Periodic magnetic geodesics on Heisenberg manifolds

Abstract

We study the dynamics of magnetic flows on Heisenberg groups, investigating the extent to which properties of the underlying Riemannian geometry are reflected in the magnetic flow. Much of the analysis, including a calculation of the Mañé critical value, is carried out for $(2n+1)$-dimensional Heisenberg groups endowed with any left invariant metric and any exact, left-invariant magnetic field. In the three-dimensional Heisenberg case, we obtain a complete analysis of left-invariant, exact magnetic flows. This is interesting in and of itself, because of the difficulty of determining geodesic information on manifolds in general. We use this analysis to establish two primary results. We first show that the vectors tangent to periodic magnetic geodesics are dense for sufficiently large energy levels and that the lower bound for these energy levels coincides with the Mañé critical value. We then show that the marked magnetic length spectrum of left-invariant magnetic systems on compact quotients of the Heisenberg group determines the Riemannian metric. Both results confirm that this class of magnetic flows carries significant information about the underlying geometry. Finally, we provide an example to show that extending this analysis of magnetic flows to the Heisenberg-type setting is considerably more difficult.

Appendix: Tangent bundle viewpoint: periodic magnetic geodesics in Heisenberg manifolds

In the presence of a Riemannian metric, the tangent and cotangent bundles are canonically identified. Structures can be defined equivalently on either vector bundle and computations can be carried out in the more convenient of the two settings. For the bulk of the paper, we found it easier to work on the cotangent bundle and this section demonstrates how the computations would proceed on the tangent bundle.

In [22], A. Kaplan introduced so-called j-maps (see Sect. 2.4 for the definition) to study Clifford modules. A metric two-step nilpotent Lie algebra is completely characterized by its associated j-maps. Since being introduced, they have proven very useful in the study of two-step nilpotent geometry. In this appendix, we show how the magnetic geodesic equations can be characterized in terms of the j-maps.

Let G be a two-step nilpotent Lie group endowed with a left-invariant metric g and an exact, left-invariant magnetic form $\Omega$. Let $\mathfrak {g} = \mathfrak {v} \oplus \mathfrak {z}$ be the decomposition of the Lie algebra into the center and its orthogonal complement. By Lemma 1, there is $\zeta _m \in \mathfrak {z}^*$ such that $\Omega = d(B\zeta _m)$.

Lemma 13

The Lorentz force associated with the magnetic field $\Omega$ satisfies $F_\mathfrak {v} = j(-BZ_m)$ and $F_\mathfrak {z} = 0$, where $Z_m = \sharp (\zeta _m)$.

Proof

Let $X \in \mathfrak {g}$, $V \in \mathfrak {v}$ and $Z \in \mathfrak {z}$. Then,

$$\begin{aligned} g(F(Z), X)&= \Omega (Z, X) = d(B \zeta _m)(Z,X) = -B\zeta _m([Z,X]) = 0, \end{aligned}$$

and

$$\begin{aligned} g(F(V), X)&= \Omega (V, X) = d(B \zeta _m)(V,X) = -B\zeta _m([V,X]) \\&= -Bg(\sharp (\zeta _m), [V,X]) = -Bg(j(Z_m)V, X) \\&= g(j(-B Z_m)V, X). \end{aligned}$$

$\square$

Because of Lemma 13, we will write $F = j(-BZ_m)$ with the understanding that F vanishes on central vectors and agrees with $j(-BZ_m)$ on vectors in $\mathfrak {v}$. Let $\gamma (t) = \exp (X(t) + Z(t))$ be a magnetic geodesic on G where $X(t) \in \mathfrak {v}$ and $Z(t) \in \mathfrak {z}$. By (13), we can express the velocity vector of $\gamma$ as $\gamma '(t) = X'(t) + \frac{1}{2}[X'(t), X(t)] + Z'(t)$. The condition for $\gamma$ to be a magnetic geodesic is $\nabla _{\gamma '(t)} \gamma '(t) = F(\gamma '(t))$. Using (15) to expand this condition and imposing the initial conditions $\gamma (0) = e$ and $\gamma '(0) = X_0 + Z_0$, the geodesic equations on $\mathfrak {v}$ and $\mathfrak {z}$ separately are

$$\begin{aligned}&X''(t) = j(Z_0 - BZ_m)X'(t) \end{aligned}$$

(48)

$$\begin{aligned}&Z'(t) + \frac{1}{2}[X'(t),X(t)] = Z_0. \end{aligned}$$

(49)

We restrict to the three-dimensional Heisenberg case and consider the magnetic geodesics in this context. Following the approach as illustrated in Prop. 3.5 on pp. 625–628 of [11], and reducing to the three-dimensional Heisenberg case, a straightforward calculation gives the following result.

Corollary 1

If $z_{0}-B=0$, (or if $z_{0}-B\ne 0$ and $x_{0}=y_{0}=0$ ), then $\sigma \left( t\right)$ is the one parameter subgroup

$$\begin{aligned} \sigma \left( t\right) =\exp \left( x\left( t\right) X+y\left( t\right) Y+z\left( t\right) Z\right) =\exp \left( t\left( x_{0}X+y_{0}Y+z_{0}Z\right) \right) \text {.} \end{aligned}$$

(50)

If $z_{0}-B\ne 0$, the solution is

$$\begin{aligned} \left( \begin{array}{c} x\left( t\right) \\ y\left( t\right) \end{array} \right) =\frac{A}{z_{0}-B}\left( \begin{array}{cc} \sin \left( t\left( \frac{z_{0}-B}{A}\right) \right) &{} -\left( 1-\cos \left( t\left( \frac{z_{0}-B}{A}\right) \right) \right) \\ 1-\cos \left( t\left( \frac{z_{0}-B}{A}\right) \right) &{} \sin \left( t\left( \frac{z_{0}-B}{A}\right) \right) \end{array} \right) \left( \begin{array}{c} x_{0} \\ y_{0} \end{array} \right) \text {,} \end{aligned}$$

(51)

and

$$\begin{aligned} \begin{array}{c} z\left( t\right) =\left( z_{0}+\frac{A\left( x_0^2+y_0^2\right) }{2\left( z_{0}-B\right) }\right) t-\frac{A^2 \left( x_0^2+y_0^2\right) }{2\left( z_{0}-B\right) ^{2}}\sin \left( t\left( \frac{z_{0}-B}{A }\right) \right) . \end{array} \end{aligned}$$

(52)

Remark 17

The coordinate functions (51) and (52) are equivalent the one obtained in (25)–(27) in the following sense. In order to obtain the magnetic geodesic through the origin determined by $(u_0, v_0, z_0)$ as in Sect. 3 take as initial tangent vector in (51) and (52) to be $(x_0, y_0, z_0) = (u_0/A, v_0/A, z_0 + B)$.

We now present some of the main results about the three-dimensional Heisenberg manifold proved in the body of the paper, but expressed using the tangent bundle, rather than the cotangent bundle.

Continuing the notation from the previous sections, we fix energy E,  magnetic strength B,  and metric parameter A. Let $\left( H, g_A, \Omega \right)$ denote a simply connected Heisenberg manifold. The theorems in this section state precisely the set of periods $\omega$ such that there exists an initial velocity $v_{p}\in TH$ such that $\sigma _{v_{p}}\left( t\right)$ is periodic with period $\omega$. We also precisely state the set of initial velocities $v_{p}$, hence the set of geodesics, that produce each period $\omega$.

Let $\Gamma$ denote a cocompact discrete subgroup of H and, as above, denote the resulting compact Heisenberg manifold by $\left( \Gamma \backslash H, g_A, \Omega \right) .$ For all $\gamma \in \Gamma ,$ we state below precisely the set of periods $\omega$ such that there exists an initial velocity $v_{p}\in TH$ such that $\sigma _{v_{p}}\left( t\right)$ is $\gamma$-periodic with period $\omega$. We also precisely state the set of initial velocities $v_{p}$, hence the set of geodesics, that produce each period $\omega$.

1.1 Periodic magnetic geodesics on the simply connected Heisenberg group

We now consider the existence of periodic geodesics in $\left( H,g_{A}, d\left( B\zeta \right) \right)$, the three-dimensional Heisenberg Lie group H with left-invariant metric determined by the orthonormal basis $\left\{ \frac{1}{\sqrt{A}}X,\frac{1}{\sqrt{A}}Y,Z\right\}$ and magnetic form $\Omega =-B\,\alpha \wedge \beta$. Recall that for a vector $v\in \mathfrak {h}$, $\sigma _{v}\left( t\right)$ denotes the magnetic geodesic through the identity with initial velocity v. Note that if $v_{p}\in T_{p}H$, then $\sigma _{v_{p}}\left( t\right)$ denotes the magnetic geodesic through $p=\sigma _{v_{p}}\left( 0\right)$ with initial velocity $v_{p}$. Also note that because $g_{A}$ and $\Omega$ are left-invariant, that $\sigma _{v_{p}}\left( t\right) =L_{p}\sigma _{v}\left( t\right)$, where $v_{p}=L_{p*}\left( v\right)$; i.e., magnetic geodesics through $p\in H$ are just left translations of magnetic geodesics through the identity. Clearly, a magnetic geodesic through $p\in H$ is periodic with period $\omega$ if and only if its left translation by $p^{-1}$ is a magnetic geodesic through the identity with period $\omega$.

Theorem 7

With notation as above, fix energy E,  magnetic strength B,  and metric parameter A.

• 1. If $B^2>E^2$, then there exists a one-parameter family of vectors $v\in \mathfrak {h}$, $\left| v\right| =E,$ such that $\sigma _{v}\left( t\right)$ is periodic. In particular, $\sigma _{v}\left( t\right)$ is periodic if and only if $z_{0}=B-\mathrm {sgn}\left( B\right) \sqrt{B^{2}-E^{2}}$ and

  $$\begin{aligned} x_{0}^{2}+y_{0}^{2}=-2z_{0}\left( z_{0}-B\right) /A. \end{aligned}$$

  The set of periods of $\sigma _{v}\left( t\right)$ is $\frac{2\pi }{\sqrt{B^{2}-E^{2}}}\mathbb {Z}_{\ne 0}$, and the smallest positive period is $\omega =\left| \frac{2\pi A}{z_{0}-B}\right| =\frac{2\pi A}{\sqrt{B^{2}-E^{2}}}$.

• 2. If $B^2\le E^2$, then there does not exist a vector v with $\left| v\right| =E$ such that $\sigma _{v}\left( t\right)$ is periodic.

1.2 Periodic geodesics on compact quotients of the Heisenberg group

We ultimately wish to consider closed magnetic geodesics on Heisenberg manifolds of the form $\Gamma \backslash H$, where $\Gamma$ is a cocompact discrete subgroup of H. As above, we proceed by considering $\gamma$-periodic magnetic geodesics on the cover H.

The purpose of this section is stating more precisely, and proving, the following, which is divided into several cases. See Theorems 9, 10 and 11 below.

Theorem 8

Consider the three-dimensional Heisenberg Lie group H with left invariant metric $g_{A}$ determined by the orthonormal basis $\left\{ \frac{1}{\sqrt{A}}X,\frac{1}{\sqrt{A}}Y,Z\right\}$ and magnetic form $\Omega =d\left( B\zeta \right) =-B\alpha \wedge \beta$. Fix $\gamma \in H$ and fix energy E,  magnetic strength B and metric parameter A. Then, we can state precisely the set of periods $\omega$ such that there exists an initial velocity $v_{p}\in TH$ with $\left| v_{p}\right| =E$ such that $\sigma _{v_{p}}\left( t\right)$ is $\gamma$-periodic with period $\omega$. We can also precisely state the set of initial velocities $v_{p}$, hence, the set of geodesics, that produce each period $\omega$.

1.2.1 Noncentral case

Let $\gamma =\exp \left( x_\gamma X+y_\gamma Y+z_\gamma Z\right) \in H$ with $x_\gamma ^{2}+y_\gamma ^{2}\ne 0$. Let $a=\exp \left( a_{x}X+a_{y}Y+a_{z}Z\right) \in H$. From (12), the conjugacy class of $\gamma$ in H is $\exp \left( x_\gamma X+ y_\gamma Y+ \mathbb {R}Z\right)$.

Theorem 9

Fix energy E,  magnetic strength B and metric parameter A. Let $\gamma =\exp \left( x_\gamma X+y_\gamma Y+z_\gamma Z\right) \in H$ with $x_\gamma ^{2}+y_\gamma ^{2}\ne 0$.

• 1. If $E^2>B^2$ (ie, if $\mu >1$ ), then there exists a two-parameter family of elements $a\in H$ such that $a\gamma a^{-1}=\exp \left( x_\gamma X+y_\gamma Y+z_\gamma ^{\prime }Z\right)$ where $z_\gamma ^{\prime }=\pm B\sqrt{A\left( x_\gamma ^{2}+y_\gamma ^{2}\right) /\left( E^{2}-B^{2}\right) }$. Letting $v=$ $\frac{B}{z_\gamma ^{\prime }}\left( x_\gamma X+y_\gamma Y+z_\gamma ^{\prime }Z\right)$, which satisfies $\left| v\right| =E$, then $\gamma$ translates the (non-spiraling) magnetic geodesic $a^{-1}\exp \left( tv\right)$ with period $\omega =\pm \sqrt{A\left( x_\gamma ^{2}+y_\gamma ^{2}\right) /\left( E^{2}-B^{2}\right) }$. These are the only magnetic geodesics with energy E translated by $\gamma$.

• 2. If $E^2\le B^2$ (ie, if $\mu \le 1$ ), then neither $\gamma$ nor any of its conjugates in H translate a magnetic geodesic with energy E.

1.2.2 Central case

Throughout this subsection, we assume that $x_\gamma =y_\gamma =0$; i.e., that $\gamma$ lies in $Z\left( H\right) ,$ the center of three-dimensional Heisenberg group H. Recall that since $\gamma$ is central, $\gamma$ translates a magnetic geodesic $\sigma \left( t\right)$ through the identity $e\in H$ with period $\omega$ if and only if for all $a\in H$, $\gamma$ translates a magnetic geodesic through a with period $\omega$. That is, without loss of generality, if $\gamma$ lies in the center, we may assume that $\sigma \left( 0\right) =e$.

Recall that magnetic geodesics in H are either spiraling or one parameter subgroups. We first consider the case of one-parameter subgroups.

Theorem 10

Fix energy E,  magnetic strength B and metric parameter A. Let $\gamma =\exp \left( z_\gamma Z\right) \in Z\left( H\right)$, with $z_\gamma \ne 0$. The element $\gamma$ translates the magnetic geodesics $\sigma \left( t\right) =$ $\exp \left( \pm tEZ\right)$ with initial velocities $v=\pm EZ$ and periods $\omega =\pm z_\gamma /E$. This pair of one-parameter subgroups and their left translates are the only straight magnetic geodesics translated by $\gamma$.

Theorem 11

Fix energy E,  magnetic strength B,  and metric parameter A. Denote $\mu =\frac{E}{\left| B\right| }$. Let $\gamma =\exp \left( z_\gamma Z\right) \in Z\left( H\right)$, $z_\gamma \ne 0$. If there exists a vector $v=x_{0}X+y_{0}Y+z_{0}Z$ and a period $\omega \ne 0$ such that the spiraling geodesic $\sigma _{v}\left( t\right)$ is $\gamma$-periodic with period $\omega$, then there exists $\ell \in \mathbb {Z}_{\ne 0}$ such that $\zeta _{\ell }=$ $\frac{z_\gamma }{\pi \ell }$ satisfies the conditions relative to $\mu$ specified in the following six cases and $A\left( x_{0}^{2}+y_{0}^{2}\right)$, $z_{0}$, and $\omega$ are as expressed below. Conversely, for every choice of $\ell \in \mathbb {Z}_{\ne 0}$ such that $\zeta _{\ell }=\frac{z_\gamma }{\pi \ell }$ satisfies the conditions in one of the cases below, there exists at least one vector v as given below such that $\sigma _{v}\left( t\right)$ is $\gamma$-periodic (spiraling) geodesic with period $\omega$ as given below. Note that Case 1 requires $E^2<B^2$. Cases 2 through 5 require $E^2>B^2,$ and Case 6 requires $E^2=B^2.$ Note that in all cases, $\zeta _\ell \ne 0.$

• 1. $\frac{-2\mu }{1-\mu }<\frac{\zeta _{\ell }}{A}<\frac{2\mu }{1+\mu }<1$,

• 2. $1<\frac{2\mu }{1+\mu }<\frac{\zeta _{\ell }}{A}<2$,

• 3. $2<\frac{\zeta _{\ell }}{A}\le \frac{2\mu }{\mu -1}$,

• 4. $2<\frac{2\mu }{\mu -1}<\frac{\zeta _{\ell }}{A}$,

• 5. $\frac{\zeta _{\ell }}{A}=2$ and $\mu >1$,

• 6. $\frac{\zeta _{\ell }}{A}=1$ and $\mu =1.$

In Cases 1 through 4, we choose any $x_{0},y_{0}\in \mathbb {R}$ so that

$$\begin{aligned} A\left( x_0^2+y_0^2\right) =B^{2}\left( \frac{\mu ^{2}-1}{\frac{\zeta _{\ell }}{A}-1}\left( \frac{\zeta _{\ell }}{A}-2\right) +2\sqrt{\frac{\mu ^{2}-1}{\frac{\zeta _{\ell }}{A}-1}}\right) \end{aligned}$$

(53)

and let

$$\begin{aligned} z_{0}=-B\left( -1+\sqrt{\frac{\mu ^{2}-1}{\frac{\zeta _{\ell }}{A}-1}}\right) \end{aligned}$$

(54)

and

$$\begin{aligned} \omega =\frac{2z_\gamma A}{\zeta _{\ell }\left( z_{0}-B\right) }=\frac{2\pi \ell \sqrt{\left| \frac{\zeta _{\ell }}{A}-1\right| }}{\sqrt{E^{2}-B^{2}}}\text {.} \end{aligned}$$

In Case 4, we may also choose any $x_{0},y_{0}\in \mathbb {R}$ so that

$$\begin{aligned} A\left( x_0^2+y_0^2\right) =B^{2}\left( \frac{\mu ^{2}-1}{\frac{\zeta _{\ell }}{A}-1}\left( \frac{\zeta _{\ell }}{A}-2\right) -2\sqrt{\frac{\mu ^{2}-1}{\frac{\zeta _{\ell }}{A}-1}}\right) \end{aligned}$$

(55)

and let

$$\begin{aligned} z_{0}=-B\left( -1-\sqrt{\frac{\mu ^{2}-1}{\frac{\zeta _{\ell }}{A}-1}}\right) \end{aligned}$$

(56)

and

$$\begin{aligned} \omega =\frac{2 z_\gamma A}{\zeta _{\ell }\left( z_{0}-B\right) }=-\frac{2\pi \ell \sqrt{\left| \frac{\zeta _{\ell }}{A}-1\right| }}{\sqrt{E^{2}-B^{2}}}\text {.} \end{aligned}$$

The conditions on $\mu ,\zeta _{\ell },x_{0}$,$y_{0\text { }}$ and $z_{0}$ imply $\frac{\mu ^{2}-1}{\frac{\zeta _{\ell }}{A}-1}>0$, $x_0^2+y_0^2>0$, $E^{2}=A\left( x_0^2+y_0^2\right) +z_{0}^{2}$, and the (spiraling) magnetic geodesic through the identity $\sigma _{v}\left( t\right)$ with initial velocity $v=x_{0}X+y_{0}Y+z_{0}Z$ is $\gamma =\exp \left( z_\gamma Z\right)$-periodic with energy E and period $\omega$ as given.

In Case 5, which only occurs if $\frac{z_\gamma }{A}\in 2\pi \mathbb {Z}_{\ne 0}$, we choose any $x_{0},y_{0\text { }}\in \mathbb {R}$ so that

$$\begin{aligned} A\left( x_0^2+y_0^2\right) =2\left| B\right| \sqrt{E^{2}-B^{2}} \end{aligned}$$

and

$$\begin{aligned} z_{0}=B-\frac{A\left( x_0^2+y_0^2\right) }{2B}\text {.} \end{aligned}$$

Then, the conditions on $\mu ,\zeta _{\ell },x_{0},y_{0}$ and $z_{0\text { }}$ imply that $E^{2}=A\left( x_0^2+y_0^2\right) +z_{0}^{2}$ and the (spiraling) magnetic geodesic $\sigma _{v}\left( t\right)$ starting at the identity with initial velocity $v=x_{0}X+y_{0}Y+z_{0}Z$ is $\gamma$-periodic with energy E and period

$$\begin{aligned} \omega =-\mathrm {sgn}\left( B\right) \frac{z_\gamma }{\sqrt{E^{2}-B^{2}}}\text {.} \end{aligned}$$

In Case 6, which only occurs if $\frac{z_\gamma }{A}\in \pi \mathbb {Z}_{\ne 0}$, we choose any $x_{0},y_{0\text { }},z_{0}\in \mathbb {R}$ so that $E^{2}=B^{2}=A\left( x_{0}^{2}+y_{0}^{2}\right) +z_{0}^{2}$ and $z_{0}\ne \pm B$. The conditions on $\mu$ and $\zeta _{\ell }$ imply that the (spiraling) magnetic geodesic $\sigma _{v}\left( t\right)$ with initial velocity $v=x_{0}X+y_{0}Y+z_{0}Z$ will yield a $\gamma$-periodic magnetic geodesic with energy E and period

$$\begin{aligned} \omega =\frac{2 z_\gamma }{z_{0}-B}. \end{aligned}$$

Remark 18

In Case 1, there are infinitely many values of $\ell$ that satisfy the conditions, hence infinitely many distinct periods $\omega .$ In particular, if $\mu <1$ and there exists $\ell _{0}\in \mathbb {Z}_{>0}$ such that $\zeta _{\ell _{0}}\in \left( \frac{-2\mu }{1-\mu },\frac{2\mu }{1+\mu }\right)$, then for all $\ell >\ell _{0}$, $\zeta _{\ell }\in \left( \frac{-2\mu }{1-\mu },\frac{2\mu }{1+\mu }\right)$. Likewise if there exists $\ell _{0}\in \mathbb {Z}_{<0}$ such that $\zeta _{\ell _{0}}\in \left( \frac{-2\mu }{1-\mu },\frac{2\mu }{1+\mu }\right)$, then for all $\ell <\ell _{0}$, $\zeta _{\ell }\in \left( \frac{-2\mu }{1-\mu },\frac{2\mu }{1+\mu }\right)$.

Remark 19

In Case 6, the magnitude of the periods takes all values in the interval $\left( \left| z_\gamma \right| /E,\infty \right)$. The period $\omega =\left| z_\gamma \right| /E$ is achieved when $v=-BZ$, which implies $\sigma _{v}$is a one-parameter subgroup; i.e., non-spiraling. The magnitude of the period approaches $\infty$ as $v\rightarrow BZ$. This behavior is in contrast to the Riemannian case; i.e., the case $B=0$. In the Riemannian case, there are finitely many periods associated with each element $\gamma .$ However, if there exists $\gamma \in \Gamma$ such that $\log \gamma \in 2\pi \mathbb {Z},$ then $\Gamma \backslash H$ does not satisfy the Clean Intersection Hypothesis, so the fact that unusual magnetic geodesic behavior occurs in this case is not unprecedented (see [15]).