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Indefinite almost paracontact metric manifolds. (English) Zbl 1187.53035

Summary: We introduce the concept of (\(\varepsilon \))-almost paracontact manifolds, and in particular, of (\(\varepsilon \))-para-Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of (\(\varepsilon \))-para Sasakian manifolds are obtained. We prove that if a semi-Riemannian manifold is one of flat, proper recurrent or proper Ricci-recurrent, then it cannot admit an (\(\varepsilon \))-para Sasakian structure. We show that, for an (\(\varepsilon \))-para Sasakian manifold, the conditions of being symmetric, semi-symmetric, or of constant sectional curvature are all identical. It is shown that a symmetric space-like (resp., time-like) (\(\varepsilon \))-para Sasakian manifold \(M^{n}\) is locally isometric to a pseudohyperbolic space \(H_{\nu }^{n}(1)\) (resp., pseudosphere \(S_{\nu }^{n}(1))\). At last, it is proved that for an (\(\varepsilon \))-para Sasakian manifold the conditions of being Ricci-semi-symmetric, Ricci-symmetric, and Einstein are all identical.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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References:

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