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

We introduce new metric structures on a smooth manifold (called “weak” structures) that generalize the almost contact, Sasakian, cosymplectic, etc. metric structures $(\phi ,\xi ,\eta ,g)$ and allow us to take a fresh look at the classical theory and find new applications. This assertion is illustrated by generalizing several well-known results. It is proved that any Sasakian structure is rigid, i.e., our weak Sasakian structure is homothetically equivalent to a Sasakian structure. It is shown that a weak almost contact structure with parallel tensor φ is a weak cosymplectic structure and an example of such a structure on the product of manifolds is given. Conditions are found under which a vector field is a weak contact vector field.