Geodesic
On a Class of Almost-Hermitian Structures on Tangent Bundles
Matematičeskie zametki, Tome 76 (2004) no. 5, pp. 732-739

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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. 
@article{MZM_2004_76_5_a8,
     author = {B. V. Zayatuev},
     title = {On a {Class} of {Almost-Hermitian} {Structures} on {Tangent} {Bundles}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {732--739},
     publisher = {mathdoc},
     volume = {76},
     number = {5},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a8/}
}
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B. V. Zayatuev. On a Class of Almost-Hermitian Structures on Tangent Bundles. Matematičeskie zametki, Tome 76 (2004) no. 5, pp. 732-739. http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a8/