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On Riemannian four-manifolds and their twistor spaces: a moving frame approach

Giovanni Catino, Davide Dameno, Paolo Mastrolia

Abstract

In this paper we study the twistor space $Z$ of an oriented Riemannian four-manifold $M$ using the moving frame approach, focusing, in particular, on the Einstein, non-self-dual setting. We prove that any general first-order linear condition on the almost complex structures of $Z$ forces the underlying manifold $M$ to be self-dual, also recovering most of the known related rigidity results. Thus, we are naturally lead to consider first-order quadratic conditions, showing that the Atiyah-Hitchin-Singer almost Hermitian twistor space of an Einstein four-manifold bears a resemblance, in a suitable sense, to a nearly Kähler manifold.

On Riemannian four-manifolds and their twistor spaces: a moving frame approach

Abstract

In this paper we study the twistor space of an oriented Riemannian four-manifold using the moving frame approach, focusing, in particular, on the Einstein, non-self-dual setting. We prove that any general first-order linear condition on the almost complex structures of forces the underlying manifold to be self-dual, also recovering most of the known related rigidity results. Thus, we are naturally lead to consider first-order quadratic conditions, showing that the Atiyah-Hitchin-Singer almost Hermitian twistor space of an Einstein four-manifold bears a resemblance, in a suitable sense, to a nearly Kähler manifold.

Paper Structure

This paper contains 7 sections, 11 theorems, 117 equations.

Table of Contents

  1. Introduction and main results
  2. Geometry of Riemannian four-manifolds
  3. The twistor space of a four-manifold
  4. Linear conditions on $J_+$ and $J_-$: proof of Theorem \ref{['TH:GeneralLinear']}
  5. Quadratic conditions: proofs of Theorems \ref{['TH:LocalQuadratic']}, \ref{['TH:LocalQuadratic2']}
  6. Data availability statements
  7. Acknowledgements

Key Result

Theorem 1.1

Let $(M,g)$ be an oriented Riemannian four-manifold and let $(Z_-,g_t,\bar{J})$ be its twistor space, with $\bar{J}=J_+$ or $\bar{J}=J_-$. Suppose that, for every $X,Y$ smooth vector fields on $Z_-$, for some $a_i\in\mathbb{R}$, $i=1,\ldots,8$, such that $a_j\neq 0$ for some $j$. Then, $M$ is self-dual.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 2.1
  • Proposition 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • ...and 12 more