A cohomological approach to immersed submanifolds via integrable systems

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.


Introduction
Integrable models and their continuous deformations under various types of dynamics have produced considerable interest in various branches of mathematics, physics, and biology (see [4,9,12,13,24,41,52,54,55,57,60] for details). In fact, the motivation for this research topic came largely from applications like the growth of crystals [52], quantum field theory models [12,52,54], or the motion of boundaries between regions of different viscosities and densities [8,60].
The progress in the analytic description of surfaces obtained from nonlinear PDEs, e.g. soliton or constant curvature surfaces [3,63], has been rapid and has resulted in many new techniques and theoretical approaches. Some of the foremost relevant developments have occurred in the study of soliton surfaces immersed in Lie algebras by using techniques from the theory of completely integrable systems [7,10,11,14,17,18,19,20,56,61,62].
The possibility of using a linear spectral problem (LSP) to represent a moving frame on a soliton surface has yielded many findings concerning their intrinsic geometric properties [2,3,34,38,50,51]. The spectral parameter in the LSP describes deformations of soliton surfaces preserving their properties in such a way that integrable surfaces come in a family [3]. These surfaces are characterized by fundamental forms whose coefficients satisfy the Gauss-Weingarten and Gauss-Mainardi-Codazzi equations. It has recently proved fruitful to apply such a characterization of soliton surfaces to CP N −1 sigma models via their immersion formulas in Lie algebras. They have also been shown to play an essential role in many other problems of a physical nature (see [2,22,23,28,46,69] and references therein).
The construction of the soliton surfaces related to the completely integrable CP N −1 sigma model has been accomplished by representing the Euler-Lagrange equations for this model as a conservation law which in turn provides a closed su(N )-valued differential one-form on the surfaces. This is the so-called generalized Weierstrass formula for immersion [32,37].
The construction of smooth orientable soliton surfaces related to completely integrable models in the sense of admitting a LSP problem was pioneered by A. Sym [61,62]. His technique exploits the conformal invariance of the zero-curvature representation of the LSP relative to the spectral parameter [3]. Another approach for determining such surfaces, formulated by Cieśliński and Doliwa [10,11,17], is based on the use of gauge symmetries of the LSP. Fokas and Gel'fand [19,20] developed a third approach by using the LSP for integrable systems and their Lie symmetries to derive families of soliton surfaces. In all these cases soliton surfaces are described through the so-called immersion formulas. Most recently, a reformulation and extension of the Fokas-Gel'fand immersion formula has been performed through the formalism of generalized vector fields and their actions on jet spaces in [25,26,27,28]. This extension has provided the necessary and sufficient conditions for the existence of soliton surfaces in terms of the symmetries of the LSP and integrable models. It also described the relations between the previously known immersion formulas.
This paper aims to provide a unifying approach to immersion formulas via cohomological and geometric techniques. Our procedure provides simple geometric proofs for the theoretical results in the previous literature on the topic, simplifies expressions for immersion formulas via geometric structures, and extends the formalism of immersion formulas to create a theory of immersion formulas for soliton submanifolds and generalized Lax pairs with many potential applications (see e.g. [1,6,44,67,68] and references therein).
Let g be a Lie algebra and let M, N be manifolds. Consider an integrable system of PDEs whose independent and dependent variables are coordinates on M and N respectively. The first idea of the paper is to give a more precise description of the standard objects appearing in the study of soliton surfaces for our system of PDEs, e.g. its LSPs, deformations, and immersion formulas, by the hereafter defined parametrized g-valued differential forms on-shell on M . In a nutshell, a parametrized g-valued differential form on-shell is a family of g-valued differential forms on M parametrized by particular solutions of the integrable system of PDEs and the spectral parameter λ of a corresponding LSP. More generally, the parametrized g-valued differential forms on M are g-valued differential forms on M described by arbitrary functions from M to N that need not be particular solutions of our initial system of PDEs. The parametrized g-valued differential forms will also play a relevant role in the description of Lax pairs and immersion formulas. If not otherwise stated, it is hereafter assumed for simplicity that the independent variables of all systems of PDEs and types of g-differential forms are defined on M .
The space of parametrized g-valued differential forms, say Ω N (M ) ⊗ g, is endowed with a differential d leading to a cochain complex (see [49] for details on cochain complexes) where Ω k N (M ) ⊗ g, with k ∈ Z, is the space of parametrized g-valued differential k-forms. A LSP problem is then proved to amount to a parametrized g-valued differential one-form ω ∈ Ω 1 N (M ) ⊗ g. This allows us to define a second operator d 2ω on the above cochain complex satisfying the condition that d 2ω ω vanishes exactly on the solutions of the system of PDEs under consideration. We then say that d 2ω ω = 0 on-shell.
The space Ω S (M ) ⊗ g of parametrized g-valued differential forms on-shell can be endowed with two differentials d and d 2ω inducing new cochain complexes where Ω k S (M ) ⊗ g, with k ∈ Z, is the space of parametrized g-valued differential k-forms on-shell. In other words, d 2 2ω = 0 and d 2 = 0 on Ω S (M ) ⊗ g. We extend the Poincaré Lemma for the standard de Rham cohomology to this new realm so as to prove that every element ϑ ∈ Ω k S (M ) ⊗ g is d 2ω -closed, i.e. d 2ω ϑ = 0, if and only if it is locally exact, i.e. there exists for any arbitrary point p ∈ M an open U ⊂ M containing p and ϕ ∈ Ω k−1 g-valued differential forms are introduced in Section 5. Subsequently, the theory of immersion formulas for general soliton submanifolds through cohomological and geometrical techniques is addressed in Section 6. The relation between different immersion formulas, e.g. the previously described multidimensional generalizations of the Sym-Tafel (ST), Cieśliński-Doliwa (CD) and Fokas-Gel'fand (FG) formulas, are detailed in Section 7. Next, Section 8 concerns the special case of immersion formulas related to Lax pairs describing solutions of PDEs obeying a certain boundary condition. Our techniques are illustrated through soliton surfaces for CP N −1 sigma models in Section 9. The main contributions of the paper and topics for further research are summarized in Section 10.
where the R i are the so-called characteristics of the vector field X and D J := D j 1 . . . D jp for where u i J,α represents the variable u i J ′ with J ′ := (j 1 , . . . , j p , α) and J is an arbitrary multi-index. Geometrically, j p X is the unique vector field on J p projecting onto X on M × N that leaves the vector fields taking the Cartan distribution C p invariant (with respect to its action by Lie brackets). Alternatively, j p X can be defined as the restriction of pr X to functions in C ∞ (J p ).
Equivalently, pr X can be written as 3) The vector fields on J p are always related to a uni-parametric group of diffeomorphisms describing their flow. Meanwhile, the vector fields on J ∞ are not generally associated with any such a group of diffeomorphisms [66]. Nevertheless, there exist relevant types of vector fields on J ∞ that admit a uni-parametric group of transformations on J ∞ , e.g. the evolutionary vector fields. An evolutionary vector field is a vector field on J ∞ that projects onto each J p and whose expression on J ∞ can be considered as the limit of its projections. For instance, the so-called total derivatives D α are evolution vector fields [66].
Let us define on J p a system of partial differential equations (PDEs) in m independent and n dependent variables of the form A particular solution of (2.4) is a map u(x) from M to N whose prolongation to J p , thought of as a section of J 0 , satisfies (2.4).
The system of PDEs (2.4) determines a region E ⊂ J p where the functions ∆ µ , with µ ∈ 1, s, vanish. The system of PDEs (2.4) is hereupon assumed to be locally solvable, namely for each j p x u ∈ E there exists a solution u(x) of the system (2.4) such that j p x u belongs to the prolongation of u(x) to J p [53, p. 158]. Additionally, it is also assumed that the system (2.4) has maximal rank, i.e. the functions ∆ µ are functionally independent and hence the space E can be considered as a submanifold of J p (cf. [53, p. 158]). These are assumptions satisfied by a large family of differential equations.
A vector field X on J 0 is a classical Lie point symmetry of the nondegenerate system of PDEs (2.4) if the prolongation of X to J p is such that Therefore, the commutator of two classical Lie point symmetries is a Lie point symmetry. Thus, the Lie point symmetries form a Lie algebra V s , which, if finite-dimensional, locally defines an action of a Lie group G on J 0 . The symmetry group G transforms solutions of (2.5) into new solutions. This means that the graph corresponding to one solution is transformed into the graph associated with another solution. If the graph is preserved by the group G or, equivalently, if the vector fields X a := ξ α a ∂ α α + ϕ i a ∂ i , with a ∈ 1, r, conforming a basis of the Lie algebra V s are tangent to the graph, then the related solution is said to be G-invariant. Invariant solutions satisfy, in addition to the equations (2.4), the characteristic equations equated to zero where the index a runs over the elements of a basis of V s . The above formalism can be extended to J ∞ by extending the system (2.4) on J p to a system of equations on J ∞ of the form Recall that ∆ µ only depends on the variables u J for |J| ≤ p. Hence, the solutions of (2.7) are the extension to J ∞ of particular solutions of ∆ µ (j p x u) = 0. The equations (2.7) define a submanifold E ∞ of J ∞ and the total derivatives, D J , are now tangent to it.
The Cartan distribution C on J ∞ is the distribution generated by the vector fields D α for α ∈ 1, m. The Cartan distribution is involutive, i.e. the Lie bracket of vector fields taking values in C also takes values in C. The vector fields taking values in the Cartan distribution C are tangent to E ∞ , which allows us to restrict C to E ∞ giving rise to the distribution C| E ∞ on E ∞ . The pair (E ∞ , C| E ∞ ) is called a diffiety [65].
It may happen that invariant solutions of (2.4) are restricted in number or trivial if the full symmetry group is small. To extend the number of symmetries, and thus of solutions, one looks for generalized symmetries. They exist only if the nonlinear equation (2.4) is integrable [51], i.e. it has been obtained as the compatibility of a Lax pair. To describe them, we make use of generalized vector fields. A generalized vector field is a vector field X R on J ∞ ([53, Definition 5.1]) of the form where the functions R i ∈ C ∞ (J ∞ ) are arbitrary. The space of generalized vector fields is isomorphic to the space sym(C) of symmetries of the Cartan distribution C modulo Cartan vector fields, namely vector fields taking values in C [66]. Indeed, every vector field in sym(C) gives rise to a generalized vector field by restricting it to C ∞ (J 0 ) and every generalized vector field X R can be extended to a unique vector field pr X on J ∞ leaving invariant the space of vector fields taking values in C, i.e. pr X induces an element of sym(C), whose action on C ∞ (J 0 ) coincides with X R . In coordinates, (2.8) A vector field X R is a generalized symmetry of the system of PDEs (2.4) if and only if prX R is tangent to E ∞ [53].

Cohomologies on parametrized g-valued differential forms
This section addresses the generalizations of standard cohomologies of differential forms, e.g. the de Rham one, to the realm of cohomologies for differential forms taking values in Lie algebras parametrized by sections of a vector bundle (M × N, M, π) and a parameter. This entails the development of analogues for these new cohomologies of the Poincaré Lemma for de Rham cohomology. These results will later play a fundamental role in studying and extending Lax pairs and immersion formulas for soliton surfaces within Lie algebras to immersion formulas for soliton submanifolds in Lie algebras. Let d stand for the standard exterior differential on differential forms. The space Ω(M ) ⊗ g of g-valued differential forms on M admits a natural exterior differential d : Ω(M ) ⊗ g → Ω(M ) ⊗ g defined by setting d(θ ⊗ v) := (dθ) ⊗ v for every v ∈ g and θ ∈ Ω(M ) and extending d by R-linearity over the whole Ω(M ) ⊗ g. Due to the definition of d in terms of d, it follows that d 2 = 0 and d induces a cochain complex: where Ω k (M ) ⊗ g, with k ∈ Z, is the space of g-valued differential k-forms on M .
Similarly to the case of de Rham cohomology, θ ∈ Ω(M ) ⊗ g is said to be d-closed if dθ = 0 and d-exact when θ = dϑ for a ϑ ∈ Ω(M ) ⊗ g. Let Z k dR (M, g) and B k dR (M, g) be the spaces of d-closed and d-exact g-valued differential k-forms, respectively. From d 2 = 0 it follows that B k dR (M, g) ⊂ Z k dR (M, g), and it makes sense to define Since a g-valued differential form on M is nothing but a family of dim g mutually independent differential forms on M , the Poincaré Lemma for standard differential forms can be straightforwardly extended to d-closed g-valued differential forms on M . Moreover, H k Let us define a new cohomology on Ω(M ) ⊗ g by using the bracket [· ∧ ·] of g-valued differential forms on M [36]. This bracket is defined in such a way that and its value is extended by C ∞ (M )-bilinearity over the whole Ω(M ) ⊗ g. This prompts us to define a new cochain complex on Ω(M ) ⊗ g given in the following theorem.
Theorem 3.1. Assume ω ∈ Ω 1 (M ) ⊗ g and define If d ω ω = 0, then d 2 2ω = 0 and d 2ω induces a cochain complex Proof. Assume that ϑ ∈ Ω k (M ) ⊗ g with k ∈ N. Since ω ∈ Ω 1 (M ) ⊗ g in view of the definition of d 2ω , and the Lie bracket on g-valued differential forms, it follows that dϑ − [ω ∧ ϑ] is a g-valued differential (k + 1)-form. Hence, the sequence of linear morphisms (3.1) is well defined. Let us prove that d 2 2ω = 0. By evaluating d 2 2ω on ϑ, we obtain Since dω = 1 2 [ω ∧ ω] by assumption, it follows that Let us set dx J := dx j 1 ∧ . . . ∧ dx j |J | . In local coordinates ω = dx α ⊗ v α and ϑ = dx J ⊗ v J for certain vectors v α , v J ∈ g. Hence, The Jacobi identity for the Lie bracket in g implies that Since the coefficients relative to the indices α, β, J of the above g-valued differential (k + 1)-form are symmetric relative to the interchange of α and β, it follows that d 2 2ω ϑ = 0. As this result remains true for any ϑ and any coordinated open subset of M , it turns out that d 2 2ω = 0.
As in the de Rham cohomology induced by d on g-valued differential forms, θ ∈ Ω(M ) ⊗ g is said to be d 2ω -closed if d 2ω θ = 0 and d 2ω -exact when θ = d 2ω ϑ for a certain ϑ ∈ Ω(M ) ⊗ g. Let Z k 2ω (M, g) and B k 2ω (M, g) be the spaces of d 2ω -closed and d 2ω -exact g-valued differential k-forms, respectively. Since d 2 2ω = 0, it follows that B k 2ω (M, g) ⊂ Z k 2ω (M, g) and we can define The Poincaré Lemma for standard differential forms can be straightforwardly extended to g-valued differential one-forms on M relative to d ω and d 2ω . The first generalization is immediate and, in particular, Meanwhile, a generalization of the Poincaré Lemma for the cohomology d 2ω is given by the following theorem.
By using the zero-curvature condition (ZCC) for this system of first-order PDEs in the unknown θ, it follows that the system has a local solution if and only if d( Hence, d 2ω θ = ϑ for a certain locally defined θ if and only if d 2ω ϑ = 0. It is relevant that the d-closedness of certain elements of Ω(M ) ⊗ g is related to the existence of solutions to certain systems of partial differential equations. For instance, if d ω ω = 0, then 2ω (M, g). The first and third cases are immediate. The second one follows from the fact that G and g are matrix Lie groups and Lie algebras, respectively, and d acts on them as a de Rham differential on their matrix entries.
The above results can be generalized to the hereafter called parametrized g-valued differential forms. These differential forms will appear naturally in the study of Lax pairs and immersion formulas. We define N := J p × Λ, where Λ is a one-dimensional submanifold of C coordinated by the variable λ. Definition 3.3. A parametrized g-valued differential form is a family of g-valued differential forms on M , say parametrized by arbitrary sections u ∈ Γ(π) and λ ∈ Λ. A parametrized g-valued differential form on-shell is a family of g-valued differential forms (3.4) where u ∈ Γ(π) is a solution of ∆(j p x u) = 0. We write Ω N (M ) ⊗ g (resp. Ω S (M ) ⊗ g) for the space of parametrized g-valued differential forms (resp. on-shell).
The space Ω N (M ) ⊗ g admits a standard grading Ω N (M ) ⊗ g = k∈Z Ω k N (M ) ⊗ g compatible with the exterior product of parametrized g-valued differential forms that is defined in the natural way. A similar grading can be applied to parametrized g-valued differential equations on-shell. The elements of each Ω k N (M ) ⊗ g (resp. Ω k S (M ) ⊗ g) are called parametrized g-valued differential k-forms (resp. on-shell).
The following theorem allows us to define two cohomologies on Ω S (M )⊗g. It is indeed a natural generalization of Theorem 3.1 to Ω N (M ) ⊗ g and Ω S (M ) ⊗ g.  Proof. In view of (3.5), the differential d of a parametrized g-valued differential k-form is a parametrized g-valued differential (k + 1)-form. Additionally, the commutativity of partial derivatives ∂ α , ∂ β , for α, β ∈ 1, m, and its action on the differential functions ϑ J (u(x), λ) for each fixed u(x) implies that [d α , d β ] = 0. Hence, because the coefficients of d 2 ϑ, i.e. d β d α ϑ J , are symmetric relative to the interchange of α and β. Additionally, Following the same ideas of Theorem 3.1 and using the previous fact, we see that d 2ω induces a cohomology on Ω S (M ) ⊗ g.
The above theorem denotes the different differentials for the parametrized g-valued differential forms in the same way as the differentials for standard g-valued differential forms. This simplifies the notation and it is not misleading since the meaning of d and d 2ω is clear taking into account on which kind of g-valued differential form the operator is acting on. Moreover, both operators are essentially the same thing, but they operate in different spaces. Similarly to previous cohomologies, one can define closeness and exactness relative to the operators d and d ω . The spaces Z k S (M, g), B k S (M, g), and H k S (M, g) of closed, open, and equivalence classes of parametrized g-valued differential forms on-shell modulo exact ones can be defined as for previous cohomologies. Similarly, one can define the corresponding spaces for parametrized g-valued differential forms. Moreover, applying Theorem 3.2 for each fixed u(x) and λ, we get the following corollary.
More easily, one has the following trivial result.

Spectral differential forms
It turns out that parametrized g-valued differential forms of the previous section can be generated, in relevant cases, by means of a new type of g-valued differential forms on spaces of infinite jets: the hereafter called spectral differential forms. Although the theoretical description of these new structures is more complicated than the description of parametrized g-valued differential forms, they offer numerous practical advantages in calculations, which justifies their introduction. In particular, this formalism permits us to use the powerful machinery of the theory of jet bundles to obtain Lie symmetries, immersion formulas, and other related structures. Relevantly, this formalism will fill some theoretical details in the jet formalism of the immersion formulas given in [25].
Definition 4.1. A spectral differential form is a family of g-valued differential forms on J ∞ parametrized by a spectral parameter λ ∈ Λ ⊂ C for a certain submanifold Λ ⊂ C and taking the form ). An spectral differential form on-shell is the restriction of a spectral differential form (4.1) to a λ-parametrized family of g-valued differential forms on E ∞ . We write Ω λ (J ∞ )⊗ g (resp. Ω λ (E ∞ )⊗ g) for the space of spectral differential forms (resp. on-shell).
The space Ω λ (J ∞ ) ⊗ g admits a standard grading Ω λ (J ∞ ) ⊗ g = k∈Z Ω k λ (J ∞ ) ⊗ g compatible with the exterior product of spectral differential forms defined in the natural way. A similar grading can be applied to spectral differential forms on-shell. The elements of each Ω k λ (J ∞ ) ⊗ g (resp. Ω k λ (E ∞ ) ⊗ g) are called spectral differential k-forms (resp. on-shell). Every spectral differential form induces a unique parametrized g-valued differential form. Indeed, given (4.1) and a section s ∈ Γ(π) of the form s(x) = (x, u(x)), a parametrized g-valued differential form is defined by ω(u(x), λ) := j ∞ s * ω λ . If u(x) is a solution of our system of PDEs, then ω(u(x), λ) becomes a parametrized g-valued differential form on-shell. Meanwhile, a parametrized g-valued differential form need not come from a spectral differential form.
Since spectral differential forms give rise to parametrized g-valued differential forms, it becomes relevant how to apply the formalism for parametrized g-valued differential forms directly to spectral differential forms. This is accomplished by the following theorem, which provides an extension of the horizontal differential on the infinite-dimensional jet bundle J ∞ (see [39]).
The space Ω λ (J ∞ ) ⊗ g admits a cohomology given by d. The space Ω λ (E ∞ ) ⊗ g admits two cohomologies given by d and d 2ω .
Proof. The formula (4.2) ensures that the differential d of a spectral differential k-form on J ∞ is a spectral differential (k + 1)-form. Since [D α , D β ] = 0 on J ∞ (cf. [53]), because the coefficients of this spectral differential (k + 2)-form, namely D β D α ϑ J , are symmetric relative to the interchange of α and β. Hence, d gives rise to a cohomology in Ω λ (J ∞ ) ⊗ g. Meanwhile, The vector fields D α are tangent to E ∞ , which allows us to define the action of D α on C ∞ λ (E ∞ ) and to restrict d and d 2ω to spectral differential forms on-shell. Moreover, since [D α , D β ] = 0 on E ∞ , it follows that d 2 = 0, which gives rise to a differential and a cochain simplex on Ω λ (E ∞ ) ⊗ g. This along with the definition (4.2) allows us to apply the ideas of the proof in Theorem 3.1 so as to prove that d 2ω gives rise to a cohomology in Ω λ (E ∞ ) ⊗ g. [21] for details).
It is worth commenting on the space on which the spectral differential form ω vanishes. The LSP for a system of PDEs gives rise to a parametrized g-valued differential form ω such that d ω ω = 0 for every λ ∈ Λ only on particular solutions of our system of PDEs. In practical applications ω is then described by a spectral differential one-form, ω λ , vanishing for every λ ∈ Λ on those points of J ∞ projecting onto E. Then, the submanifold E ∞ is contained in this space and d ω λ ω λ = 0 on E ∞ .
Let us prove the following Poincaré Lemma-type result, which is an extension of the Poincaré Lemma for the cohomology of horizontal forms on diffieties and infinite-dimensional jet bundles J ∞ [66]. Theorem 4.3. Let J 0 and E ∞ be homotopic to R m . Then every d-closed spectral differential form or spectral differential form on-shell is exact. Every d 2ω -closed spectral differential one-form on-shell is locally exact.
Proof. The differential operators d and d 2ω induce cochain complexes with respect to the gradings we have the cochain complexes For every fixed λ, the space Ω k λ (J ∞ ) ⊗ g is isomorphic to dim g-copies of the so-called horizontal spaceΛ of horizontal differential forms on J ∞ . Moreover, these spaces give rise to a cochain complex for the so-called horizontal differential d h of the variational complex (see [53,Section 5.4]), which coincides with d when dim g = 1.
If a horizontal form is closed relative to d h , then it is locally exact (cf. [53,Theorem 5.82]). Hence, a d-closed spectral differential form will be always locally d-exact.
The previous result can be extended to spectral differential forms on-shell by using the diffiety (E ∞ , C| E ∞ ). In fact, it is useful to note that, for a trivial differential equation E = J p , one has that (E ∞ , C| E ∞ ) = (J ∞ , C) and the following commentaries retrieve the results of the previous paragraph. The space of spectral differential forms on-shell for a particular λ is isomorphic to dim g-copies of the space of horizontal forms of the diffiety, Λ h (E ∞ ), and the restriction of the differential d to Ω λ (E ∞ ) ⊗ g is indeed the horizontal differential on the diffiety E ∞ (see [65] for details on this and following commentaries). As every closed horizontal differential form is locally closed relative to the differential of the diffiety, it follows that every d-closed form on E ∞ is locally exact. Since E ∞ is homotopic to R m , every d-closed spectral differential form on-shell is exact.
The proof for the local exactness of d 2ω -closed spectral differential one-forms follows from writing the partial differential equations Υ = d 2ω F , for Υ ∈ Ω 1 λ (E ∞ ) ⊗ g in terms of d. The obtained differential equation in F is integrable if the system of PDEs on E ∞ is analytic and formally integrable, which amounts to the fact that d 2ω Υ = 0 (cf. [40]). Hence, a local solution F can be obtained and Υ becomes locally exact.
Previous theorems denote the differentials for the parametrized spectral differential forms in the same way as those for standard parametrized and standard g-valued differential forms. This simplifies the notation and it does not lead to a mistake: the meaning of d and d 2ω is clear taking into account which kind of g-valued differential form the operator is acting on. Moreover, both operators are essentially the same thing, but they operate in different spaces.

LSPs and g-valued differential forms
To analyse immersion formulas for soliton surfaces in Lie algebras and to generalize them to immersed soliton submanifolds in Lie algebras, we will describe integrability conditions in terms of cohomological techniques.
Let us consider an integrable (in the sense of having a LSP) system of PDEs on J p of the form for a function ∆ : J p → R s . As in Section 2, the above system is assumed to have maximal rank and to be locally solvable. Recall that the maximal rank assumption implies that the zeros of ∆ give rise to a submanifold E ⊂ J p . Let G be a matrix Lie group and let g be its matrix Lie algebra. Consider that the LSP problem related to (5.1) is given by a (λ, u(x))-parametrized family of systems of linear PDEs 2) whose integrability condition, the Zero-Curvature Condition (ZCC), i.e.
is satisfied if and only if u(x) is a solution to (5.1). Let us describe this result cohomologically. In this respect, we define a parametrized g-valued differential one-form ω(u(x), λ) := U α (u(x), λ)dx α . Then, the Zero-Curvature Condition (ZCC) for (5.1) amounts to looking for the curves u(x) : M → N ensuring that This equivalence appears as a direct consequence of writing (5.4) in coordinates, namely for all α, β ∈ 1, m and λ ∈ Λ. Hence, the above expression vanishes for a certain u(x) if and only if (5.3) holds.
Example 5.1. The complex projective space CP 1 can be understood via the Hopf fibration h : S 3 → CP 1 ≃ S 3 /S 1 ≃ S 2 as the space of orbits in the three-dimensional unit sphere S 3 ⊂ C 2 (relative to the canonical Hermitian product in C 2 ) with respect to the natural action of S 1 , understood as the Lie group of complex numbers with module one, on S 3 (see [35,64] for details). Equivalently, CP 1 amounts to the space of rank-one Hermitian projectors P onto C 2 , namely P † = P , P 2 = P and trP = 1 [18,23]. The solutions for the CP 1 sigma model are given by mappings P : S 2 → CP 1 where S 2 is understood as the compactification of C by gluing the infinity point topologically. The differential equations for the CP 1 sigma model are given by where [·, ·] stands for the commutator of operators. Hence, the CP 1 sigma model can be understood as a system of PDEs on the jet bundle J 2 relative to the projection onto the second factor π : CP 1 × S 2 → S 2 . This system admits a LSP problem with a purely imaginary spectral parameter λ ∈ C. To clearly show the geometric properties of this LSP, it is convenient to introduce the new variable θ := i(P − Id 2 /2) ∈ su (2). Then, the LSP reads This gives rise to a real parametrized su(2)-valued differential one-form on C of the form Observe that θ(x, y), ∂ x θ(x, y), ∂ y θ(x, y) are understood as particular curves in su (2).
The LSP of the CP 1 model, its parametrized su(2)-valued differential one-form, and the corresponding system of PDEs can be understood as structures on a jet bundle. This motivates the following geometric approach to the LSP equations and related structures based upon jets and spectral differential forms taking values in su(2).
The system of partial differential equations (5.1) can be naturally extended to a manifold E ∞ ⊂ J ∞ by assuming [u] ∈ J ∞ and adding the conditions D J [∇ 2 P, P ] = 0 for an arbitrary multi-index J. This allows us to define a new submanifold E ∞ ⊂ J ∞ where all previous functions on J ∞ vanishes. Then, the total derivatives D x , D y become tangent to E ∞ .
Let us now provide a new formalism of the LSP problem in the language of jets. Essentially, this is a reinterpretation of the algebraic approach given in [20] in terms of functions on rings and an improvement of the method on spaces of jets detailed in [25].
Consider that the hereafter called jet LSP for the differential equation (5.1) is a system of PDEs on J ∞ of the form The system of PDEs (5.6) must be involutive to be integrable, i.e. D α D β Φ([u], λ) = D β D α Φ([u], λ). Nevertheless, this system of partial differential equations is such that the unknown is defined on an infinite-dimensional jet space J ∞ . Hence, it is not obvious that this condition is sufficient to ensure integrability. Nevertheless, it can be proved under quite general conditions, e.g. the system must be analytic [40], that the involutivity of the system ensures its integrability.
The following theorem enables us to extend the notion of integrability on finite-dimensional manifolds to J ∞ and the diffiety E ∞ . Proof. Let us prove the direct part. If (5.6) has a solution on E ∞ , then on E ∞ , whereupon the direct part follows. The converse is immediate by taking into account that involutivity and analiticity of the system (5.6) on E ∞ gives rise to a solution (cf. [40]).
It is worth noting that the pull-back by an arbitrary section s(x) := (x, u(x)) of J 0 of the structures related to spectral differential forms permit us to recover the formalism for parametrized g-valued differential forms. In particular, j ∞ s * d ω ω = 0 retrieves (5.3). Moreover, j ∞ s * Φ([u], λ) provides a solution for the pull-back of the system (5.2), namely In other words, the infinite-dimensional jet approach and spectral differential forms represent a way of representing the standard formalism for LSP when the structures appearing can be defined on infinite-dimensional jet manifolds. This gives as a submanifold E sm ⊂ J 2 . If the additional conditions are assumed then a submanifold E ∞ sm ⊂ J ∞ is obtained and the derivatives D x , D y are tangent to it. To describe (5.8), a LSP on J ∞ is introduced: (2). The corresponding spectral differential one-form on J ∞ is given by Hence Composing this with the infinite-prolongation j ∞ s of a section s(x, y) := (x, y, θ(x, y)) related to an arbitrary θ(x, y), we obtain that j ∞ s * d ω ω = 0 amounts to Finally, let us consider an assumption that will be useful in the following sections and that appears in different practical applications of immersion formulas in the literature.
Subsequently, we will assume that there exists a particular on-shell solution Φ([u], λ) to (5.2) satisfying the so-called on-shell condition where Id N is an N × N identity matrix.
Example 5.4. A solution to the jet LSP for the CP 1 sigma model can be written (cf. [26]) as for [u] ∈ E ∞ , λ := it, and t ∈ R.

Immersion formulas
Now we are interested in looking for a simultaneous infinitesimal deformation of the LSP and the zero curvature condition. This will be done in a geometric manner, which will recover the case of immersed soliton surfaces as a particular instance, and it will enable us to generalize the procedure to any immersed submanifold in Lie algebras. Our formalism will use spectral differential forms, which is very practical in applications. The formalism for parametrized g-valued differential forms can be easily obtained by the hint given by considering the pull-back of given expressions for particular sections u ∈ Γ(π). Subsequently, the dependence of all structures on the corresponding variables will be omitted to simplify the notation. We recall that G and g are assumed to be finite-dimensional matrix Lie groups and Lie algebras respectively.
Consider a deformation Although the above spectral differential forms may admit extensions to Ω λ (J ∞ ) ⊗ g, e.g. the above ω is given, in applications, by the restriction to E ∞ of the spectral differential form related to a LSP, we will be mainly concerned with their values on-shell. The first condition in (6.1) implies that The latter is satisfied if and only if d ω ω = 0 on-shell and one has that where Υ := A α dx α in coordinates. Under previous conditions, it can be proved that dF = Ad Φ −1 Υ for a certain F ∈ C ∞ λ (E ∞ ) ⊗ g (cf. [25,26]). The main task now is to provide methods to obtain F from a certain d 2ω -closed Υ.
Since our deformation does not transform λ, the singularity structure of the ω ǫ in the spectral parameter λ remains untouched. Each ω and Υ satisfying the above equations generate a submanifold immersed in the Lie algebra g as explained next in this section. It is worth noting that our geometric approach allows us to simplify previous proofs in the literature (see for instance [19,25,26]). Moreover, our proof is general for any arbitrary M and deals not only with M = R 2 as is standard in the literature [10,11,17,19,25,26].
We then say that Φ is an on-shell spectral solution of the LSP problem when Φ([u], λ) is a particular solution of (5.6) for every λ and every [u] ∈ E ∞ . We may assume that Φ is extended to other [u] out of E ∞ but in that case we cannot ensure that Φ([u], λ) will be a solution of (5.6) for all values of λ ∈ Λ. In such a case, we say that Φ([u], λ) is an off-shell spectral solution of (2.4).
Lemma 6.1. Let Φ be an on-shell solution of the jet LSP (5.6) and let ω be the associated element in Ω 1 λ (J ∞ ) ⊗ g satisfying d ω ω = 0 for every λ ∈ Λ on-shell. Then, Proof. Let us prove (6.2) for a spectral differential k-form, say ϑ : where v 1 , . . . , v r stands for a basis of g. The validity of equality (6.2) on the whole Ω λ (E ∞ ) ⊗ g follows from this by linearity. Now, Lemma 6.2. Let Φ be an on-shell spectral solution of the jet LSP (5.6) and let ω be its associated spectral differential one-form in Ω 1 λ (J ∞ ) ⊗ g. If Υ is a spectral differential one-form on-shell, then there exists a locally defined if and only if d 2ω Υ = 0 on-shell.
Proof. From Lemma 6.1 and the present assumptions, it follows that on-shell. Hence, Ad Φ −1 Υ is d-closed (on-shell) if and only if d 2ω Υ = 0 on-shell. From the properties of the cohomology d on Ω λ (E ∞ ) ⊗ g, the spectral differential 1-form on-shell Ad Φ −1 Υ is closed if and only if there exists locally an F ∈ C ∞ λ (E ∞ ) ⊗ g such that (6.3) is satisfied. This finishes the proof.
In order to fix accurately the notation of our paper, let us introduce the following notion. Definition 6.3. We call F in (6.3) an immersion formula for the jet LSP (5.6). Lemma 6.2 shows the interest of finding the on-shell spectral differential one-forms Υ satisfying the equation d 2ω Υ = 0 on-shell for a certain ω ∈ Ω 1 λ (J ∞ ) ⊗ g satisfying the condition d ω ω = 0 on-shell. There are certain known methods to obtain Υ. The work [37] employs an exact differential one-form, the generalized Weierstrass embedding formula, to obtain an immersion formula. The articles [10,61] show that the admissible symmetries of the ZCC (5.3) when m = 2 include a conformal transformation of the spectral parameter λ, a gauge transformation of the wavefunction Φ in the LSP (5.2) and generalized symmetries of the integrable system (5.1). All these symmetries can be used to determine explicitly an immersion formula F . Most works deal with applications where the image of F is a two-dimensional surface. Less commonly applications deal with an F whose image is an immersed submanifold in a Lie algebra [6]. All such results can be found in a simpler and more general way, e.g. allowing the image of F to be a general immersed submanifold in g, by using the following theorem.
Theorem 6.4. Let Φ be an on-shell spectral solution of the associated LSP (5.6) and let ω ∈ Ω 1 λ (J ∞ ) ⊗ g be the associated spectral differential form. Assume that Υ ∈ Ω 1 λ (E ∞ ) ⊗ g takes the form on-shell Here, β(λ) is an arbitrary scalar function depending only on λ, the function S is an arbitrary element of C ∞ λ (E ∞ ) ⊗ g, and the vector field X R := R i ∂ u i is a generalized symmetry of ∆[j p x u] = 0. Then, there exists an immersion formula F ∈ C ∞ λ (E ∞ ) ⊗ g given by the formula Proof. Due to historical reasons, let us prove that the three terms of the spectral g-valued differential one-form on-shell (6.4) are such that the action of Ad Φ −1 on each one is the differential (relative to d) of the g-valued spectral functions on-shell respectively. This proves that the latter g-valued spectral functions are immersion formulas for (6.4). Let us start by studying the Sym-Tafel spectral differential 1-form, namely Υ ST . Using the fact that Φ is an on-shell spectral solution of (5.2), we obtain on-shell Let us now study the Cieśliński-Doliwa 1-form, i.e. Υ CD = d 2ω S. From Lemma 6.1, it follows that The Fokas-Gel'fand 1-form, i.e. Υ F G = L prX R ω, induces a function F F G ∈ C ∞ λ (E ∞ ) ⊗ g. To show this it is enough to observe that since [D α , prX R ] = 0 one has on-shell that 7 Relations between immersion formulas.
The work [26] showed that there exists a gauge transformation mapping the Sym-Taffel and Cieśliński-Doliwa immersion formulas to each other when the potential, S, of one of them is invertible. This result is an immediate consequence of our cohomology approach, as shown next.
It is worth noting that although d ω ω = 0 on-shell, D λ d ω ω may not vanish, as these terms give the infinitesimal variation of d ω ω along a one-parametric family of curves (u ǫ (t), λ ǫ ) generated by the flow of D λ . Only u 0 (t) must be a solution to ∆(j p x u) = 0, while d ω ω may not vanish on (u ǫ (x), λ ǫ ). If β(λ)L D λ ω = dS ST on-shell, then (7.2) vanishes and L D λ d ω ω = 0 on-shell, i.e. D λ is symmetry of the ZCC condition. Conversely, if D λ is a symmetry of ZCC condition for the ∆(j p x u), then the expression (7.2) vanishes on-shell and, due to the properties of the d 2ω cohomology, there exists a It is worth noting that the cohomological proof of this theorem is almost trivial. Meanwhile, its counterpart in coordinates in [26] is much longer and obscure.
Analogously, it is possible to see that the remaining terms within the g-valued spectral differential one-form Υ in Theorem 6.4 are induced from other types of gauges. Then, these gauges allow us to map one term into another. Proposition 7.2. A vector field X R is a generalized symmetry of the ZCC d ω ω of the jet LSP associated with an integrable system ∆[j p x u] = 0 if and only if there exists a spectral function Proof. Using the Lemma 6.1, we obtain that d 2ω If Υ F G = d 2ω S F G on-shell, then L pr X R d ω ω = 0 on-shell and the generalized symmetry X R is a symmetry of ∆(j p x u). Conversely, if X R is a generalized symmetry of ∆(j p x u), then d 2ω Υ F G = 0 Let us prove a last result allowing us to map different immersion maps among themselves.
, a so-called gauge transformation, mapping the immersion formula S 1 of Υ 1 onto the immersion formula S 2 of Υ 2 or vice versa, whenever S 1 or S 2 are invertible matrices of g, respectively.
Proof. It is immediate that under the given assumptions, is the required element for invertible S 1 or S 2 , respectively.
In particular, the application of the above corollary for the Sym-Tafel and Fokas-Gelf'and terms is summarised in the following diagram. Φ ∈ G P P P P P P P P P P P q Figure 1: Representation of the relations between the wavefunction Φ ∈ G and the g-valued ST and FG immersion formulas for soliton submanifolds.

Immersion formulas for PDEs with initial conditions
It frequently happens in applications that an LSP only describes a subspace of particular solutions of a system of PDEs. For instance, it may happen that the the ZCC of the LSP just recovers the particular solutions of the system of PDEs satisfying a particular initial condition. The aim of this section is to recover immersion formulas and other results in this case. To illustrate the problem under consideration, let us analyse the partial differential equation in J 2 for π : (u; x, y) ∈ R × R 2 → (x, y) ∈ R 2 given by for some function f : u ∈ R → f (u) ∈ R. It is straightforward to prove that the function u 2 x − f (u) is a constant of motion of (8.1). Nevertheless, not every solution of our PDE satisfies u 2 x = f (u), which can be understood as a boundary condition. For instance, if f (u) = 0 the function u(x, y) = x satisfies (8.1) without obeying the given initial condition. The PDE (8.1), along with the previous initial condition, admits a Lax pair The system (8.1) admits the generalized Lie symmetries Since pr Q 1 − D x = ∂ x , the vector field Q 1 induces a translation in the x variable, which is a symmetry of the initial condition u 2 x − f (u) and (8.1). Therefore, we can redefine our PDE by adding the initial condition. This enables us to apply all our previous results to this new problem.
On the other hand, a straightforward calculation shows that Q 2 is an infinitesimal symmetry only of (8.1), i.e. it does not leave invariant the initial condition. As a consequence, the formalism given in the previous section does not apply to this new case. Despite this, it is possible to rewrite all previous results to deal with it.
Let us consider a system of PDEs given by (2.4) and a LSP which only applies to solutions of (2.4) with some additional particular conditions. Let the new system by described by S ′ ⊂ J p . The parametrized g-valued differential forms of Ω S (M ) ⊗ g can be reduced to the set S ′ . The restrictions of Theorem 3.4 and Corollaries 3.5 and 3.6 to S ′ remain true. This allows us to restrict our cohomologies, d and d 2ω to Ω S ′ (M ) ⊗ g. Moreover, Lemmas 6.1 and 6.2 remain true when Φ is a solution of the LSP for curves u(x) in S ′ .
Let us now comment on spectral differential forms. If we consider the initial condition as a part of the initial system of PDEs, the space of solutions of the new system of PDEs with the initial conditions induces a submanifold E 2 ⊂ E ⊂ J p and a new prolongation E ∞ 2 . Hence, all previous results for E now apply to E 2 . Since they can be trivially restricted to the systems of PDEs on E 2 , the restriction of the Cieliński-Doliwa or the Sym-Tafel immersion formulas remain valid for the restriction of the initial system.
In spite of previous arguments, there exists one important difference relative to previous immersion formulas in the case of the restriction. The symmetries of E do not necessarily give rise to symmetries of the associated LSP. Consequently, the FG immersion formula must be modified accordingly. This will be solved by adding a new term in the Fokas-Gelf'and one-form. This is explained in the next theorem.
Theorem 8.1. Let Φ be an on-shell spectral solution of the associated LSP (5.2) defined over a submanifold of solutions of the initial systems of PDEs and let ω ∈ Ω 1 λ (J ∞ ) ⊗ g be the associated g-valued spectral differential form. Assume that Υ F G ′ ∈ Ω 1 λ (J ∞ ) ⊗ g takes the form on-shell Here, the vector field X R is a generalized symmetry of ∆(j p x u) = 0. Then, there exists an immersion formula for Υ F G ′ given by Proof. As in Theorem 6.4, let us prove that is such that the action of Ad Φ −1 on it gives the differential (relative to d) of the g-valued spectral function (8.3). To show this it is enough to recall that since [D α , prX R ] = 0 and then 9 Soliton surfaces for CP N −1 sigma models Let us now illustrate our techniques by studying immersion formulas for the general CP N −1 sigma model. Based on the Gram-Schmidt orthogonalization procedure, a method for constructing an entire class of solutions on S 2 admitting the finite action of the CP N −1 sigma model was proposed by A. Dim, Z Horvarth and D. Zakrzewski [15,16] and latter studied by R. Sasaki [59]. In particular, we focus on the case N = 3, whose immersion formulas will allow us to study the so-called Veronese surfaces and the mixed type soliton surfaces [2,5,70]. A final type of surface induced by the aforesaid will be also briefly commented on (for details on these models see [2,34,45,69,70]). The most fruitful technique for the study of integrable surfaces for CP N −1 models has been formulated through descriptions of the model via rank-one Hermitian projectors. A matrix P (z,z), with z ∈ R := C ∪ {∞}, is said to be a rank-one Hermitian projector if P 2 = P, P = P † , tr P = 1.
The image of the projector P is determined by a complex one-dimensional space within C N . Therefore, there exists a C N -valued function The equations of motion for the CP N −1 model read For our purposes of investigating infinitesimal symmetries and immersion formulas related to this model, a more appropriate form of the above expression is given by the differential equation on the second-order jet J 2 of the bundle π : (θ, ξ) ∈ su(N ) × R → ξ ∈ R given by where θ := i(P − Id N /N ) ∈ su(N ). This differential equation induces in the standard way a submanifold of E ⊂ J 2 [53]. To apply our approach, it is necessary to extend this differential equation to J ∞ by adding, as standard, the conditions D J (∂ 2 1 + ∂ 2 2 )θ, θ = 0, for an arbitrary multi-index J [65]. This gives rise to a differential equation on J ∞ given by which in turn gives rise to a submanifold E ∞ of J ∞ [65]. The LSP associated with (9.4) is given by [50,69] [69,70]. This system is related to the su(N )-valued spectral differential one-form In fact, the integrability condition for (9.5) reads as One sees that the above vanishes exactly on the space of solutions θ(ξ) corresponding to (9.4). Additionally, the above spectral su(N )-differential form vanishes on-shell, i.e. on the submanifold To study the immersion submanifolds for (9.3), it is interesting to obtain a spectral solution of (9.5). This can be done by adapting to J ∞ the rising and lowering projectors for CP N −1 models , where the traces in the denominators are different from zero for θ = 0 unless the whole matrix is zero and D := (D x + iD y )/2 andD := (D x − iD y )/2. Then, a family of spectral solutions Φ k ([θ], λ) to the LSP (9.5) read as (cf. [26]) where , λ) is anti-holomorphic and the remaining ones k = 1, N − 2 are mixed.
For any real functions f and g of one variable, the equations of motion (9.3) admit the Lie symmetries given by the prolongations to J 2 of the vector fields which are called conformal as they vanish on functions depending only on the independent variables x, y. The integrated form of the immersed surface in su(N ) induced by the above symmetry is given by the FG formula [30, p. 15]. Using (9.7) for k = 0, we obtain There is another possible approach to take into account. It turns out that the spectral su(N )-valued differential form satisfies d 2ω Υ = 0 on-shell. As a consequence, it provides a deformation giving rise to an immersion surface of the form dF = −[∂ x θ, θ]dx + [∂ y θ, θ]dy. Hence, F is given by integrating this form This is indeed the expression in the set of coordinates {x, y} on R of the Weirstrass immersion formula for CP N −1 models [37].

Soliton surfaces associated with the CP 2 sigma model
Let us analyse the geometric properties of immersion surfaces for the particular CP N −1 sigma model with N = 3 by using our techniques. The obtained surface is one of the elements of the so-called Veronese sequence [5,14]. In this case, the projector P given by (9.1) for k = 0 can be written in terms of the holomorphic function f 0 := (1, √ 2ξ, ξ 2 ). One obtains that Hence, X 0 = i (Id 3 /3 − P 0 ) and therefore This is an immersion surface which is an anti-Hermitian solution of the equation: Id 3 = 0. (9.10) By considering the basis S i for su(3) given by the orthonormalization of the basis in ([31, p. 11]) relative to the Killing metric on su(3) of the form A, B := −Tr(AB)/2, we obtain that every X 0 = 8 α=1 x α S α satisfying the matrix equation (9.10) is such that its coordinates obey the following independent equations: 14) The first one is obtained by considering the trace of (9.10). The second and third ones come from analyzing the diagonal elements of (9.10). Other conditions on the coordinates of X 0 , e.g.
arise from analyzing all entries of the equation (9.10). Previous conditions allow us to determine the shape of the immersed surface. The equation (9.11) represents a sphere in R 8 . To obtain the Veronese surface one has to intersect the sphere (9.11) by an ellipsoidal cylinder described by (9.12) (the generatrix lines parallel to the (x 1 , x 2 , x 3 ) axes) and a degenerate hyperbola given by (9.13). The equation (9.12) shows that −1/(2 √ 3) ≤ x 4 ≤ 1/ √ 3 while (9.14) allows us to determine a discrete set of values of x 3 for those possible values of x 4 . Once particular values of x 3 and x 4 are determined, equations (9.11)-(9.14) enable us to ensure that the coordinates x 1 , x 2 , x 6 , x 8 , x 5 , x 7 of our surface are contained in the two-dimensional spheres of the form , for certain functions F 1 , F 2 , F 3 . The final expression for the immersed surfaces can be obtained by using the remaining conditions, e.g. (9.15), (9.16), resulting from (9.10).
Instead of following the above procedure, let us now analyse the immersed surface by using global and local invariants for surfaces. The parametrization of the surface in terms of the complex variable ξ = x + iy is conformal, namely the Lie derivative in terms of ∂ξ of the fundamental form I = 4(1 + x 2 + y 2 ) −2 (dx ⊗ dx + dy ⊗ dy) = 2(1 + |ξ| 2 ) −2 (dξ ⊗ dξ + dξ ⊗ dξ), which is the metric of the sphere, vanishes.
Some simple calculations permit us to obtain that the Gaussian curvature of our immersed surface is κ = 2 while the Euler-Poincaré character is χ = 2. The mean curvature tensor of the surface, H, is anti-Hermitian and orthogonal, relative to the metric on su(3), to ∂X 0 ,∂X 0 and X 0 . The norm of H is 16. As a consequence, the Veronese surface has positive constant mean curvature. Meanwhile, the Willmore functional Q = 2 indicates that the winding number of a submanifold of the surface is at least equal to 2. This happens for the projection by π : (x 1 , . . . , x 8 ) ∈ R 8 ≃ su(3) → (x 4 , x 6 , x 8 ) ∈ R 3 of the Veronese surface. Let us analyse a second possible immersed surface induced by a different spectral solution. It is well known that f 1 := (∂P 0 )f 0 gives rise to a new solution P 1 of the CP 2 model. In other words, the function f 1 = 1 1 + |ξ| 2 (−2ξ, √ 2(1 − |ξ| 2 ), 2ξ).
gives rise to the particular solution This leads to the immersion formula which is an anti-Hermitian solution to X 3 1 + X 1 = 0.
In the already used Gell-Mann orthonormal basis {S α } α∈1,8 of su (3), the coordinates of the above X 1 satisfy x 3 = x 4 / √ 3, x 6 = 0, x 8 = 0, x 3 1 + 1 = 0. (9.17) Since X 1 = 0, the set of equations describing a non-singular manifold reduce to In this case, one obtains that the six previous independent conditions (9.17)-(9.20) on su (3), which is eight-dimensional, give rise to a surface. The condition (9.20) shows that the immersed surface is a closed fourth degree curve in the coordinates x 1 , x 2 parametrized by where the above bounds for x 4 are a consequence of (9.19). The metric of the surface is that one of the sphere, as in the previous example, with Gaussian curvature κ = 2. However, the mean curvature is H, H = 4, where H is the mean curvature tensor of the surface. This implies a smaller value of the Willmore functional W = 2π relative to the Veronese surface, but a topological charge Q = 0.
A last anti-holomorphic solution to the CP 2 model can be found which leads to an immersed surface X 2 : = −i(P 2 + 2(P 0 + P 1 )) + i 5 3 Since this surface, which appears as a consequence of the application of the standard techniques for CP N −1 sigma models, is a linear combination of previous ones, its analysis will be not accomplished.

Concluding remarks and outlook
This paper has presented a cohomological approach to the study of immersed soliton surfaces for integral systems of PDEs through several types of spectral g-valued differential forms. This has revealed several properties of such surfaces while providing an extension of known methods to general soliton submanifolds for more general systems of integrable PDEs. Our work has also shed some light on the jet approach to the study of immersed soliton surfaces. The CP N −1 sigma models have been studied through our techniques.
In the future, we aim to expand our analysis in several directions. First, it is interesting to investigate whether soliton surfaces are stable and they can therefore be observable in nature. This future study would attempt to devise an appropriate perturbation theory and suggest the development of good approximate solutions. Second, it is worth noting that analytic expressions for surfaces may reveal quality features that otherwise might be difficult to detect numerically. In this respect, we aim to develop computer techniques for the visualization of soliton surfaces. It is also natural to ask how the integrable characteristics -Hamiltonian structures, conserved quantities, singularities structures and so on -are manifest on surfaces. Third, we are interested in problems whose soliton surfaces are well-known experimentally but the associated physical system is not fully developed. Hence, we propose to use the variational problem of the geometric functional, i.e. the Willmore functional interpreted as an action functional, to compute the Euler-Lagrange equations determining the surface. This approach has been accomplished for biological membrane models obtained via the generalized Weierstrass representation [37], but not, to our knowledge, for immersion formulas for soliton surfaces using our approach. Finally, the cohomological approach may be further developed and the properties of the used cohomologies must be studied in detail, e.g. the description of more general Poincaré Lemmas. This will be the topic of a future work.