Abstract
We construct a new almost-Hermitian structure of anti-invariant type on tangent bundles and deduce criteria for this structure to belong to all the Gray--Hervella classes. In particular, we prove that the tangent bundles over Kählerian and semi-Kählerian manifolds carry, respectively, a Kählerian and a semi-Kählerian structure.
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Zayatuev, B.V. On a Class of Almost-Hermitian Structures on Tangent Bundles. Mathematical Notes 76, 682–688 (2004). https://doi.org/10.1023/B:MATN.0000049667.15551.02
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DOI: https://doi.org/10.1023/B:MATN.0000049667.15551.02