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The Hopf-Rinow theorem and the Mañé critical value for magnetic geodesics on odd-dimensional spheres

Peter Albers, Gabriele Benedetti, Levin Maier

Abstract

The subject of this article are magnetic geodesics on odd-dimensional spheres endowed with the round metric and with the magnetic potential given by the standard contact form. We compute the Mañé's critical value of the system and show that a value of the energy is supercritical if and only if all pairs of points on the sphere can be connected by a magnetic geodesic with that value of the energy. Our methods are explicit and rely on the description of the submanifolds invariant by the flow and of the symmetries of the system, which we define for a general magnetic system and call totally magnetic submanifolds and magnetomorphisms, respectively. We recover hereby the known fact that the system is super-integrable: the three-spheres obtained intersecting the ambient space with a complex plane are totally magnetic and each magnetic geodesic is tangent to a two-dimensional Clifford torus. In our study the integral of motion given by the angle between magnetic geodesics and the Reeb vector field plays a special role, and can be used to realize the magnetic flow as an interpolation between the sub-Riemannian geodesic flow of the contact distribution and the Reeb flow of the contact form.

The Hopf-Rinow theorem and the Mañé critical value for magnetic geodesics on odd-dimensional spheres

Abstract

The subject of this article are magnetic geodesics on odd-dimensional spheres endowed with the round metric and with the magnetic potential given by the standard contact form. We compute the Mañé's critical value of the system and show that a value of the energy is supercritical if and only if all pairs of points on the sphere can be connected by a magnetic geodesic with that value of the energy. Our methods are explicit and rely on the description of the submanifolds invariant by the flow and of the symmetries of the system, which we define for a general magnetic system and call totally magnetic submanifolds and magnetomorphisms, respectively. We recover hereby the known fact that the system is super-integrable: the three-spheres obtained intersecting the ambient space with a complex plane are totally magnetic and each magnetic geodesic is tangent to a two-dimensional Clifford torus. In our study the integral of motion given by the angle between magnetic geodesics and the Reeb vector field plays a special role, and can be used to realize the magnetic flow as an interpolation between the sub-Riemannian geodesic flow of the contact distribution and the Reeb flow of the contact form.

Paper Structure

This paper contains 20 sections, 14 theorems, 144 equations, 3 figures.

Table of Contents

  1. Introduction and statement of results
  2. Magnetomorphisms and totally magnetic submanifolds
  3. Contact angle, Clifford tori and Hopf fibration.
  4. Applications to Hamiltonian PDE's and Hofer--Zehnder capacities:
  5. Mañé's critical value for magnetic Lagrangians
  6. Three equivalent definitions
  7. Mañé's critical value for Killing magnetic Lagrangians
  8. Mañé's critical value for $K$-contact Lagrangians
  9. Clifford tori and magnetic geodesics on ${\mathbb{S}^{2n+1}}$
  10. The Hopf--Rinow Theorem for $({\mathbb{S}^{2n+1}},g,{\mathrm{d}}\alpha)$
  11. Proof of Corollary \ref{['c:connecting']}
  12. Proof of Theorem \ref{['t:mane']} (Second part)
  13. Magnetic geodesics and the Hopf fibration
  14. The proof of Theorem \ref{['t:hopf']}
  15. Restriction of $({\mathbb{S}^3},g,{\mathrm{d}}\alpha)$ around a Hopf fibre
  16. ...and 5 more sections

Key Result

Theorem 1.1

The Mañé's critical value of the system is The Mather set of the system is Let $q_0$ and $q_1$ be two points on ${\mathbb{S}^{2n+1}}$ and denote by $\langle q_0,q_1\rangle$ their Hermitian product. For every $k>0$, let $\mathcal{G}_k(q_0,q_1)$ be the set of magnetic geodesics with energy $k$ connecting $q_0$ and $q_1$. We have the following three cases

Figures (3)

  • Figure 1: The sets $C_s([0,\pi])$ for $s=1,2,3$.
  • Figure 2: This picture illustrates Theorem \ref{['t:hopf']} and the fact that we can connect the Reeb orbit $\{z_2=0\}$ and the Reeb orbit $\{z_1=0\}$ with a unit speed magnetic geodesic of strength $s$ if and only if $s\in[0,2)$, see Theorem \ref{['t:mane']}. (Picture by Ana Chavez Caliz.)
  • Figure 3: ${\mathbb{S}^3}$ is the union of two solid tori. The magnetic geodesic $\gamma(t)$ is contained in the Clifford torus which is the common boundary of the two solid tori. The magnetic geodesic spirals around the two Reeb orbits $\{z_2=0\}$ and $\{z_1=0\}$ which are the souls of the solid tori. (Picture made by Ana Chavez Caliz.)

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 30 more