1 Introduction
2 Preliminaries
3 The rigidity of Sasakian structure and characteristic of a cosymplectic structure
4 Weak $K$-contact structure
5 Weak Nearly Sasakian/Cosymplectic Manifolds
6 Splitting of weak nearly cosymplectic manifolds
7 Characterization of Sasakian manifolds
8 Weak $\beta$-Kenmotsu manifolds