Relating the curvature tensor and the complex Jacobi operator of an almost Hermitian manifold

Abstract

Abstract. Let J be a unitary almost complex structure on a Riemannian manifold (M, g). If x is a unit tangent vector, let π := Span{x, Jx} be the associated complex line in the tangent bundle of M . The complex Jacobi operator and the complex curvature operators are defined, respectively, by J (π) := J (x) + J (Jx) and R(π) := R(x, Jx). We show that if (M, g) is Hermitian or if (M, g) is nearly Kähler, then either the complex Jacobi operator or the complex curvature operator completely determine the full curvature operator; this generalizes a well known result in the real setting to the complex setting. We also show this result fails for general almost Hermitian manifolds.

Geometric Realizations of Para-Hermitian Curvature Models

“…Theorem 5.3). Furthermore, either the complex Jacobi operator or the complex curvature operator completely determine the components in W ⊥ 7 of a curvature tensor [8]; the algebraic condition determining W 7 also plays a role in the study of Jacobi-Ricci commuting curvature tensors [36].…”

Geometric realizations of Hermitian curvature models

“…Among these factors W 7 , see Theorem 2.1 for a precise definition, plays a central role in our discussion, since the space of vectors satisfying the Gray condition is exactly W ⊥ 7 as will be discussed in Section 3. We note that either the complex Jacobi operator or the complex curvature operator completely determine the components in W ⊥ 7 of a curvature tensor [3]. Also note that the algebraic condition determining W 7 plays a role in the study of Jacobi-Ricci commuting curvature tensors [12].…”

Complex Osserman Kähler manifolds in dimension four

“…Basic results. We refer to [4] for the proof of the following Lemma: Lemma 3.1. Let H be a Kähler model.…”