assume that the Bott integral has only simple atoms. We leave for the Appendix the case of arbitrary $3$-atoms, where we also discuss the topological classification of the moduli of Arnold fluids with Morse-Bott Bernoulli function. Taking into acccount this appendix, we get a proof of Theorem \ref{main2}.

\subsection{Topology of Bott integrable systems}
\label{ssec:blocks2}

We will describe in this subsection some aspects of the topological classification of integrable systems with Bott integrals in isoenergy surfaces of dimension three. For more details on this theory, we confer the reader to \cite{F1} which we will mainly follow in the discussion. 


Consider a symplectic manifold $(M,\omega)$ of dimension four with an integrable Hamiltonian system $F=(H,f)$. That is, they satisfy $dH\wedge df \neq 0$ almost everywhere and they commute with respect to the Poisson bracket $\{f,H\}=0$. Denote by $Q$ a three dimensional regular isoenergy hypersurface of $H$, and assume that $f$ restricts to $Q$ as a Morse-Bott function. 


Let us denote by $(H)$ the class of orientable closed three manifolds that are isoenergy hypersurfaces of some integrable Hamiltonian system with the properties described above, in some four dimensional symplectic manifold with boundary. Similarly, we denote by $(G)$ the class of orientable graph manifolds and $(Q)$ the class of three manifolds that can be decomposed into the sum of "elementary bricks" which are solid tori $D^2\times S^1$, a torus times an interval $T^2\times I$ or $N^2 \times S^1$. Here $N^2$ denotes a disk with two holes. In a series of papers \cite{BF,F2,FZ1,FZ2}, it was proved that all three classes coincide.
\begin{theorem}[Brailov-Fomenko, Fomenko, Fomenko-Zieschang]\label{FZ}
The three classes coincide, i.e. we have $(H)=(Q)=(G)$.
\end{theorem}

Even better, with the assumption we took on the critical level set of $f$, up to five types of blocks describe the topology of the foliation induced by the Bott integral. These five blocks are:
\begin{itemize}
\item Type $I$: The solid torus $S^1 \times D^2$.
\item Type $II$: The thick torus $T^2 \times [1,2]$.
\item Type $III$: The space $N^2 \times S^1$, where $N^2$ is a $2$-dimensional disk with two holes.
\item Type $IV$: The mapping torus of $N^2$, with a rotation of angle $\pi$ that we will denote $N^2 \tilde \times S^1$.
\item Type $V$: The mapping torus of $S^1 \times [-1,1]$ by the diffeomorphism $\varphi(\theta,t)=(-\theta,-t)$. 
\end{itemize}

Denote by $f$ the Bott integral and consider the following integers: $m$ stable periodic orbits (minimum or maximum), $p$ critical tori (minimum or maximum), $q$ the number of unstable critical circles with orientable separatrix diagram, $s$ the number of unstable critical circles with non-orientable separatrix diagram, and $r$ the number of critical Klein bottles (minimum or maximum). Then the manifold $M$ can be represented as $M=mI+pII+qIII+sIV+rV$, gluing the elementary blocks by certain diffeomorphisms of the torus boundary components. 


If we further indicate how the blocks are connected by means of edges (if we want, oriented with respect to the increase direction of the function), we obtain a complete topological description of the level sets of the Bott integral.


The topology of the function is then determined by some graph, as similarly discussed in \cite{Cas}. The graph satisfies that each vertex has one, two or three edges. That is because blocks $I$ and $V$ have one torus in the boundary, blocks $II$ and $IV$ have two boundary tori and blocks $III$ has three boundary tori. In order to fix the topology of the ambient manifold, one needs to specify the mapping class of each gluing diffeomorphism: the coefficient of a Dehn twist. As described in \cite[Section 4.1]{F1}, there is a family of natural choices of framings (all equivalent) in each boundary torus, and hence the coefficient of a Dehn twist determines the gluing isotopy class. Fixing a graph with Dehn coefficients in each edge determines both the topology of the manifold and the topology of the function. 


The whole theory applies also to Arnold fluids with a Morse-Bott Bernoulli function. The topological properties originally follow from the analysis of the neighborhood of the critical set of the Morse-Bott integral of some non-vanishing vector field, and hence works analogously in this context. This was already used in \cite{EG0}.


\begin{example}
Take for example the Arnold fluid constructed in Theorem \ref{main} for Figure \ref{fig:emb}. Asssume we take a height function that only has a critical point in each value: assume for example that the critical value joining $\partial D_2$ and $\partial \tilde D_1$ is lower than the critical value joining $\partial D_1$ and $\partial D_2$. A representation of the graph associated to such topological decomposition would be Figure \ref{fig:graph}. We took a framing in the boundary of the Klein bottle neighborhood for which the gluing is trivial as described in Section \ref{sec:pre}. Whenever the Dehn coefficients are trivial in some gluing, nothing is indicated in the edge. The coefficients $(\alpha_i,\beta_i)$ are indicated by the Seifert invariants.

\begin{figure}[!ht]
\begin{center}
\begin{tikzpicture}
     \node[scale=0.8] at (-0.8,-2,0) {$I$};
     \node[scale=0.8] at (1,-2,0) {$I$};
     \node[scale=0.8] at (2.8,-2,0) {$V$};
     
     \node[scale=0.8] at (1.9,-1) {$III$};
     \node[scale=0.8] at (1,0) {$III$};
     
     \node[scale=0.8] at (1,1,0) {$III$};
     \node[scale=0.8] at (1,2,0) {$III$};
     \node[scale=0.8] at (1,3,0) {$III$};
     \node[scale=0.8] at (1,4,0) {$III$};
     \node[scale=0.8] at (1,5,0) {$I$};
     
     \draw (-0.7,-1.8) -- (0.8,-0.2);
     \draw (1,0.2) -- (1,0.8);
     \draw (2.7,-1.8) -- (2,-1.2);
	 \draw (1.1,-1.8) -- (1.8,-1.2);   
	 \draw (1.8,-0.8) -- (1.1,-0.2); 
	  
     
     \draw (1.1,1.2) arc(-50:50:0.4);
     \draw (0.9,1.8) arc(130:230:0.4);
     
     \draw (1,2.2)--(1,2.8);
     
     \draw (1.1,3.2) arc(-50:50:0.4);
     \draw (0.9,3.8) arc(130:230:0.4);
     
     \draw (1,4.2) -- (1,4.8);
     
     \node[scale=0.7] at (-0.5,-0.9) {$(\alpha_1,\beta_1)$};
     \node[scale=0.7] at (0.85,-1.45) {$(\alpha_2,\beta_2)$};     
     
     \draw [->] (4,-2) -- (4,5);
     \node at (5,1.5) {$B$};
     
     \node at (-3,0) {};
\end{tikzpicture}
\end{center}
\caption{Example of graph representation}
\label{fig:graph}
\end{figure}

\end{example}

\subsection{Topological realization of Arnold fluids}


Using similar arguments as we did to prove Theorem \ref{main}, we can construct an Arnold fluid realizing each topogical configuration. Let us start by constructing an Arnold fluid in each of the "elementary blocks".

\begin{prop} \label{ArnBl}
All blocks admit an Arnold fluid with the following properties. For type $I$, the longitudinal core circle is a minimum or maximum of the Bernoulli function. For type $II$, the torus $T^2 \times \{3/2\}$ is a minimum or maximum of the Bernoulli function. For type $III$, critical set is a figure eight times a circle: the central circle is of saddle type. For type $IV$, exactly as for type $III$ but with a non orientable separatrix diagram for the critical circle. For type $V$, the core Klein bottle is a minimum or maximum of the Bernoulli function. In all cases, the boundary components are regular level sets of the Bernoulli function. For blocks $III$ and $IV$, we can assume that the Bernoulli function decreases (or increases) outwards in the exterior of $N^2$ boundary component and respectively increases (or decreases) in the other boundary components.
\end{prop}

\begin{proof}
We give construct in each block a vector field $X$ and a one form $\alpha$ such that $\iota_X d\alpha=-dh$ for some function $h$ satisfying the mentioned properties. In all cases, it is easy to check that the vector field is volume preserving as in Subsection \ref{ss:volcrit}.

Types $I$ and $V$ follow from the discussion in Section \ref{sec:arn}. In the first one, the vector field is the longitudinal flow $\pp{}{\theta}$ with one form $v(r)d\theta$, where $\theta$ is the longitudinal coordinate of the solid torus and $r$ the radial coordinate in $D^2$.  The function $v(r)$ is equal to $(\varepsilon + r^2)$ close to $r=0$ and equal to $ r$ close to the boundary $\{r=1\}$ if it is a minimum. If it a maximum we can take for example $v(r)=1+\varepsilon -r^2$ close to $r=0$ and $v(r)=1+\varepsilon - r$ close to $r=1$.  


Type $V$ is given by the mapping torus with core Klein bottle introduced in Subsection \ref{ssec:blocks}. The vector field is given by the mapping torus direction $\pp{}{\theta}$, and again the one form is $v(r^2)d\theta$, where the function $v$ is equal to $\varepsilon + r^2$ close to $r=0$ and $r^2$ close to $r=1$. Similarly for a maximum, take $1+\varepsilon -r^2$ close to $r=0$ and $1+\varepsilon - r^2$ close to $r=1$. Recall that $r$ is the coordinate in $[-1,1]$, for the mapping torus obtained by the diffeomorphism $\varphi: S^1 \times [-1,1] \rightarrow S^1 \times [-1,1]$.

For the type $II$ block, consider the standard coordinates $(\theta, \varphi,t)$ in $T^2 \times [1,2]$. Take the vector field $X=\pp{}{\theta}$ and the one form $\alpha=v(t)d\theta$. If we want a minimum, we choose as function $v(t)=+\varepsilon + (t-3/2)^2)$, which implies that the Bernoulli function is $h=\int_t v'(t)= t^2-3t $. If we want it as a maximum then $v(t)=1+\varepsilon-(t-3/2)^2$ and $h=-t^2+3t$. We have $\iota_Xd\alpha=-dh$ and $\alpha(X)>0$.

For the type $III$ and type $IV$, denote by $\theta$ the coordinate in the $S^1$ component. Take a one form $\alpha=v(x,y)d\theta$, where $(x,y)$ are coordinates in $N^2$. Taking the function $v$ such that it has a saddle point in $(0,0)$ and two minima or maxima in $(\pm 1,0)$ is enough. For example, take $v(x,y)=K \pm \frac{1}{4}(y^2-x^2+1/2x^4)$, for some big enough constant $K>0$ added to the Hamiltonian of the Duffing equation. Then as Bernoulli function take similarly $h(x,y)= C \pm \frac{1}{4}(y^2-x^2+1/2x^4)$, for some constant $C$. We have
$$\iota_Xd\alpha=\iota_X (\pp{v}{x}dx\wedge d\theta + \pp{v}{y}dy\wedge d\theta)=-\pp{v}{x}dx-\pp{v}{y}dy,$$
 which is exactly equal to $-dh$. It is also satisfied that $\alpha(X)>0$. Depending on the sign, we obtain that $h$ is decreasing or increasing (outwards with respect to the boundary, that we take to be a level set of $h$) in the interior boundary components, and respectively increasing or decreasing in the exterior boundary components. Observe that the defined one form and function $h$ are well defined in the mapping torus in the case of type $IV$ blocks. This is because the Duffing potential is invariant with respect to the rotation of angle $\pi$, which is easily seen in polar coordinates. Hence if we denote by $p$ the projection of $N^2 \tilde \times S^1$ into $S^1$, the function $p^*h$ is well defined. This covers the case of block $IV$.
 

The $N^2$ copy that we take is the one given by the level sets of the function $h$ or $v$: i.e. the boundary and holes we take are given by some of the regular level sets of these functions. Figure \ref{fig:level} gives a representation of the critical level set and the boundary level sets given by the function.
\begin{figure}
\begin{center}
\begin{tikzpicture}
     \node[anchor=south west,inner sep=0] at (0,0) {\includegraphics[scale=0.16]{MorseEight.png}};
\end{tikzpicture}
\end{center}

\caption{Level sets of $h$}
\label{fig:level}
\end{figure}
 
 In all cases, the vector field is volume preserving and Lemma \ref{eul} concludes.
\end{proof}


Combining this result with the interpolation lemma, we can realize any configuration graph as Figure \ref{fig:graph}, and so any possible topology is realized by an Arnold fluid. We also state it in the general case of a molecule with gluing coefficients. 
\begin{theorem}\label{Mole}
Given a graph with blocks $I-V$ and Dehn coefficients, there exist an Arnold fluid with Morse-Bott Bernoulli function realizing it. In general, given a molecule with gluing coefficients there is an Arnold fluid realizing it.
\end{theorem}
\begin{proof}
As in the whole section, we restrict to graphs with simple atoms, i.e. blocks of the form $I-V$, and leave for the appendix the general case.


Take a graph with Dehn coefficients and oriented edges. Each vertex indicates the type of block (and hence of the neighborhood of some connected component of a critical level set) of the Bernoulli function. The amount of up-directed edges for type $III$ blocks indicates if in the interior boundary components or in the exterior one the Bernoulli function is increasing. We start from the bottom and construct in the minima blocks an Arnold fluid using Proposition \ref{ArnBl}. We proceed by induction.

Assume we have an Arnold fluid in a manifold with boundary $N$ realizing a subgraph of the given marked molecule. Denote by $B$ the Bernoulli function in $N$. In a neighborhood of a torus boundary component of some of its blocks, there exist coordinates $(t,\theta)$ such that the one form is $\alpha=td\theta$. We attach the following block, that we assume to be of type $III$ or $IV$, via a Dehn twist with the coefficients indicated by the edge of the graph. Using Proposition \ref{ArnBl}, we endow the block with an Arnold fluid. Up to choosing well the constants $K$ and $C$ in Proposition \ref{ArnBl}, we can make sure that the minimal value of the Bernoulli function is higher than the maximal value of $B$ in $N$. Denote by $A$ and $B$ the maximal value of $B$ in $N$ and the minimal value of the Bernoulli function in the new block. Hence in a neighborhood of the gluing locus $U(T^2) \cong T^2 \times [A,B]$ we can assume that in each boundary component $T^2 \times \{A\}$ and $T^2 \times \{B\}$ we have the Arnold fluids respectively of $N$ and the glued block. We are in the hypotheses of the interpolation Lemma \ref{int}, since the vector fields are always linear in the torus boundaries. In a neighborhood $U(T^2)$, we obtain a globally defined non-vanishing vector field $X$, and a one form $\alpha$ such that $\alpha(X)>0$ and $\iota_X d\alpha=-dB'$. Here $B'$ is a function which coincides with $B$ in $N$ except at the neighborhood where we applied the interpolation lemma. 

The cases of attaching the last blocks of type $I$, $II$ or $V$ containing a maximum are done analogously.

At the end, the Bernoulli function realizes the given graph. Furthermore, by construction, the vector field is volume preserving in each block and hence globally preserves some volume form. This proves, by Lemma \ref{eul}, that the construction yields a globally defined Arnold fluid realizing the initial graph with coefficients.



In the general case, when we can have more than one critical circle in the same connected component of the critical level set, the neighborhood of a singular leaf is a $3$-atom as described in \cite{F1}. In the Appendix we explain how to construct an Arnold fluid in the case of an arbitrary $3$-atom, in the sense of any possible foliation around a singular leaf. Then using the interpolation lemma as above proves that given any graph with arbitrary $3$-atoms of any complexity as vertices, there is an Arnold fluid realizing it. 
\end{proof} 

Theorem \ref{ArnTop} stated in the introduction is just a reformulation of the realization of any marked molecule.

\begin{Remark}
By the same arguments as in the previous section, we can in fact assume that the Bernoulli function is analytic.
\end{Remark}

\subsection{Symplectization of fluids}
In this last subsection, we will show how the Arnold fluids we constructed can be symplectized in to give rise to a Bott integrable system.

In order to transform the constructed Arnold fluids into integrable systems, we will need a one form with some properties. Recall that a stable Hamiltonian structure is a pair $(\alpha,\omega)$ of a one form and a two form such that $\alpha\wedge \omega>0$ and $\ker \omega \subset \ker d\alpha$. The equations $\iota_R \omega=0$ and $\alpha(R)=1$ uniquely define the Reeb field of the stable Hamiltonian structure. It was already proved in \cite{CV} that any non-vanishing steady solution to the Euler equations with non constant analytic Bernoulli function can be rescaled to the Reeb field of a stable Hamiltonian structure. 


For some rescaling of the solutions we constructed (with $B$ of Morse-Bott type and eventually analytic) we can explicitely construct a one form $\beta$ satisfying the condition $\iota_X d\beta=0$ and $\beta(X)>0$, but with an additional property: that it vanishes when evaluated at the curl of $X$.

\begin{lemma}\label{beta}
In Theorem \ref{ArnTop} denote by $Y$ the curl (for the constructed metric) of the steady flow $X$. Then, there exist a one form $\beta$ such that
\begin{itemize}
\item $\beta(X)>0$,
\item $\iota_{X} d\beta=0$,
\item $\beta(Y)=0$.
\end{itemize}
\end{lemma}

\begin{proof}
Let us keep the simplifying assumption that the marked molecule has only simple atoms i.e. blocks of type $I-V$. We have a solution as constructed in Theorem \ref{ArnBl}: a given Arnold fluid in each block using Proposition \ref{ArnBl}, and interpolations in each gluing locus using Lemma \ref{int}. 


For a type $I$ block of the form $S^1\times D^2$ with coordinates $(\theta,r,\varphi)$, the preserved volume form is $\mu=rdr\wedge d\varphi\wedge d\theta$. For the type $II$ block, the volume form is $dt\wedge d\varphi\wedge d\theta$ for coordinates in $T^2 \times [1,2]$. In type $III$ and $IV$, the volume form is $\mu_N\wedge d\theta$, where $\mu_N$ is an area form in the disk with two holes and $\theta$ a coordinate of the mapping torus. Finally, type $V$ block has as volume form $\mu_s\wedge d\theta$ where $\mu_S$ is an area form in $S^1 \times [-1,1]$. We want to study the curl $Y$ of the solutions constructed in Proposition \ref{ArnBl}. In cases $I$ and $II$, the curl is of the form $Y=H(r) \pp{}{\varphi}$. In the three remaining cases, the curl equation writes
\begin{align}
\iota_Y \mu &= d\alpha \\
 			&= df \wedge d\theta,\label{base}
\end{align} 
where $f$ is some function in the base space of the mapping torus. We shall prove that the form $d\theta$ satisfies $d\theta(Y)=0$.

Assume the converse, that is that $d\theta(Y)$ is not vanishing everywhere. Denote by $B$ the base of the mapping torus and $i$ the inclusion into the mapping torus total space. Then we have that $i^*\iota_Y \mu$ is not vanishing. But this contradicts equation \eqref{base}. From the whole discussion, it follows that the one form $\beta=d\theta$ satisfies $\beta(X)>0$, $\iota_X d\beta=0$ and $\beta(Y)=0$. 



Finally, we only need to extend $\beta$ in the interpolation areas. In any of the applications of the interpolation lemma we did in Theorem \ref{ArnTop}, observe that in the boundary $U=[1,2]\times T^2$ we have $\beta|_{t=1}=d\theta$ and $\beta|_{t=2}= Cd\theta +Dd\varphi$. Let us prove that in an arbitrary interpolation, we can find a $\beta$ satisfying the boundary conditions and the required conditions inside. For the volume form $dt\wedge d\theta \wedge d\varphi$, which is preserved by $X$ and extends as a globally preserved volume form, we can compute the curl of $X$. It satisfies the condition $\iota_Y \mu=d(t\alpha)$. By writing such an equation in every interval (1)-(7) of Lemma \ref{int}, we find the following expression of $Y$.

\begin{equation*}
Y=
\begin{cases}
\pp{}{\varphi} - [tH_1'+H_1] \pp{}{\theta} \qquad &t\in I_1 \\
\pp{}{\varphi} - \pp{}{\theta} \qquad &t\in I_2\\
\pp{}{\varphi} - \pp{}{\theta} \qquad &t\in I_3\\
[1+H_4(C-1)+tH_4'(C-1)]\pp{}{\varphi} - \pp{}{\theta}\qquad &t\in I_4\\
C\pp{}{\varphi} - \pp{}{\theta} \qquad &t\in I_5\\
C\pp{}{\varphi} - [1+H_6(D-1)+tH_6'(D-1)]\pp{}{\theta} \qquad &t\in I_6\\
C\pp{}{\varphi}- D\pp{}{\theta}\qquad &t\in I_7
\end{cases}
\end{equation*}
Hence, we define $\beta$ as the following one form.
\begin{equation*}
\beta= 
\begin{cases}
 d\theta + [tH_1'+H_1] d\varphi \qquad &t\in I_1 \\
d\theta + d\varphi \qquad &t\in I_2\\
d\theta + d\varphi \qquad &t\in I_3\\
[1+H_4(C-1)+tH_4'(C-1)]d\theta + d\varphi \qquad &t\in I_4\\
Cd\theta+d\varphi \qquad &t\in I_5\\
Cd\theta + [1+H_6(D-1)+tH_6'(D-1)]d\varphi \qquad &t\in I_6\\
Cd\theta + Dd\varphi\qquad &t\in I_7
\end{cases}
\end{equation*}
Such one form clearly satisfies $\beta(Y)=0$. Furthermore, looking at the expression of $X$ in Lemma \ref{int}, we have that $\beta(X)>0$ and $\iota_Xd\beta=0$.
\end{proof}

 The interpolation Lemma \ref{int} was adjusted so that one can also find the one form $\beta$.


Take $M$ to be a graph manifold and $X$ an Arnold fluid constructed as in Theorem \ref{main}, so that for some metric $g$ we have 
\begin{equation*}
\begin{cases}
\iota_X d\alpha&=-dB \\
d\iota_X\mu&=0
\end{cases},
\end{equation*}
where $\alpha=g(X,\cdot)$ and $\mu$ is the Riemannian volume. Denote $Y$ the curl of $X$ with respect to $g$. By Lemma \ref{beta} we know there is a one form $\beta$ such that $\beta(X)>0$, $\beta(Y)=0$ and $\iota_X d\beta=0$. 

\begin{Remark}
Note that for this $\beta$, the vector field $X$ is a rescaling of the Reeb field of the stable Hamiltonian structure $(\beta, \iota_X\mu)$.
\end{Remark}

Consider in $M\times \mathbb{R}$, with coordinate $t$ in the second component, equipped with the two form
$$ \omega= dt \wedge \beta +td\beta + \iota_X \mu. $$
For $t$ small enough, it is clearly a symplectic form. This is in fact the symplectization (cf. \cite{CV2}) of the stable Hamiltonian structure $(\beta,\iota_X\mu)$. The Euler flow $X$ and its curl can be seen as some Hamiltonian system with Bott integral in the symplectic manifold $M\times (-\varepsilon,\varepsilon)$ equipped with $\omega$ as symplectic form.

\begin{prop}\label{sym}
The pair $F=(t,-B)$ defines an integrable system in $M\times [-\varepsilon,\varepsilon]$. The Hamiltonian vector fields of $t$ and $B$ in $M\times \{0 \}$ are respectively $X$ and its curl. 
\end{prop}

\begin{proof}

The vector field $X$ satisfies that
$$ \iota_X \omega= -dt+t\iota_X d\beta= -dt,$$
which implies that $X$ is the Hamiltonian vector field of the function $H=t$. Furthermore, contracting $Y$ with the symplectic form we obtain
$$ \iota_Y \omega= \iota_Y \iota_X \mu + t\iota_Yd\beta. $$
Recall that $Y$ satisfies that $\iota_Y\mu=d\alpha$, so we have $\iota_Y\iota_X \mu= -\iota_X d\alpha=dB$. If $X_B$ denotes the Hamiltonian vector field of the function $-B$, we have that $X_B|_{t=0}=Y$. 


It remains to check that $F=(t,B)$ define an integrable system. Clearly, $dt\wedge dB \neq 0$ almost everywhere, since $dB$ vanishes in zero measure stratified sets. Additionally, we have
\begin{align*}
\omega(X_t,X_{B})&=-\omega(X_B,X_t)\\
				&= -\iota_{X_B}\omega (X_t) \\
				&= -dB(X) \\
				&=0.
\end{align*}
The last equality follows from the first Euler equation: the fact that $\iota_X d\alpha=-dB$.
\end{proof}

We obtain an alternative proof that any topological configuration of a Bott integrable system can be realized, with the additional property that the Hamiltonian vector field is, up to rescaling, the Reeb field of a stable Hamiltonian structure. The result is stated in the introduction as Theorem \ref{maincor}. The realization theorem for Bott integrable systems was originally proved by Bolsinov-Fomenko-Matveev \cite{BFM}.


Proposition \ref{sym} unveils an example of an explicit (and expected) relation between Arnold's structure theorem and the classical Arnold-Liouville theorem in the theory of integrable systems. However the symplectization procedure to obtain integrable systems is \emph{ad hoc}. In general, for a non-vanishing flow with an analytic or even Morse-Bott Bernoulli function, it is not possible to find a one form as in Lemma \ref{beta}. 

In a point of the critical set of the Bernoulli function, we have that $\iota_Xd\alpha=0$ and $\iota_Yd\alpha=0$. This implies that either $d\alpha$ vanishes and so does $Y$, or $Y$ is non-vanishing and parallel to $X$. It is clear that in the second case one cannot find a one form such that $\beta(X)>0$ and $\beta(Y)=0$. It is possible to find examples where this happens, using Example 4.4 in \cite{KKP}.

\begin{example}
Consider the three torus $T^3$ with the standard metric on it $g=d\theta_1^2+ d\theta_2^2 +d\theta_3^2$. We take the volume preserving vector field 
$$ X= \sin^2 \theta_3 \pp{}{\theta_1} + \cos \theta_3 \pp{}{\theta_2}, $$
which is tangent to the tori obtained by fixing the third coordinate. The curl of $X$ is given by $Y=\sin \theta_3 \pp{}{\theta_1} + 2\sin \theta_3 \cos \theta_3 \pp{}{\theta_2}$. The dual form to $X$ is $\alpha= \sin^2 \theta_3 d\theta_1 + \cos \theta_3 d\theta_2$, from which we can deduce that the analytic Bernoulli function is $B=\frac{1}{2}(\sin^4\theta_3 + \cos^2\theta_3)$. Along the torus $\theta_3=\pi/2$, we have that the derivative of the Bernoulli function vanishes. However, both $X$ and $Y$ are non-vanishing and parallel. Hence in such example one cannot find a one form as in Lemma \ref{beta}.
\end{example}

We can also produce an example with a Morse-Bott Bernoulli function.

\begin{example}
Consider the solid torus as in the block of Section \ref{ssec:blocks}. Take coordinates $(\theta,x,y)$ in $S^1\times D^2$, and denote by $(r,\varphi)$ polar coordinates in $D^2$. We consider the one form
$$ \alpha= (r^2+\varepsilon)d\theta + \varphi(r)xdy, $$
where $r=x^2+y^2$, the function $\varphi(r)$ is constantly equal to $1$ close to $0$ and equal to $0$ for $r\geq \delta$. The vector field will still be $X=\pp{}{\theta}$ and the volume form is $\mu=rdr\wedge d\varphi \wedge d\theta=dx\wedge dy \wedge d\theta$. We have
$$ d\alpha= 2rdr\wedge d\theta + (\pp{\varphi}{x}2x^3+\varphi)dx\wedge dy. $$
As before, we have that $\iota_Xd\alpha=d(r^2)$, so the Bernoulli function is Morse-Bott $B=r^2$. However, constructing a metric with Lemma \ref{eul}, the curl of $X$ is no longer $\pp{}{\varphi}$. For $r>\delta$, we have $Y=\pp{}{\varphi}$. For $r$ very close to $0$ we have that $\varphi(r)=1$ and hence $d\alpha=2rdr\wedge d\theta + dx\wedge dy$.  For such a form, the curl of $X$ is 
$$ Y=2\pp{}{\varphi} + \pp{}{\theta}, $$
which doesn't vanish at $r=0$. The construction in the solid torus can be completed to a compact manifold as done in Section \ref{sec:arn}.

\end{example}

In both cases the one form $\beta$ cannot be constructed, and in fact these fluids cannot be seen as integrable systems, as long as we ask the natural compatibility conditions that $X$ and $Y$ are respectively the Hamiltonian vector fields of the integrals $t$ and $B$. Indeed if $Y$ was the Hamiltonian vector field of $B$, it should always vanish at the critical points of $B$, since it would be defined by the equation $\iota_Y \omega=dB$ for some symplectic form $\omega$. Such condition is not satisfied in the previous examples.

\section{Singular Morse-Bott Arnold fluids}

In this last section we study the most general case of Euler flows with a Morse-Bott Bernoulli function. Those are singular Arnold fluids, and by singular we mean that we allow the vector field $X$ to have stagnation points. We will prove that these fluids do not exist in non graph manifolds.

\subsection{Critical sets of the Bernoulli function}

Let us first analyze the level sets of a Morse-Bott Bernoulli function. A first lemma is the non-existence of non-degenerate critical points.

\begin{lemma}\label{lem:nosad}
Let $X$ be a steady Euler flow with smooth Bernoulli function $B$. Then $B$ does not have any non-degenerate critical point.
\end{lemma}  
\begin{proof}
Assume there is a critical point $p$ which is not of saddle type. By the Morse lemma there is a local chart $(U,(x,y,z))$ around $c$ such that the functon is $B=x^2+y^2+z^2$ or $B=-x^2-y^2-z^2$. But then either $B^{-1}(\varepsilon)$ or $B^{-1}(-\varepsilon)$ is a regular level set diffeomorphic to a sphere. This is a contradiction with Arnold's theorem, which ensures that all regular level sets are tori. It only remains the case of a saddle point.\hfill \newline


Now let $p$ be a saddle point of $B$. Again by the Morse lemma, there are coordinates $(x,y,z)$ such that 
$$ B=x^2+y^2-z^2. $$
In these coordinates, we can write 
$$X=X_1\pp{}{x}+X_2\pp{}{y}+X_3\pp{}{z}$$
and 
$$ d\alpha=Ady\wedge dz + Bdx\wedge dz + Cdx\wedge dy, $$
where $X_1,X_2,X_3,A,B,C$ are functions depending on $x,y,z$. The first Euler equation $\iota_Xd\alpha=-dB$ implies
\begin{equation}\label{eq:system}
\begin{cases}
X_1B-X_2A=z\\
X_1C-X_3A=-y\\
-X_3B-X_2C=-x
\end{cases}
\end{equation}

We claim that $d\alpha|_p\neq 0$. Assume the converse, that $d\alpha|_p=0$. Because $B$ is an integral of $X$, we have that $X$ vanishes at $p$ necessarily. There are several ways to see this. Take coordinates $(x,y,z)$ around $p$ and denote by $X_1,X_2,X_3$ the components in each coordinates of $X$. Similarly, denote $B_1,B_2,B_3$ the derivatives of $B$ with respect to $x,y$ and $z$. The fact that $B$ is an integral of $X$ implies that $X_iB_i=0$. Deriving this equation and restricting to the critical point we get
$$(X_1,X_2,X_3) D^2B (X_1,X_2,X_3)^T|_p=0, $$
where $D^2B$ is the Hessian of $B$. This matrix is non-degenerate at $p$ and hence we deduce that $(X_1,X_2,X_3)|_p=0$.





Taking the Taylor expansions of the functions $X_1,X_2,X_3$ and $A,B,C$, they all have a vanishing coefficient of order $0$. In particular, the combinations of the system \eqref{eq:system} yield functions that vanish at order two at least. This contradicts the system of equations.


We deduce that $d\alpha|_p\neq 0$. However, we know that the vorticity $Y$ is determined by the equation
$$ \iota_Y \mu=d\alpha,  $$
which implies that $Y|_p\neq 0$. However $B$ is also an integral of $Y$ because $\iota_YdB=-\iota_Y\iota_Xd\alpha=\iota_X \iota_Yd\alpha=0$ and by the previous argument this implies that $Y|_p=0$. We obtain a contradiction, and conclude that a saddle point cannot exist.
\end{proof}

The previous lemma applies in full generality, however we will see that in the case of a non graph manifold this is enough to prove the non-existence of Bott integrable fluids. We can deduce the structure of the critical level set of $B$.

\begin{lemma}\label{lem:stru}
Let $X$ be an Euler flow with Morse-Bott Bernoulli function. Let $c$ be a critical submanifold. Then $c$ is either a circle, a torus or a Klein bottle. If $c$ is a circle of saddle type, denote by $Z$ a regular component of the critical level set containing $c$. Then $Z$ is an orientable finitely punctured surface with finite genus.
\end{lemma}

\begin{proof}
If $c$ is two dimensional, it has to be a compact surface. Furthermore, the regular level sets in a trivial neighborhood of $c$ must be tori because of Arnold's theorem. This implies that $c$ is either a torus or a Klein bottle. If $c$ is one dimensional, it is compact and hence a circle. 
The last case is when $c$ is a critical point. 


For the second part of the lemma, denote by $Z$ a $2$-dimensional strata of a critical level set: it is an open embedded surface. The fact that $Z$ is orientable follows from the fact that $dB\neq0$ everywhere in $Z$ and is transverse to it. Then the gradient of $B$, which satisfies $g(\operatorname{grad}B,\cdot)=dB$, is a vector field everywhere transverse to $Z$. This implies that $Z$ is an open orientable surface. By compactness, it has a finite amount of punctures (approaching the critical points or circles) and has finite genus.
\end{proof}



\subsection{Non existence of Bott integrable fluids}

We proceed with the proof of Theorem \ref{thm:sing}.

\begin{proof}[Proof of Theorem \ref{thm:sing}]

We will first show that a stratified Bernoulli function has necessarily a non-empty $0$-strata if $M$ is not of graph type. A function is stratified \cite{EG0} if its critical values are isolated and the critical level sets are Whitney stratified sets of codimension greater than zero. This includes both analytic and Morse-Bott functions. The claim follows easily from the theory of tame functions introduced in \cite{FM}.

Assume that there are no $0$-strata. The $1$-strata are necessarily critical circles, by compactness. By Arnold's theorem, every regular level set is a torus. Then the function $B$ is a tame function in the sense of \cite{FM}, and $M$ has to be a graph manifold: this is a contradiction.

Hence, if $B$ is a Morse-Bott Bernoulli function of some steady Euler flow in a non-graph manifold, it necessarily has an isolated critical point. By Lemma \ref{lem:nosad}, this is not possible and we conclude that such steady flow cannot exist.
\end{proof}
The Morse-Bott assumption was key in the proof, and so for the case of an analytical Bernoulli function the problem of existence of integrable steady fluids in non graph manifolds remains open.
