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We study the torsion of the \(\alpha \) -connections defined on the density manifold in terms of a regular Riemannian metric. In the case of the Fisher-Rao metric our results confirm the fact that all \(\alpha \) -connections are torsion free. For the \(\alpha \) -connections \(\nabla ^{(\textrm{O}, \alpha )}\) obtained by the Otto metric, we show that, except for \(\alpha = -1\) , they are not torsion free and that \(\nabla ^{(\textrm{O}, 0)}\) is compatible with the Otto metric, but not its Levi-Civita connection. In fact, we derive an explicit formula for this torsion and show that the \(\nabla ^{(\textrm{O}, 0)}\) -geodesics differ from those of the Otto metric.

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Torsion of  \(\alpha \) -Connections on the Density Manifold

  • 会议论文

  • 发表日期：

    2026年

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  • Nihat Ay,
  • Lorenz J. Schwachhöfer

摘要

We study the torsion of the \(\alpha \) -connections defined on the density manifold in terms of a regular Riemannian metric. In the case of the Fisher-Rao metric our results confirm the fact that all \(\alpha \) -connections are torsion free. For the \(\alpha \) -connections \(\nabla ^{(\textrm{O}, \alpha )}\) obtained by the Otto metric, we show that, except for \(\alpha = -1\) , they are not torsion free and that \(\nabla ^{(\textrm{O}, 0)}\) is compatible with the Otto metric, but not its Levi-Civita connection. In fact, we derive an explicit formula for this torsion and show that the \(\nabla ^{(\textrm{O}, 0)}\) -geodesics differ from those of the Otto metric.