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Relating the curvature tensor and the complex Jacobi operator of an almost Hermitian manifold
Miguel Brozos-Vazquez, Eduardo Garcia-Rio and Peter B. Gilkey
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.