In this post I begin discussing/translating a rigidty result proven in Grognet for magnetic flows.

Introduction

Recall that there are many equivalent definitions for the magnetic flow (see this post). One reason to study magnetic flows is that they behave like twisted geodesics (magnetic flows are sometimes referred to as twisted geodesic flows, per Paternain). It’s therefore natural to try to see whether any result we have on geodesic flows applies to magnetic geodesic flows. One important result is that of marked length spectrum rigidty. As seen in Otal’s theorem information on the geodesic flow (and thus information on geodesics) translates to geometric information: Knowing that the lengths of corresponding geodesics are the same tells us that the metrics are the same up to isometry in the very nice case of a negatively curved surface.

The key to Otal’s theorem is that we have negative curvature, as this forces the geodesic flow to be Anosov and we can therefore apply Livschitz’s theorem. Applying the above philosophy on magnetic flows, the natural question to try to ask is the following: On a fixed closed orientable surface \(M\), if we have two magnetic flows which are Anosov and they are conjugate to one another, is it true that the underlying metrics are isometric? Moreover, do we get any information on the closed \(2\)-form (correspondingly, the magnetic field)? Grognet answers this in the affirmative in a specific case, namely if we have a magnetic flow which is conjugate to a geodesic flow.

The goal is to create a series of posts which both translates Grognet to English and gives some personal perspective on the paper. In this post we cover Sections 2-4 from the paper.

Set Up

Magnetic Flows

To start, recall that the magnetic flow has a variety of different definitions which are all equivalent. We list the relevant ones now, again following Merry and Paternain’s notes. Throughout, fix a closed orientable surface \(M\).

1) Given a metric \(g\), we can define the Hamiltonian

\[H : TM \rightarrow \mathbb{R}, \ \ H(x,v) := g_x(v,v).\]

Recall that we have the canonical \(1\)-form (sometimes the Liouville \(1\)-form) \(\alpha\) on \(T^*M\), given by

\[\alpha_{(x,v)}(\xi) := v( d_{(x,v)} \tau(\xi)),\]

where \(\tau : T^*M \rightarrow M\) is the footprint map. Then we have the canonical symplectic form given by

\[\omega := -d\alpha.\]

Using the musical isomorphism given to us by the metric, we can push this onto the tangent bundle via \(\hat{g}^{-1}(\omega) = \omega'\). Given a closed \(2\)-form \(\sigma\) on \(M\), we can pull it back via the footprint map \(\pi : TM \rightarrow M\) to get a closed \(2\)-form on \(TM\). Adding this to the canonical symplectic form, we get the magnetic symplectic form

\[\omega_\sigma := \omega' + \pi^*(\sigma).\]

It’s not a hard calculation to check that this is a symplectic \(2\)-form (non-degeneracy is the hard part, and you can verify this in coordinates). If we take the symplectic gradient of the Hamiltonian above, we get a vector field \(F\) satisfying

\[i_F(\omega_\sigma) = dH.\]

This vector field generates the magnetic flow on the tangent bundle.

Remark: Grognet doesn’t really use this definition in the paper, but I’ve included it since it’s generally how the magnetic flow is defined.

2) Let \(\kappa \in C^\infty(M).\) Then a curve \(\gamma : \mathbb{R} \rightarrow M\) is a magnetic geodesic if it satisfies the following differential equation:

\[\frac{\dot{\gamma}(t)}{dt} = \kappa(\gamma(t)) i \dot{\gamma}(t),\]

where \(i\) is the almost complex structure induced by the metric \(g\) (that is, it rotates a vector by \(\pi/2\)). We can then obtain a flow by restricting to different energy surfaces and looking at solutions to the differential equation with given initial conditions. The function \(\kappa\) is sometimes referred to as the magnetic field.

We remark that in both cases we are given flows on the entire tangent bundle. Generally when we think of the magnetic flow on the surface, we restrict it to the unit tangent bundle with respect to it’s metric. We have the following claim. We will also sometimes denote a magnetic flow either by \((M,g,\sigma)\) where \(\sigma\) a closed \(2\)-form or \((M,g,\kappa)\) where \(\kappa\) a magnetic field.

It is not hard to see that any magnetic flow preserves the Liouville \(1\)-form on \(SM\) (we’ll be sloppy and use the same notation for the Liouville \(1\)-form, even though it’s technically different from before). One observation to make (using Cartan’s structural equations) is that the vector field generating any magnetic flow is going to be \(F = X + \kappa V\), where \(\kappa\) the corresponding magnetic field, \(X\) the vector field generating the geodesic flow, and \(V\) the vector field corresponding to vertical vectors. We have that \(X\) preserves the Liouville form and it’s not too hard to see \(V\) preserves the Liouville form, so

\[\mathcal{L}_F(\alpha) = 0.\]

Furthermore we get that the flow preserves the so-called Liouville measure, which is the measure \(\mu\) characterized by

\[\int_{SM} f d\mu = -\int_{SM} f (\alpha \wedge \omega) \text{ for all } f \in C^0(SM).\]

Orbit Equivalence and Quasi-geodesics

We recall what a quasi-geodesic is (Definition 3.1 in the paper).

Definition: If \((E,d)\) and \((E',d')\) are two metric spaces, then a map \(f : E \rightarrow E'\) is said to be a quasi-isometry if there exists \(C_1, C_2 > 0\) so that

\[\frac{1}{C_1}d(x,y) - C_2 \leq d'(f(x), f(y)) \leq C_1 d(x,y) \text{ for all } x,y \in E.\]

Examining a Riemannian manifold \((M,g)\) diffeomorphic to \(\mathbb{R}^n\), we say that a curve \(c : \mathbb{R} \rightarrow M\) is a quasi-geodesic if it is a quasi-isometry from \(\mathbb{R}\) to \(M\).

We also recall what an orbit equivalence is (Definition 3.2 in the paper).

Definition: Two flows \(\phi_t\) and \(\psi_t\) on a manifold \(M\) are said to be orbit equivalent if there exists a homeomorphism \(h : M \rightarrow M\) and a continuous map \(\tau : M \times \mathbb{R} \rightarrow \mathbb{R}\) such that

1) For all fixed \(x\), the map \(\tau(x, \cdot)\) is a homeomorphism from \(\mathbb{R}\) to itself.

2) For all \((x,t) \in N \times \mathbb{R}\), one has

\[h(\phi_t(x)) = \psi_{\tau(x,t)}(h(x)).\]

Jacobi Fields

Following Grognet, we diverge from the case where we’re dealing with a surface and just consider an \(n\)-manifold (since this is a more general construction anyways). Consider a family of \(C^3\) curves \(\{c_s : \mathbb{R} \rightarrow M \}\) which come from a magnetic flow \((M,g, \kappa)\) on the unit tangent bundle \(SM\). We can define two vector fields:

\[S := \frac{\partial c_s}{\partial s}, \ \ T := \frac{\partial c_t}{\partial t}.\]

Letting \(E_0^s(0) = T\), choose an orthonormal basis \(\{E_0^s(0), \ldots, E_{n-1}^s(0)\}\) of the vector space \(T_{c_s(0)}M.\) We consider the following differential equation:

\[\nabla_T V = \kappa \cdot i V.\]

For any \(s\), we see the above differential equation defines a vector field \(V^s(t)\) along the curve \(c_s(t)\). Thus by uniqueness and existence of solutions, we can find vector fields so that

\[V_i^s(t) = E_i^s(t), \ \ V_i^s(0) = E_i^s(0) \text{ for } i \in \{0, \ldots, n-1\}.\]

Since we have a basis, we can write

\[S = x T + \sum_{i=1}^n y_i E_i^s = \sum_{i=0}^n y_i E_i, \text{ where } y_0 = x.\]

We see that

\[\begin{split} \nabla_T S = \sum_{i=0}^{n-1} \nabla_T(y_i E_i^s) \\ = \sum_{i=0}^{n-1} \nabla_T(y_i V_i^s) \\ = \sum_{i=0}^{n-1} [\nabla_T(y_i) V_i^s + y_i \nabla_T(V_i^s)] \\ = \sum_{i=0}^{n-1} [y_i' V_i^s + y_i \kappa i V_i^s] \\ = \sum_{i=0}^{n-1} y_i' E_i^s + \kappa i S, \end{split}\]

where \(y_i' = \frac{dy_i}{dt}.\)

Grognet then defines an operator \(D\) on the vector fields along the curve \(c\) by

\[D(V) = \nabla_T V - i \kappa V.\]

By the above calculation, we observe that

\[D(S) = \sum_{i=0}^{n-1} y_i' E_i^s.\]

We then define the space orthogonal to the trajectory to be the span of \(\{E_1(t), \ldots, E_{n-1}(t)\}\), and we define a projection \(\Pi\) on the space of vector fields along the curve \(c\) to the space orthogonal to the trajectory in the obvious way. We denote by \(Y\) the orthogonal component of the Jacobi field \(S\). That is,

\[Y := \Pi(S) = \sum_{i=1}^n y_i E_i^s.\]

Following Grognet, we’ll drop the notation of \(s\) and it should be clear from context what we’re referring to.

Let \(v_1, v_2\) be two vector fields along the curve \(c_s(t)\). We can then write

\[v_1 = \sum_{i=0}^{n-1} a_i E_i, \ \ v_2 = \sum_{i=0}^{n-1} b_i E_i.\]

By orthonormality, we have

\[\langle v_1, v_2 \rangle = \sum_{i=0}^{n-1} a_ib_i.\]

Thus

\[\frac{d}{dt} \langle v_1, v_2 \rangle = \sum_{i=0}^{n-1} (a_i' b_i + a_i b_i').\]

Now use the earlier observation to get that

\[D(v_1) = \sum_{i=0}^{n-1} a_i' E_i,\]

and likewise

\[D(v_2) = \sum_{i=0}^{n-1} b_i' E_i.\]

Thus

\[\frac{d}{dt} \langle v_1, v_2 \rangle = \langle D(v_1), v_2 \rangle + \langle v_1, D(v_2) \rangle.\]

Let \(R\) be the metric tensor associated to \(g\), so

\[\langle R(a,b)c, d\rangle = R(a,b,c,d).\]

Then we have three identities:

(1) \([S,T] = 0.\)

(2) \(\nabla_T S = \nabla_S T.\)

(3) \(\nabla_T \nabla_S T = \nabla_S \nabla_T T - R(T,S)T.\)

To see the first identity, notice that it follows by letting \(\alpha(s,t) = c_s(t)\) and observing that \(T = d\alpha(d/dt),\) \(S = d\alpha(d/ds),\) so

\[[S,T] = d\alpha\left(\left[\frac{d}{dt}, \frac{d}{ds} \right]\right) = 0.\]

The second identity and third identity follow by mostly routine calculations. As a consequence, we get the following:

\[\nabla_T \nabla_T S + R(T,S)T = \nabla_S (i\kappa T) = (\nabla_S i \kappa) T + i\kappa \nabla_T S.\]

Replacing \(\nabla_T\) with \(D + i\kappa\), we get

\[D^2S + i\kappa DS + (\nabla_T(i\kappa)) S + R(T,S)T = (\nabla_S i \kappa) T.\]

We obtain the tangent component of the Jacobi field by taking the dot product of the above equation with \(T\). Recalling that \(x\) is the component of \(\langle T, S\rangle,\) we see that

\[x'' + i \kappa \langle DS,T \rangle + \langle (\nabla_T i \kappa) S, T \rangle = 0,\]

or

\[x'' + \frac{d}{dt} \langle i \kappa S, T \rangle = 0.\]

Now we have

\[\langle T, T \rangle = 1,\]

so taking the derivative with respect to \(s\) we get

\[x' + \langle i \kappa S, T \rangle = 0.\]

Taking the derivative of this with respect to \(t\) derives the earlier equation. If we let

\[A = \Pi(i \kappa),\]

\(B Y = -(\nabla_Y b) T + \Pi(\nabla_T(b) Y) + \langle Y, i \kappa T \rangle i\kappa T + R(T,Y)T\),

then we can rewrite the above to get Proposition 4.1. The Jacobi field along some family of curves must satisfy the following differential equation:

\[\begin{split} x' + \langle i \kappa S, T \rangle = 0, \\ D^2(Y) + A D(Y) + B Y = 0. \end{split}\]

In fact, pretty much everything about Jacobi fields for geodesics applies to Jacobi fields for magnetic flows. This is the content of Proposition 4.2. After fixing a real number \(a\) and a curve \(c\), we have

(1) The space of Jacobi fields along the curve \(c\) is isomorphic to an orthogonal direct sum

\[T_{c(a)}M \oplus T_{c(a)}M\]

via the mapping

\[J \mapsto (J(a), DJ(a)).\]

(2) The space of Jacobi fields come from a variation is isomorphic to the space

\[T_{c(a)}M \oplus \frac{dc}{dt}(a)^\perp.\]

(3) Every Jacobi field along a curve comes from a variation.

This leads us to our next definition (Definition 4.2).

Definition: We say that two points \(a\) and \(b\) on the manifold \(M\) are conjugate for the flow associated to the magnetic field if there exists a closed curve of the magnetic flow joining them together for which the Jacobi field along this curve is \(0\) at \(a\) and \(b\) but is not identically \(0\).

Recalling what we set \(Y\) to before, let

\[f_s(t) = \|Y(t)\|^2 = \sum_{i=1}^{n-1} y_i^2(t).\]

To match Grognet, we’ll also write

\[b = i \kappa.\]

We now have the following result.

Lemma 4.1: Supposing that the curves \(\{c_s(t)\}\) are parameterized with unit speed, we have the following inequality:

\[\frac{1}{2} f_s''(t) \geq \left( \frac{5}{4} \|b\|^2 + \|D_\bullet b T\|^2 \right) \|Y\|^2 - R(T,Y,T,Y).\]

Proof: First, we have

\[f_s(t) = \langle Y, Y \rangle.\]

Taking the derivative with respect to time, we observe

\[f_s'(t) = 2 \langle Y, D(Y) \rangle.\]

Doing it again, we have

\[f_s''(t) = 2 \langle Y, D^2(Y)\rangle + 2 \langle D(Y), D(Y) \rangle.\]

Use the Jacobi field equation and dot it with \(Y\) to get the following:

\[\langle D^2(Y), Y \rangle + \langle b D(S), Y \rangle + \langle \nabla_T(b) (xT), Y \rangle + R(T,Y,T,Y) = \langle D_S(b) T, Y \rangle.\]

Recall we have

\[\langle b D(S), Y \rangle = \langle b D(Y), Y \rangle + \langle b (x'T), Y \rangle.\]

Since everything is parameterized with unit speed, we get

\[\langle b D(S), Y \rangle = \langle b D(Y), Y \rangle + \langle b Y, Y \rangle^2.\]

Also recall that we have

\[\langle \nabla_S b T, Y \rangle - \langle \nabla_T b (xT), Y \rangle = \langle \nabla_Y b T, Y \rangle.\]

Substitute this in to get

\[\begin{split} \frac{1}{2} f_s''(t) = \langle D(Y), D(Y) \rangle - \langle b D(Y), Y \rangle \\ + \langle \nabla_Y b T, Y \rangle - \langle Y, b T \rangle^2 - R(T,Y,T,Y). \end{split}\]

Finally, we conclude the proof by noting that

\[\langle D(Y), D(Y) \rangle - \langle b D(Y), Y \rangle \geq \frac{1}{4} \|b (Y)\|^2.\]

As Grognet points out, what we are doing is eliminating the “dissipative” terms.

Using this, we can almost immediately conclude the following.

Proposition 4.3: If the sectional curvature \(K\) satisfies the following pinching:

\[-k_0^2 \leq K \leq k_1^2 < 0,\]

and if the first jet of the magnetic field \(b\) satisfies the following inequality:

\[\frac{5}{4} \|b\|^2 + \|D_\bullet b\| \leq k_1^2,\]

then the flow has no conjugate points.

Proof: Using the above lemma, we see that the norm of the Jacobi fields is a convex function, therefore it cannot be canceled more than once unless the Jacobi represents a simple change of parameterization of the orbit.

One of the biggest results in the paper comes up at this point.

Theorem 4.1: Suppose \(M\) is a compact Riemannian manifold with constant negative sectional curvature satisfying

\[-k_0^2 \leq K \leq -k_1^2 < 0.\]

If the magnetic field satisfies the following inequality:

\[\frac{5}{4} \|b\|_\infty^2 + \|Db\|_\infty < k_1^2,\]

then the magnetic flow is Anosov.

Grognet gives the following remarks:

(1) The condition required is of non-empty interior in the \(C^1\) topology for the magnetic field and in the \(C^2\) topology for the metric.

(2) For surfaces, the coefficient \(5/4\) can be improved to just \(1\). This turns out to be the best case scenario, as the horocyclic flow of surfaces with curvature \(-1\) gives an example where this cannot be improved.

(3) A necessary condition is given by an integral:

\[\int_{SM} \|b v\|^2 d\lambda < -(n-1) \int_{SM} \text{Ric}(v)d \lambda.\]

As a consequence of this theorem and results on Anosov flows, we get the following result.

Proposition 4.4: The Liouville measure is ergodic for an Anosov magnetic flow of class \(C^2\).