We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
135/2006
Relating the curvature tensor and the complex Jacobi operator of an almost Hermitian manifold
Miguel Brozos-Vazquez, Eduardo Garcia-Rio and Peter B. Gilkey
Abstract
Let J be a unitary almost complex structure on a Riemannian manifold (M,g). If x is a unit tangent vector, let P be the complex line in the tangent bundle which is spanned by x and by Jx. The complex Jacobi operator is defined by Jc(P)=J(x)+J(Jx) and the complex curvature operator is defined by Rc(P)=R(x,Jx). We show that if (M,g) is Hermitian or if (M,g) is nearly Kaehler, then either the complex Jacobi operator or the complex curvature operator completely determines the full curvature operator. This generalizes a well known result in the real setting to the complex setting. We also show that this result fails for general almost Hermitian manifolds.
