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

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

2006-11-20

arxiv.org/abs/math/0611605v1

Mathematics

Differential Geometry

Scientific paper

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 associated complex line spanned by x and by Jx. We show that if (M,g) is Hermitian or if (M,g) is nearly Kaehler, then either the complex Jacobi operator (JC(P)y=R(y,x)x+R(y,Jx)Jx) or the complex curvature operator (RC(P)y=R(x,Jx)y) 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 manifold.

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