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Almost complex structure; curvature operator; integrability; tangent bundle
In this paper, the standard almost complex structure on the tangent bunle of a Riemannian manifold will be generalized. We will generalize the standard one to the new ones such that the induced $(0,2)$-tensor on the tangent bundle using these structures and Liouville $1$-form will be a Riemannian metric. Moreover, under the integrability condition, the curvature operator of the base manifold will be classified.
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