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