  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 at
the classical theory. We demonstrate this statement by generalizing several
well-known results. We prove that any Sasakian structure is rigid, i.e., our
weak Sasakian structure is homothetically equivalent to a Sasakian structure.
We show that a weak almost contact structure with parallel tensor $\varphi$ is
a weak cosymplectic structure and give an example of such a structure on the
product of manifolds. We find conditions for a vector field to be a weak
contact infinitesimal transformation.
