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<p>Using the notion of magnetic curvature recently introduced by the first author, we extend E. Hopf’s theorem to the setting of magnetic systems. Namely, we prove that if the magnetic flow on the <i>s</i>-sphere bundle is without conjugate points, then the total magnetic curvature is non-positive, and vanishes if and only if the magnetic system is magnetically flat. We then prove that magnetic flatness is a rigid condition, in the sense that it only occurs when either the magnetic form is trivial and the metric is flat, or when the magnetic system is Kähler, the metric has constant negative sectional holomorphic curvature, and <i>s</i> equals the Mañé critical value.</p>

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Magnetic Flatness and E. Hopf’s Theorem for Magnetic Systems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年1月11日

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  • Valerio Assenza,
  • James Marshall Reber,
  • Ivo Terek

摘要

Using the notion of magnetic curvature recently introduced by the first author, we extend E. Hopf’s theorem to the setting of magnetic systems. Namely, we prove that if the magnetic flow on the s-sphere bundle is without conjugate points, then the total magnetic curvature is non-positive, and vanishes if and only if the magnetic system is magnetically flat. We then prove that magnetic flatness is a rigid condition, in the sense that it only occurs when either the magnetic form is trivial and the metric is flat, or when the magnetic system is Kähler, the metric has constant negative sectional holomorphic curvature, and s equals the Mañé critical value.