HYPERSURFACES OF ALMOST Γ-Paracontact RIEMANNIAN MANIFOLD ENDOWED WITH a QUARTER SYMMETRIC METRIC CONNECTION

Abstract

Abstract. We define a quarter symmetric metric connection in an almost r-paracontact Riemannian manifold and we consider invariant, noninvariant and anti-invariant hypersurfaces of an almost r-paracontact Riemannian manifold endowed with a quarter symmetric metric connection.

On Recurrent Lightlike Hypersurface of Kenmotsu Manifold

“…In case ( , m) = (1, 0) in (1.1) and ( , m) = (0, 1) in (1.2): The above connection ∇ reduces to a quarter-symmetric non-metric connection. The notion of quartersymmetric non-metric connection was introduced by Golab [5] and then, studied by Sengupta-Biswas [4] and Ahmad-Haseeb [3]. In case ( , m) = (0, 0) in (1.1) and ( , m) = (0, 1) in (1.2): The above connection ∇ will be a quarter-symmetric metric connection.…”

“…where ∇ and ∇ are the induced linear connections on T M and S(T M ) respectively, B and C are the local second fundamental forms on T M and S(T M ) respectively, A N and A N are the shape operators on T M and S(T M ) respectively, and τ and σ are 1-forms on M . For a lightlike hypersurface M of Kenmotsu manifold ( M , g ), it is known[3] that J(Rad(T M )) and J(tr(T M )) are subbundles of S(T M ), of rank 1 such that J(Rad(T M )) ∩ J(tr(T M )) = 0. Thus there exist two non-degenerate almost complex distributions D 0 and D on M with respect to J, i.e., J(D 0 ) = D 0 and J(D) = D, such that…”