_g $$ computed with the help of the symplectic structure $\omega$. Let me remind that no conjugate points condition for the orbit $\vartheta$ of $g^t$ means that for any two points $x, y \in \vartheta$ $$ g^t_* V(x)\pitchfork V(y) $$ where $t$ is a time difference between $y$ and $x$ and $V(x)$, $V(y)$ are vertical subspaces at $x,y$. %Theorem 2.1 \newtheorem{theorem}{Theorem} \begin{theorem} \label{T21} Suppose that the Riemannian metric $g$ is conformally flat. Then there always exist orbits of $g^t$ on the level of $\{ H= 1/2 \}$ with conjugate points unless the 2-form $\beta$ vanishes identically. \end{theorem} % Remark 2.2 \newtheorem{remark}{Remark} \begin{remark} \label{R22} In other words, no conjugate points condition implies $\beta \equiv 0$ and then it follows from \cite{[C-F]} and \cite{[K]} that the metric is flat too. Actually, we will see it once more by our computations later. \end{remark} The proof which is suggested in this paper follows the original scheme by E. Hopf. However, it might need serious modifications if one tries to generalize the result for arbitrary Riemannian metric. The first ingredient of the proof is to construct measurable field of Lagrangian subspaces $l(x) \subseteq T_x (T^* {\Bbb T}^n)$ for any $x \in \{ H= 1/2 \}$. This field can be constructed by the following limit procedure: $$ l(x) = lim_{t \rightarrow + \infty} g^t_* V \left( g^{-t} (x) \right) $$ It was first used by E. Hopf \cite{[H]} and L. Green \cite{[G]} for Riemannian case. We refer the reader to recent paper \cite{[C-I]} for the proof in a general optical case. It follows from the very construction of $l$ that the field $l$ is invariant under $g^t$ and is transversal to the vertical field $V$ everywhere. With the construction of $l$ Theorem \ref{T21} is a corollary of the following %\ref{T23} % Theorem 2.3 \begin{theorem} \label{T23} Let $l$ be a measurable field of Lagragian subspaces invariant under the flow $g^t$. If $l$ is transversal to $V$ everywhere then $\beta$ vanishes identically, and the metric $g$ is flat. \end{theorem} This last theorem has the following dynamical interpretation. % Theorem 2.4 \begin{theorem} \label{T24} Suppose that the energy shell $\{ H= 1/2 \}$ is smoothly foliated by Lagrangian tori homologous to the zero section of $T^* {\Bbb T}^n$. Then the 2-form $\beta$ vanishes identically and the metric is flat. \end{theorem} % Remark 2.5 \begin{remark} \label{R25} In the paper \cite{[K]}, such a situation is called total integrability and simple examples of totally integrable magnetic geodesic flows are given. Let me remark that the Lagrangian torii in all these examples cannot be homologous to the zero section as it follows from Theorem \ref{T24}. It is an interesting, completely open problem to characterise totally integrable magnetic geodesic flows. \end{remark} \noindent{\large \bf Acknowledgements} \noindent I was introduced to the subject of magnetic fields by Viktor Ginzburg (UC Santa Cruz). He suggested to me the question on the rigidity for magnetic fields. I am deeply grateful to him for many very useful discussions. The results of this paper were presented at the Symplectic Geometry meeting in Warwick 1998 and on the Arthur Besse Geometry Seminar. I am thankful to D. Salamon, F. Laudenbach and P. Gauduchon for inviting me to speak there. I would also like to thank the EPSRC and Arc-en-Ciel for their support. \section{Proofs} Let me explain first how Theorem \ref{T24} follows from Theorem \ref{T23}. Since the torii are Lagrangian and homologous to the zero section then the 2-form $\beta$ must be exact. Denote by $\alpha$ the primitive 1-form. Then one can easily see that the flow $g^t$ is equivalent to the Hamiltonian flow $\tilde g^t$ of the function $\tilde H = 1/2 < p - \alpha, p - \alpha>_g$ with respect to the standard structure $\omega_0$. This equivalence is given by the diffeomorphism $(q,p) \rightarrow (q,p + \alpha)$ which is fiber preserving. Note that the function $\tilde H$ is strictly convex with respect to $p$. It follows from generalised Birkhoff theorem (see \cite{[B-P2]} for its most general form and for the survey and discussions) that all the Lagrangian torii are the sections of the cotangent bundle. But then the distribution of their tangent spaces meets the conditions of Theorem \ref{T23}. {\bf Proof of Theorem \ref{T23}} We shall work in standard coordinates $(q,p)$ on $T^* {\Bbb T}^n$, such that the Riemannian metric $g$ is given by $$ ds^2 = {1 \over 2 \lambda} \left( dq^2_1 + \cdots + dq^2_n \right). $$ Then \begin{equation} H(p,q) = {\lambda \over 2} \left( p^2_1 + \cdots + p^2_n \right) \label{31} \end{equation} Write $\beta = d \alpha + \gamma$, where $\alpha$ is a 1-form and $\gamma$ is a 2-form, having constant coefficients in the coordinates $(q_1 \cdots q_n)$, $\gamma = \Sigma_{i