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Torsion of $α$-connections on the density manifold

Nihat Ay, Lorenz J. Schwachhöfer

Abstract

We study the torsion of the $α$-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 $α$-connections are torsion free. For the $α$-connections obtained by the Otto metric, we show that, except for $α= -1$, they are not torsion free.

Torsion of $α$-connections on the density manifold

Abstract

We study the torsion of the -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 -connections are torsion free. For the -connections obtained by the Otto metric, we show that, except for , they are not torsion free.

Paper Structure

This paper contains 6 sections, 5 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. The affine structure of $P^\infty_+(M)$
  3. Regular metrics on $P^\infty_+(M)$
  4. The $\alpha$-connections of a regular metric on $P^\infty_+(M)$
  5. The Otto metric
  6. Acknowledements

Key Result

lemma 1

We have the identity where using the notation (direction-deriv).

Theorems & Definitions (12)

  • definition 1
  • lemma 1
  • proof
  • definition 2
  • theorem 1
  • proof
  • corollary 1
  • proof
  • proposition 1
  • theorem 2
  • ...and 2 more