In this study, the authors focus on quasi-hemi-slant submanifolds ($qhs$-submanifolds) of $(\alpha ,\beta )$-type almost contact manifolds, also known as trans-Sasakian manifolds. Essentially, we give sufficient and necessary conditions for the integrability of distributions using the concept of quasi-hemi-slant submanifolds of trans-Sasakian manifolds. We also consider the geometry of foliations dictated by the distribution and the requirements for submanifolds of trans-Sasakian manifolds with quasi-hemi-slant factors to be totally geodesic. Lastly, we give an illustration of a submanifold with a quasi-hemi-slant factor and discuss its application to number theory.

In this paper, we analyze a silver Riemannian structure on a Riemannian manifold. We compute some fundamental properties of the induced structure on submanifolds immersed in a silver Riemannian manifold and also obtain some results for induced structures on submanifolds of codimension 2. Moreover, we explore the conditions for totally geodesic and minimal submanifolds in a silver Riemannian manifold. Finally, we also give an example of submanifolds in a silver Riemannian manifold.