Results in Mathematics

, Volume 72, Issue 4, pp 2037–2056 | Cite as

Interaction of Codazzi Couplings with (Para-)Kähler Geometry

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Abstract

We study Codazzi couplings of an affine connection \(\nabla \) with a pseudo-Riemannian metric g, a nondegenerate 2-form \(\omega \), and a tangent bundle isomorphism L on smooth manifolds, as an extension of their parallelism under \(\nabla \). In the case that L is an almost complex or an almost para-complex structure and \((g, \omega , L)\) form a compatible triple, we show that Codazzi coupling of a torsion-free \(\nabla \) with any two of the three leads to its coupling with the remainder, which further gives rise to a (para-)Kähler structure on the manifold. This is what we call a Codazzi-(para-)Kähler structure; it is a natural generalization of special (para-)Kähler geometry, without requiring \(\nabla \) to be flat. In addition, we also prove a general result that g-conjugate, \(\omega \)-conjugate, and L-gauge transformations of \(\nabla \), along with identity, form an involutive Abelian group. Hence a Codazzi-(para-)Kähler manifold admits a pair of torsion-free connections compatible with the \((g, \omega , L)\). Our results imply that any statistical manifold may admit a (para-)Kähler structure as long as one can find an L that is compatible to g and Codazzi coupled with \(\nabla \).

Keywords

Codazzi coupling conjugate connection gauge transformation Kähler structure Para-Kähler structure statistical manifold Torsion 

Mathematics Subject Classification

32Q15 32Q60 53B05 53B35 53D05 62B10 
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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Columbia UniversityNew YorkUSA
  2. 2.University of MichiganAnn ArborUSA

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