Geodesic
Some Para-Hermitian Related Complex Structures and Non-existence of Semi-Riemannian Metric on Some Spheres
Serdica Mathematical Journal, Tome 25 (1999) no. 1, pp. 83-90.

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It is shown that the spheres S^(2n) (resp: S^k with k ≡ 1 mod 4) can be given neither an indefinite metric of any signature (resp: of signature (r, k − r) with 2 ≤ r ≤ k − 2) nor an almost paracomplex structure. Further for every given Riemannian metric on an almost para-Hermitian manifold with the associated 2-form φ one can construct an almost Hermitian structure (under certain conditions, two different almost Hermitian structures) whose associated 2-form(s) is φ.
Keywords: Hermitian, Para-Hermitian and Indefinite Metric 
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Erdem, Sadettin. Some Para-Hermitian Related Complex Structures and Non-existence of Semi-Riemannian Metric on Some Spheres. Serdica Mathematical Journal, Tome 25 (1999) no. 1, pp. 83-90. http://geodesic.mathdoc.fr/item/SMJ2_1999_25_1_a7/