  A geometric approach to immersion formulas for soliton surfaces is provided
through new cohomologies on spaces of special types of $\mathfrak{g}$-valued
differential forms. This leads us to introduce Poincar\'e-type lemmas for these
cohomologies, which appropriately describe the integrability conditions of Lax
pairs associated with systems of PDEs. Our methods clarify the structure and
properties of the deformations and soliton surfaces for the aforesaid Lax
pairs. Our findings also allow for the generalization of the theory of soliton
surfaces in Lie algebras to general soliton submanifolds. Techniques from the
theory of infinite-dimensional jet manifolds and diffieties enable us to
justify certain common assumptions of the theory of soliton surfaces.
Theoretical results are illustrated through $\mathbb{C}P^{N-1}$ sigma models.
