Some Symmetry Properties of Almost S-Manifolds

Manifold theory is an important topic in differential geometry. Riemannian manifolds are a wide class of differentiable manifolds.  Riemannian manifolds consist of two fundamental class, as contact manifolds and complex manifolds. The notion of globally framed metric -manifold is a generalization of these fundamental classes. Almost -manifolds which are globally framed metric -manifold generalize some contact manifolds carrying their dimension to . On the other hand, classification is important for Riemannian manifolds with respect to some intrinsic and extrinsic tools as well as all sciences. Moreover, symmetric manifolds play an important role in differential geometry. There are a lot of symmetry type for Riemannian manifolds with respect to different arguments. Under these considerations, in the present paper  we study some symmetry conditions on almost -manifolds . We investigate weak symmetries and -symmetries of these type manifolds. We obtain some necessary and sufficient conditions to characterize of their structures . Firstly, we prove that the existence of weakly symmetric and weakly Ricci symmetric almost -manifolds under some special conditions. Then, we show that every -symmetric almost -manifold verifying the -nullity distribution is an -Einstein manifold of globally framed type. Finally, we get some necessary and sufficient condition for a -Ricci symmetric almost -manifold verifying the -nullity distribution to be an -Einstein manifold of globally framed type.