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Codazzi tensor fields in reductive homogeneous spaces

James Marshall Reber, Ivo Terek

Abstract

We extend the results about left-invariant Codazzi tensor fields on Lie groups equipped with left-invariant Riemannian metrics obtained by d'Atri in 1985 to the setting of reductive homogeneous spaces $G/H$, where the curvature of the canonical connection of second kind associated with the fixed reductive decomposition $\mathfrak{g} = \mathfrak{h}\oplus\mathfrak{m}$ enters the picture. In particular, we show that invariant Codazzi tensor fields on a naturally reductive homogeneous space are parallel.

Codazzi tensor fields in reductive homogeneous spaces

Abstract

We extend the results about left-invariant Codazzi tensor fields on Lie groups equipped with left-invariant Riemannian metrics obtained by d'Atri in 1985 to the setting of reductive homogeneous spaces , where the curvature of the canonical connection of second kind associated with the fixed reductive decomposition enters the picture. In particular, we show that invariant Codazzi tensor fields on a naturally reductive homogeneous space are parallel.
Paper Structure (3 sections, 6 theorems, 34 equations)

This paper contains 3 sections, 6 theorems, 34 equations.

Table of Contents

  1. Preliminaries
  2. The Codazzi compatibility condition in $\mathfrak{m}$
  3. Codazzi tensors versus difference curvatures

Key Result

Lemma 1.1

For a $G$-invariant connection $\nabla$ and a $G$-invariant $k$-times covariant tensor field $\varTheta$ on $G/H$, corresponding to $\alpha$ and $\theta$ on $\mathfrak{m}$ under eqn:nomizu--eqn:alpha_and_nabla and eqn:equivariant_sections, the covariant differential $\nabla\varTheta$ is also $G$-inv for all $X,Y_1,\ldots, Y_k\in\mathfrak{m}$.

Theorems & Definitions (14)

  • Lemma 1.1
  • proof
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • proof
  • Example 3.2
  • ...and 4 more