This paper investigates Lagrangian-like submanifolds, the analogues of Lagrangian submanifolds in cosymplectic geometry and lays the underlying work by studying their linear algebraic properties, including an analysis of the Lagrangian-like Grassmannian \(U(n)/O(n)\) . Moving into the manifold setting, we establish geometric results concerning the local structure of these submanifolds. In particular, we prove a relative Moser theorem that ensures the stability of cosymplectic structures under deformations and a cosymplectic Weinstein theorem that provides a canonical local model, \((T^*L \times \mathbb {R}, \omega _{\text {can}}, \eta _{\text {can}})\) , near any Lagrangian-like submanifold \(L\) . Applications of these results include the construction of a Weinstein-like chart near the identity in the group of cosymplectomorphisms, thus bridging the geometry with topological and dynamical invariants via fixed points and a newly introduced co-flux homomorphism for cosymplectomorphisms.
