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Twists, Codazzi Tensors, and the $6$-sphere

David N. Pham

Abstract

Let $(M,g,J,ω)$ be an almost Hermitian manifold. Given an automorphism $ψ\in \mathrm{Aut}(TM)$, the existing structure can be twisted to obtain a new almost Hermitian manifold $(M,g^ψ,J^ψ,ω^ψ)$. In the current paper, we study these $ψ$-twisted almost Hermitian structures with particular emphasis on questions regarding the integrability of $J^ψ$ and the Riemannian geometry of $g^ψ$. By studying the latter, we identity a certain class of $\mathrm{Aut}(TM)$ with nice transformation properties. We call these automorphisms $g$-\textit{Codazzi maps} because of their close relationship with Codazzi tensors. The aforementioned results are ultimately applied to the standard nearly Kähler structure on the $6$-sphere where we prove a nonintegrability result for the class of $g$-Codazzi maps.

Twists, Codazzi Tensors, and the $6$-sphere

Abstract

Let be an almost Hermitian manifold. Given an automorphism , the existing structure can be twisted to obtain a new almost Hermitian manifold . In the current paper, we study these -twisted almost Hermitian structures with particular emphasis on questions regarding the integrability of and the Riemannian geometry of . By studying the latter, we identity a certain class of with nice transformation properties. We call these automorphisms -\textit{Codazzi maps} because of their close relationship with Codazzi tensors. The aforementioned results are ultimately applied to the standard nearly Kähler structure on the -sphere where we prove a nonintegrability result for the class of -Codazzi maps.
Paper Structure (4 sections, 18 theorems, 193 equations)

This paper contains 4 sections, 18 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. The Riemannian geometry of twisted metrics
  3. $g$-Codazzi maps on the $n$-sphere
  4. Integrability and $g$-Codazzi maps

Key Result

Proposition 2.1

Let $\psi\in \hbox{Aut}(TM)$ and let $h:=g^\psi$. Then In particular, if $(\nabla^g_X\psi^{-1})Y=(\nabla^g_Y\psi^{-1})X$, then

Theorems & Definitions (43)

  • Definition 1.1
  • Conjecture 1.2
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3: Hicks, Hicks1965
  • proof
  • Example 2.4
  • Proposition 2.5: Hicks, Hicks1965
  • proof
  • ...and 33 more