On Riemannian 4-manifolds and their twistor spaces: a moving frame approach (Q6656200)

This paper explores the twistor spaces of oriented Riemannian four-manifolds using the moving frame formalism. The authors focus on Einstein manifolds in the non-self-dual setting, providing new insights into their structure. The primary results demonstrate that general first-order linear conditions on the almost complex structures of the twistor space necessitate the underlying manifold's self-duality. Moreover, the authors extend their analysis to quadratic conditions, revealing that the Atiyah-Hitchin-Singer almost Hermitian twistor space of an Einstein four-manifold shares structural similarities with nearly Kähler manifolds.\N\NIn particular, Theorem 1.1 establishes that any general linear conditions imposed on the covariant derivative of the almost complex structures \(J\) on the twistor space \(Z\) force the underlying four-manifold \(M\) to be self-dual. Specifically, the conditions involve linear combinations of terms such as \((\nabla_X J)(Y)\), \((\nabla_{JX} J)(Y)\), and their interactions. This result provides an alternative pathway to classical results by Atiyah-Hitchin-Singer, demonstrating the strong interplay between the geometry of \(M\) and the properties of \(Z\).\N\NTheorem 1.2 provides a novel quadratic condition that is both necessary and sufficient for an oriented Riemannian four-manifold \(M\) to be Einstein. For any orthonormal frame on the negatively oriented frame bundle \(O(M)^-\), the authors show that a specific quadratic equation involving components of the covariant derivative of \(J\) must hold:\N\[\N\sum_{t=1}^6 (J^t_{p,q} + J^t_{q,p})(J^t_{p,p} - J^t_{q,q}) = 0, \quad \forall p, q.\N\]\NTheorem 1.4 focuses on the holomorphic scalar curvature of the twistor space \(Z^-\) associated with an Einstein four-manifold \(M\). The authors compute the Ricci* tensor to derive bounds on the difference between the scalar curvature \(S\) and the holomorphic scalar curvature \(S^*\):\N\[\N-\frac{1}{2} |\nabla J|^2 \leq S - S^* \leq |\nabla J|^2.\N\]\NEquality holds if and only if \(M\) is self-dual. This result highlights how \(S - S^*\) measures the deviation of the twistor space from being nearly Kähler or self-dual.\N\N.