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We introduce new metric structures on a smooth manifold (called "weak"structures) that generalize the almost contact, Sasakian, cosymplectic, etc.metric structures $(\varphi,\xi,\eta,g)$ and allow us to take a fresh look atthe classical theory. We demonstrate this statement by generalizing severalwell-known results. We prove that any Sasakian structure is rigid, i.e., ourweak Sasakian structure is homothetically equivalent to a Sasakian structure.We show that a weak almost contact structure with parallel tensor $\varphi$ isa weak cosymplectic structure and give an example of such a structure on theproduct of manifolds. We find conditions for a vector field to be a weakcontact infinitesimal transformation.

On the rigidity of the Sasakian structure and characterization of cosymplectic manifolds