Pseudo-Reimannian manifolds endowed with an almost para f-structure

Abstract

Let M˜(U,Ω˜,η˜,ξ,g˜) be a pseudo-Riemannian manifold of signature (n+1,n). One defines on M˜ an almost cosymplectic para f-structure and proves that a manifold M˜ endowed with such a structure is ξ-Ricci flat and is foliated by minimal hypersurfaces normal to ξ, which are of Otsuki's type. Further one considers on M˜ a 2(n−1)-dimensional involutive distribution P⊥ and a recurrent vector field V˜. It is proved that the maximal integral manifold M⊥ of P⊥ has V as the mean curvature vector (up to 1/2(n−1)). If the complimentary orthogonal distribution P of P⊥ is also involutive, then the whole manifold M˜ is foliate. Different other properties regarding the vector field V˜ are discussed.

Keywords

• pseudo-Riemannian manifold
• cosymplectic manifold
• para f-structure
• minimal hypersurface.