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Transformations and Coupling Relations for Affine Connections

  • James Tao
  • Jun ZhangEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9389)

Abstract

The statistical structure on a manifold \(\mathfrak {M}\) is predicated upon a special kind of coupling between the Riemannian metric g and a torsion-free affine connection \(\nabla \) on the \(T\mathfrak {M}\), such that \(\nabla g\) is totally symmetric, forming, by definition, a “Codazzi pair” \(\{\nabla , g\}\). In this paper, we first investigate various transformations of affine connections, including additive translation (by an arbitrary (1,2)-tensor K), multiplicative perturbation (through an arbitrary invertible operator L on \(T\mathfrak {M}\)), and conjugation (through a non-degenerate two-form h). We then study the Codazzi coupling of \(\nabla \) with h and its coupling with L, and the link between these two couplings. We introduce, as special cases of K-translations, various transformations that generalize traditional projective and dual-projective transformations, and study their commutativity with L-perturbation and h-conjugation transformations. Our derivations allow affine connections to carry torsion, and we investigate conditions under which torsions are preserved by the various transformations mentioned above. Our systematic approach establishes a general setting for the study of Information Geometry based on transformations and coupling relations of affine connections – in particular, we provide a generalization of conformal-projective transformation.

Keywords

Conformal Transformation General Transformation Projective Transformation Invertible Operator Leibniz Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgment

This work was completed while the second author (J.Z.) was on sabbatical visit at the Center for Mathematical Sciences and Applications at Harvard University under the auspices and support of Prof. S.T. Yau.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Harvard UniversityCambridgeUSA
  2. 2.University of MichiganAnn ArborUSA

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