Higher dimensional holonomy map for ruled submanifolds in graded manifolds

The deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [J. Differential Geom., 36(3):551-589, 1992], and we provide a characterization for singularities as well as a deformability criterion.


Introduction
The goal of this work is to study the deformability of a some particular kind of submanifolds immersed in an equiregular graded manifold (N, H 1 , . . . , H s ), that is a smooth manifold endowed with a filtration of sub-bundles of the tangent bundle Given p ∈ N , a vector v ∈ T p N has degree i if v ∈ H i p but v / ∈ H i−1 p . When we consider an immersed submanifold Φ :M → N and we set M = Φ(M ), the interaction between the tangent space TpM = (dΦ)p(TpM ), where (dΦ)p denotes the differential of Φ atp, and the filtration H 1 p ⊂ H 2 p ⊂ · · · ⊂ H s p is embodied by the induced tangent flag (1.1) TpM ∩ H 1 p ⊂ · · · ⊂ TpM ∩ H s p , where p = Φ(p),p ∈M . The smooth submanifold M equipped with the induced filtration pointwise described by (1.1) inherits a graded structure, that is no more equiregular. M. Gromov in [25] consider the homogeneous dimension of the tangent flag (1.1) to define the pointwise degree by deg M (p) = s j=1 j(m j −m j−1 ), wherem 0 = 0 andm j = dim(TpM ∩ H j p ). In an alternative definition provided in [35], the authors write the m-tangent vector to M = Φ(M ) as linear combination of simple m-vectors X j1 ∧· · · ∧X jm where (X 1 , . . . , X n ) is an adapted basis of T N , see [5] or (2.3). Then the pointwise degree is the maximum of the degree of the simple m-vectors whose degree is in turn given by the sum of the degrees of the single vectors appearing in the wedge product. The degree deg(M ) of a submanifold M is the maximum of the pointwise degree among all points inM .
In [35] V. Magnani and D. Vittone introduced a notion of area for submanifolds immersed in Carnot groups that later was generalized by [14] for immersed submanifolds in graded structures. Given a Riemannian metric g in the ambient space N , the area functional A d (M ) in [14] is obtained by a limit process involving the Riemannian areas of M associated to a sequence of dilated metrics g r of the original one g. The density of this area is given by the projection of the mvector e 1 ∧ . . . ∧ e m tangent to M onto the space of m-vectors of degree equal to d = deg(M ), see equation (2.8). The central issue is that the area functional depends on the degree deg(M ) of the immersed submanifold M . Thus, if we wish to compute the first variation formula for this area functional we need to deform the original submanifold by variations Γ(p, τ ) that preserve the original degree deg(M ). This constraint on the degree gives rise to a first order system of PDEs that defines the admissibility for vector fields on M .
The simplest example of immersion is given by a curve γ : I ⊂ R → N , with γ ′ (t) = 0 at every t ∈ I. The pointwise degree of γ(I) at γ(t) is the degree of its tangent vector γ ′ (t) at every t ∈ I. In this particular case the admissibility system is a system of ODEs along the curve γ. This restriction on vector fields produces the phenomenon of singular curves, that do not admit enough compactly supported variations in the sub-bundles determined by the original degree of γ. This issue has been addressed by L. Hsu in [29] and R. Bryant and L. Hsu in [10]. These two works are based on the Griffiths formalism [23] that studies variational problems using the geometric theory of exterior differential system [8,9] and the method of moving frames developed by E. Cartan [11]. In Carnot manifolds (N, H), that are a particular case of graded manifolds where the flag of sub-bundles is produced by a bracket generating distribution H, the usual approach to face this problem is by means of the critical points of the endpoint map [38]. The presence of singular curves is strongly connected with the existence of abnormal geodesics, firstly established by R. Montgomery in [36,37]. In the literature many papers concerning this topic have been published, just to name a few we cite [2,1,33,31,39,3,44]. The paper [33] by E. Le Donne, G.P. Leonardi, R. Monti and D. Vittone is specially remarkable because of the new algebraic characterization of abnormal sub-Riemannian extremals in stratified nilpotent Lie groups.
More precisely, L. Hsu [29] defines the singular curves as the ones along which the holonomy map fails to be surjective. This holonomy map studies the controllability along the curve restricted to [a, b] ⊂ I of a system of ODEs embodying the constraint on sub-bundles determined by the degree. In [13,Section 5] the authors revisited this construction and defined an admissible vector field as a solution of this system. A powerful characterization of singular curves in terms of solutions of ODEs is given by [29,Theorem 6]. On the other hand, when a curve γ is regular restricted to [a, b], [29,Theorem 3] ensures that for any compactly supported admissible vector field V on [a, b] there exists a variation, preserving the original degree of γ, whose variational vector field is V . Then, only for regular curves this deformability theorem allows us to compute the first variation formula for the length functional deducing the geodesic equations ([13, Section 7]), whereas for singular curves the situation is more complicated.
The deformability problem of a higher dimensional immersion Φ :M → N has been first studied in [14]. The admissibility system of first order linear PDEs expressing this condition in coordinates is not easy to study. Nonetheless, [14,Proposition 5.5] shows that only the transversal part V ⊥ of the vector field V = V ⊤ + V ⊥ affects the admissibility system. Therefore, in the present work we consider an adapted tangent basis E 1 , . . . , E m for the flag (1.1) and then we add transversal vector fields V m+1 , . . . , V n of increasing degrees so that a sorting of {E 1 , . . . , E m , V m+1 , . . . , V n } is a local adapted basis for N . Then we consider the metric g that makes E 1 , . . . , E m , V m+1 , . . . , V n an orthonormal basis. Hence we obtain that the admissibility system is equivalent to for i = m + k + 1, . . . , n and deg(V i ) > deg(E j ). In equation (1.2) the integer k, defined in (3.2), separates the horizontal control of the systems The presence of isolated submanifolds and a mild deformability theorem under the strong regularity assumption are showed in [14]. However, the definition of singularity for immersed submanifolds, analogous to the one provided by [29] in the case of curves, is missing. Therefore the natural questions that arise are: • is it possible to define a generalization of the holonomy map for submanifolds of dimension grater than one? • Under what condition does the surjection of these holonomy map still imply a deformability theorem in the style of [29,Theorem 3]?
In the present paper we answer the first question in the cases of ruled mdimensional submanifolds whose (m − 1) tangent vector fields E 2 , . . . , E m have degree s and the first vector field E 1 has degree equal to ι 0 , where 1 ι 0 s − 1. The resulting degree is deg(M ) = (m − 1)s + ι 0 . Therefore the ruled submanifold is foliated by curves of degree ι 0 out of the characteristic setM 0 , whose points have degree strictly less than deg(M ). Then, under an exponential change of coordinates x = (x 1 ,x), the admissibility system (1.2) becomes where ∂ x1 is the partial derivative in the direction E 1 , G are the horizontal coordinates V H = m+k h=m+1 g h V h , F are the vertical components given by V V = n r=m+k+1 f r V r and A, B are matrices defined at the end of Section 4. Therefore, this system of ODEs is easy to solve in the direction ∂ x1 perpendicular to the (m−1) foliation generated by E 2 , . . . , E m . We consider a bounded open set Σ 0 ⊂ {x 1 = 0} in the foliation, then we build the ε-cylinder Ω ε = {(x 1 ,x) :x ∈ Σ 0 , 0 < x 1 < ε} over Σ 0 . We consider the horizontal controls G in the space of continuous functions compactly supported in Ω ε . For each fixed G , F is the solution of (1.3) vanishing on Σ 0 . Then we can define a higher dimensional holonomy map H ε M , whose image is the solution F , evaluated on the top of the cylinder Ω ε . We say that a ruled submanifold is regular when by varying the controls G the image of the holonomy map is a dense subspace, that contains a Schaulder basis of the Banach space of continuous vertical functions on Σ ε vanishing at infinity. This Banach space is the closure with respect to the supremum norm of the space of compactly supported vertical functions on Σ ε . Namely an immersion is regular if we are able to generate all possible continuous vertical functions vanishing at infinity on Σ ε ⊂ {x 1 = ε} by letting vary the control G in the space of continuous horizontal functions vanishing at infinity inside the cylinder Ω ε . The main difference with respect to the one dimensional case is that the target space of the holonomy map is now the Banach space of continuous vertical vector vanishing at infinity on the foliation, instead of the finite vertical space of vectors at the final point γ(b) of the curve. In Theorem 5.8 we provide a nice characterization of singular ruled submaifolds in analogy with [29,Theorem 6].
For general submanifolds there are several obstacles to the construction of a satisfactory generalization of the holonomy map. The main difficulty is that we do not know how to verify a priori the compatibility conditions [26,Eq. (1.4), Chapter VI], that are necessary and sufficient conditions for the uniqueness and the existence of a solution of the admissibility system (1.2) (see [26,Theorem 3.2,Chapter VI]). In Example 3.5 we show how we can deal with these compatibility conditions in the particular case of horizontal immersions in the Heisenberg group.
In order to give a positive answer to the second question, we need to consider two additional assumptions on the ruled submanifold: the first one (i) is that the vector fields E 2 , . . . , E m of degree s fill the grading H 1 ⊂ . . . ⊂ H s from the top, namely dim(H s ) − dim(H s−1 ) = m − 1, and the second one (ii) is that the ruled immersion foliated by curves of degree ι 0 verifies the bound s − 3 ι 0 s − 1. Under these hypotheses the space of m-vector fields of degree grater than deg(M ) is reasonably simple, thus in Theorem 6.6 we show that each admissible vector field on a regular immersed ruled submanifold is integrable in the spirit of [29,Theorem 3]. This result is sharper than the one obtained for general submanifolds [14,Theorem 7.3], where the authors only provide variations of the original immersion compactly supported in an open neighborhood of the strongly regular point. Indeed, since we solve a differential linear system of equations along the characteristics curves of degree ι 0 , we obtain a global result. On the other hand in [14,Theorem 7.3] the admissibility system is solved algebraically assuming a pointwise full rank condition of the matrix A(p). To integrate the vector field V (p) on Ω ε we follow the exponential map generating the non-admissible compactly supported variation Γ τ (p) = exp Φ(p) (τ V (p)) of the initial immersion Φ, where supp(V ) ⊂ Ω ε . By the Implicit Function Theorem there exists a vector field Y (p, τ ) on Ω ε vanishing on Σ 0 such that the perturbationsΓ τ (p) = exp Φ(p) (τ V (p) + Y (τ,p)) of Γ are immersions of the same degree of Φ for each τ small enough. In generalΓ does not move points on Σ 0 but changes the values of Φ on Σ ε . Finally, the regularity condition on Φ allows us to produce the admissible variation that fixes the values on Σ ε and integrate V . On the other hand, when the bundle of m-vector fields of degree greater than deg(M ) for a general ruled submanifold is larger than the target space of the higher dimensional holonomy, we lose the surjection in Implicit Function Theorem that allows us to perturb the exponential map to integrate V .
A direct consequence of this result is that the regular ruled immersions of degree d that satisfy the assumption (i) and (ii) are accumulation points for the domain of degree d area functional A d (·). Therefore it makes sense to consider the first variation formula computed in [14,Section 8]. An interesting strand of research is deducing the mean curvature equations for the critical points of the area functional taking into account the restriction embodied by the holonomy map. Contrary to what can be expected, we exhibit in Example 6.7 a plane foliated by abnormal geodesics of degree one that is regular and is a critical point for the area functional (since its mean curvature equation vanishes).
Furthermore these ruled surfaces appear in the study of the geometrical structures of the visual brain, built by the connectivity between neural cells [16]. A geometric characterization of the response of the primary visual cortex in the presence of a visual stimulus from the retina was first described by the D. H. Hubel and T. Wiesel [30], that discovered that the cortical neurons are sensitive to different features such as orientation, curvature, velocity and scale. The so-called simple cells in particular are sensitive to orientation, thus G. Citti and A. Sarti in [15] proposed a model where the original image on the retina is lifted to a 2 dimensional surface of maximum degree into the three-dimensional sub-Riemannian manifold SE(2), adding orientation. In [17] they shows how minimal surfaces play an important role in the completion process of images. Adding curvature to the model, a four dimensional Engel structure arises, see § 1.5.1.4 in [42] and [19]. When in Example 6.8 we lift the previous 2D surfaces in this structure we obtain surfaces of codimension 2, but their degree is not maximum since we need to take into account the constraint that curvature is the derivative of orientation. Nevertheless these surfaces are ruled, regular and verify the assumption (i) and (ii), therefore by Theorem 6.6 they can be deformed. Hence, there exists a notion of mean curvature associated to these ruled surfaces and we might ask if the completion process of images improved for SE(2) based on minimal surfaces can be generalized to this framework. Moreover, if we lift the original retinal image to higher dimensional spaces adding variables that encode new possible features, as suggested in [40] following even a non-differential approach based on metric spaces, we may ask if the lifted surfaces are still ruled and regular.
The paper is organized as follows. In Section 2 we recall the definitions of graded manifolds, degree of a submanifold, admissible variations and admissible vector fields. In Section 3 we deduce the admissibility system (1.2). In Section 4 we provide the definition of ruled submanifolds. Section 5 is completely devoted to the description of the higher-dimensional holonomy map and characterization of regular and singular ruled submanifolds. Finally, in Section 6 we give the proof of Theorem 6.6.
Acknowledgement. I warmly thank my Ph.D. supervisors Giovanna Citti and Manuel Ritoré for their advice and for fruitful discussions that gave rise to the idea of higher dimensional holonomy map. I would also like to thank Noemi Montobbio for an interesting conversation on proper subspaces of Banach spaces and the referee for her/his useful comments.

Preliminaries
Let N be an n-dimensional smooth manifold. Given two smooth vector fields X, Y on N , their commutator or Lie bracket is defined by [X, Y ] := XY − Y X. An increasing filtration (H i ) i∈N of the tangent bundle T N is a flag of sub-bundles Moreover, we say that an increasing filtration is locally finite when (iii) for each p ∈ N there exists an integer s = s(p) satisfying H s p = T p N . The step at p is the least integer s that satisfies the previous property. Then we have the following flag of subspaces ) is a smooth manifold N endowed with a locally finite increasing filtration, namely a flag of sub-bundles (2.1) satisfying (i),(ii) and (iii). For the sake of brevity a locally finite increasing filtration will be simply called a filtration. Setting n i (p) := dim H i p , the integer list (n 1 (p), · · · , n s (p)) is called the growth vector of the filtration (2.1) at p. When the growth vector is constant in a neighborhood of a point p ∈ N we say that p is a regular point for the filtration. We say that a filtration (H i ) on a manifold N is equiregular if the growth vector is constant in N . From now on we suppose that N is an equiregular graded manifold.
Given a vector v in T p N we say that the degree of v is equal to ℓ if v ∈ H ℓ p and v / ∈ H ℓ−1 p . In this case we write deg(v) = ℓ. The degree of a vector field is defined pointwise and can take different values at different points.
Let (N, (H 1 , . . . , H s )) be an equiregular graded manifold. Take p ∈ N and consider an open neighborhood U of p where a local frame {X 1 , · · · , X n1 } generating H 1 is defined. Clearly the degree of X j , for j = 1, . . . , n 1 , is equal to one since the vector fields X 1 , . . . , X n1 belong to H 1 . Moreover the vector fields X 1 , . . . , X n1 also lie in H 2 , we add some vector fields X n1+1 , · · · , X n2 ∈ H 2 \ H 1 so that (X 1 ) p , . . . , (X n2 ) p generate H 2 p . Reducing U if necessary we have that X 1 , . . . , X n2 generate H 2 in U . Iterating this procedure we obtain a basis of T M in a neighborhood of p Given an adapted basis (X i ) 1 i n , the degree of the simple m-vector field X j1 ∧ . . . ∧ X jm is defined by Any m-vector X can be expressed as a sum where J = (j 1 , . . . , j m ), 1 j 1 < · · · < j m n, is an ordered multi-index, and X J := X j1 ∧ . . . ∧ X jm . The degree of X at p with respect to the adapted basis It can be easily checked that the degree of X is independent of the choice of the adapted basis and it is denoted by deg(X).
If X = J λ J X J is an m-vector expressed as a linear combination of simple m-vectors X J , its projection onto the subset of m-vectors of degree d is given by and its projection over the subset of m-vectors of degree larger than d by In an equiregular graded manifold with a local adapted basis (X 1 , . . . , X n ), defined as in (2.3), the maximal degree that can be achieved by an m-vector, m n, is the integer d m max defined by (2.5) d m max := deg(X n−m+1 ) + · · · + deg(X n ).

2.1.
Degree of a submanifold. Let Φ :M → N be a C 1 immersion in an equiregular graded manifold (N, (H 1 , . . . , H s )) such that dim(M ) = m < n = dim(N ). Following [32,35], we define the degree of M = Φ(M ) at a pointp ∈M by where v 1 , . . . , v m is a basis of TpM = (dΦ)p(TpM ) and dΦ. We denote by TpM = (dΦ)p(TpM ) the tangent space at p = Φ(p), where (dΦ)p is the differential of Φ at p ∈M . We use this notation in order to emphasize that we consider the tangent space of the image Φ . Namely, the degree is the homogenous dimension of the flag As we pointed out in [14, Section 3] the area functional associated to an immersed sumbanifold depends on the degree.  Section 5] we recall the notions of admissible variation, its variational vector field, admissible and integrable vector field.
is an immersion of the same degree as Φ(M ) for small enough t, and (iii) Γ t (q) = Φ(q) forq outside of a given compact set ofM .
Definition 2.3. Given an admissible variation Γ, the associated variational vector field is defined by Let X 0 (M , N ) be the space of compactly supported smooth vector fields onM with value in N . Since it turns out that variational vector fields associated to an admissible variations satisfy the system (2.10) (see [14,Section 5]) we are led to the following definition Definition 2.4. Given an immersion Φ :M → N , a vector field V ∈ X 0 (M , N ) is said to be admissible if it satisfies the system of first order PDEs . . , e m is basis of TqM , for each q inM such that q = Φ(q).
Definition 2.5. We say that an admissible vector field V ∈ X 0 (M , N ) is integrable if there exists an admissible variation such that the associated variational vector field is V .
3. Intrinsic coordinates for the admissibility system of PDEs Let Φ :M → N be a smooth immersion in a graded manifold, M = Φ(M ) and d = deg(M ). By [14,Proposition 6.4] we realize that the admissibility of a vector field V is independent of the metric. Therefore we can use any metric in order to study the system. Letp be a point inM such that Reducing O if necessary, following the same argument of Section 2, there exists a local adapted basis (Ẽ 1 , . . . ,Ẽ m ) to the filtratioñ H 1 ⊂ . . . ⊂H s . For each j = 1, . . . , m we set deg(Ẽ j ) = ℓ j , then we can extend each vector fieldẼ j in a neighborhood U of N around p so that the extensions E j lie in H ℓj . Finally we complete this basis of vector fields (E 1 , . . . , E m ) to a basis of the ambient space T U adding the vector fields V m+1 , . . . , V n of increasing degree such that a sorting of {E 1 , . . . , E m , V m+1 , . . . , V n } is an adapted basis of T U . Then we consider the metric g = ·, · that makes E 1 , . . . , E m , V m+1 , . . . , V n an orthonormal basis in a neighborhood U of p. We will denote by (Y 1 , . . . , Y n ) the local adapted basis generated by this sorting of E 1 , . . . , E m , V m+1 , . . . , V n . From now on we will denote also denote (Ẽ 1 , . . . ,Ẽ m ) by (E 1 , . . . , E m ) with a little abuse of notation.
3) is equal to zero thanks to the orthogonal assumption of the basis 3) is equal to zero by orthogonality assumption of the basis E 1 , . . . , E m . Then, denoting by σ j i the permutation caused by the reordering and by sgn(σ j i ) = ±1 its sign, we have and only if i = m + k + 1, . . . , n, where k is defined in (3.2). Therefore we deduce that the only m-vectors Y ℓ1 ∧ · · · ∧ Y ℓm of degree strictly greater than d such that (3.3) is different from zero are f r V r ) the horizontal projection on H (resp. the vertical projection on V). For h = m + 1, . . . , m + k and r = m + k + 1, . . . , n, g h , f r are smooth functions on O and when we evaluate the vector field V ⊥ atq ∈Ō we mean Therefore, locally we can consider the vector field V ⊥ defined on O and extend V ⊥ to the open neighborhoood U ⊂ N . Then, putting V ⊥ in (2.10) we have By Remark 3.2 we have to consider the scalar product only with the m-vector and sgn(σ α i ) = ±1 is the sign of the permutation σ α i caused by the reordering. By substituting the expression (3.4) of V ⊥ in equation (3.5), we obtain that (2.10) is equivalent to for t = m + 1, . . . , n, r = m + k + 1, . . . , n, h = m + 1, . . . , m + k, α = 1, . . . , m, i = m + k + 1, . . . , n and deg(V i ) > deg(E α ). Then we have thatc ijtα is equal to 1 for i = t > m + k, α = j and deg(V i ) > deg(E j ) or equal to zero otherwise. Moreover, we notice by Remark 3.2 thatã ijhα andb ijrα are different from zero only when α = j and in particular we have for h = m + 1, . . . , m + k, i = m + k + 1, . . . , n, deg(V i ) > deg(E j ) and for i, r = m + k + 1, . . . , n and deg  [20]. Therefore it would be interesting to consider C 1,1 immersions and deducing the admissibility system (3.9) in a weak formulation using the tools of first order differential calculus for general metric measure spaces, developed in recent years by [12,27,22,4]. 2. Even in this smooth setting we realize that in the admissibility system (3.9) we can consider the functions f m+k+1 , . . . , f n to be continuously differentiable on O and g m+1 , . . . , g m+k in the class of continuous functions on O.
Example 3.5 (Horizontal submanifolds). Given n > 1 we consider the Heisenberg group H n , defined as R 2n+1 endowed with the distribution H generated by The Reeb vector fields is provided by T = ∂ t = [X i , Y i ] for i = 1, . . . , n and has degree equal to 2. Let g = ·, · be the Riemannian metric that makes (X 1 , . . . , X n , Y 1 , . . . , Y n , T ) an orthonormal basis. Let Ω be an open set of R m , with m n. Here we consider a smooth immersion Φ : Ω → H n such M = Φ(Ω) is a horizontal submanifold. Let E 1 , . . . , E m be an orthonormal local frame, then we have ..,n j=1,...,m has full rank equal to m, for eachp ∈ Ω. Since M is horizontal we also have that that is equivalent to Therefore a vector field V = n l=1 g l X i + g l+n Y l + f T is admissible if and only if it satisfies the system (3.9), that in this case is given by for j = 1, . . . , m. A straightforward computation shows that this system is equivalent to (3.12) E j (f ) = n i=1 β ji g i − α ji g i+n for j = 1, . . . , m.
A necessary and sufficient conditions for the uniqueness and the existence of a solution of the admissibility system (3.12) (see [ for each j, ν = 1, . . . , m. These are the so called integrability condition [26, Eq. (1.4), Chapter VI]. A straightforward computation shows that the right hand side of is equal to Moreover, the left hand side is equal to Therefore the compatibility (or integrability) conditions are given by for each ν, j = 1, . . . , m. Moreover, taking into account (3.11), the equation (3.15) is equivalent to Remark 3.6. Notice that if we want to find a solution f of (3.12), the controls g i , . . . , g 2n have to verify the compatibility conditions (3.16). Therefore to obtain a suitable generalization of the holonomy map (defined for curves in [13, Section 5]) we need to consider the subspace of the space of horizontal vector fields on M that verify (3.16). We recognize that studying the holonomy map for these horizontal immersions is engaging problem that have been investigated by [24,41], but in the present work we will consider different kind of immersions that allow us to forget these compatibility conditions in the construction of the high dimensional holonomy map.

Ruled submanifolds in graded manifolds
In this section we consider a particular type of submanifolds for which the admissibility system reduces to a system of ODEs along the characteristic curves, that rule these submanifolds by determining their degree since the other adapted tangent vectors tangent to M have highest degree equal to s. Then we follow the construction described in Section 3 to provide the metric g and the orthonormal basis E 1 , . . . , E m , V m+1 , . . . , V n whose sorting is a local adapted basis of T U . Since deg(E j ) deg(V i ) for each j = 2, . . . , m and i = m + k + 1, . . . , n, the only derivative that appears in (3.9) is E 1 . Therefore we deduce that a vector field V ⊥ , given by equation (3.4), is admissible if and only if it satisfies and f r ∈ C 1 (O), g h ∈ C(O). Given p in M each point q in a local neighborhood O of p in M can be reached using the exponential map as follows On this open neighborhood O ⊂ M we consider the local coordinates x = (x 1 , x 2 , . . . , x m ) given by the inverse map Ξ of the exponential map defined in (4.3). In the literature, these coordinates are commonly called exponential or canonical coordinates of the second kind, see [28,5]. We setx := (x 2 , . . . , x m ). Given a relative compact open subset Ω ⊂⊂ Ξ(O) we consider Then there exists ε > 0 so that the closure of the cylinder x ∈ Σ 0 } is the top of the cylinder. Since dΞ(E 1 ) = ∂ x1 in this exponential coordinates of the second kind the admissibility system (4.2) is given by where we set and we denote by B the (n − m − k) square matrix whose entries are b i1r , by A the (n − m − k) × k matrix whose entries are a i1h .

The high dimensional holonomy map for ruled submanifolds
For ruled submanifolds the system (3.9) reduces to the system of ODEs (4.2) along the characteristic curves. Therefore, a uniqueness and existence result for the solution is given by the classical Cauchy-Peano Theorem, as in the case of curves in [13,Section 5].
Let Φ :M → N be a ruled immersion in a graded manifold. Let Ω ε be the open cylinder defined in (4.5) and T Σ0 (f ) = f (0, ·) and T Σε (f ) = f (ε, ·) be the operators that evaluate functions at x 1 = 0 and at x 1 = ε, respectively.
Let C 0 (Ω ε ) the Banach space of continuous functions on Ω ε vanishing at the infinity, that is the closure of the space of compactly supported function on Ω ε , see [45,Theorem 3.17]. We always consider for each f ∈ C 0 (Ω ε ) the supremum norm We will denote byΩ ε the closure of the open set Ω ε ⊂ R m and by C(Ω ε ) the Banach space of continuous functions on the compactΩ ε . Then we consider the following Banach spaces: is the space of continuous functions on Σ ε vanishing at the infinity. Notice that the respective norms of these Banach spaces are given by where F and G are defined in (4.7). Therefore the existence and the uniqueness of the solution of the Cauchy problem allows us to define the holonomy type map , in the following way: we consider a horizontal compactly supported continuous vector field and we fix the initial condition Y V (0,x) = 0. Then there exists a unique solution of the admissibility system (4.6) with initial condition Y V (0,x) = 0. Letting be the evaluating operator for vertical vectors fields at x 1 = ε defined by T Σε (V ) = V (ε, ·), we define H ε M (Y H ) = T Σε (Y V ). Definition 5.1. We say that Φ restricted toΩ ε is regular if the image of the holonomy map H ε M is a dense subspace of V(Σ ε ), that contains a normalized Schauder basis of V(Σ ε ) (see [46,Definition 14.2]) .
The following result allows the integration of the differential system (4.6) to explicitly compute the holonomy map.
Proposition 5.2. In the above conditions, there exists a square regular matrix Proof. Lemma 5.3 below allows us to find a regular matrix D( Integrating between 0 and ε, taking into account that F (0,x) = 0 for eachx ∈ Σ 0 , and multiplying by D(ε,x) −1 , we obtain (5.2). Proof. By the Jacobi formula we have where adjD is the classical adjoint (the transpose of the cofactor matrix) of D and Tr is the trace operator. Therefore D(t, λ))D(t, λ)B(t, λ)) = det D(t, λ) Tr (B(t, λ)).
Proof. Fix y inȲ . Then there exists {y n } n∈N ⊆ Y such that y n → y as n → +∞. Since L is continuous we have L(y n ) n→+∞ − −−−− → L(y). On the other hand, by assumption L(y n ) = 0, then we conclude that L(y) = 0. Therefore we have L(y) = 0 for each y ∈Ȳ . Assume by contradiction that Y is dense in X, i.e. Y = X. Therefore we have L(x) = 0 for each x ∈ X, that implies L ≡ 0, that is absurd.
Proposition 5.6. The immersion Φ restricted toΩ ε is regular if and only if A(x 1 ,x) is linearly full in R n−m−k .
For the reader's convenience, in Lemma 5.7 we recall a classical result of calculus of variations, see for instance [6,Corollary 4.24] or [34,Exercise 4.14]. Proof. First of all we claim that for each compact set Fix a compact K ⊂ Ω and consider a sequence of continuous compactly supported functions h n 1 on Ω, h n ≡ 1 on K, vanishing out of small open neighborhood U of K such that supp(h n+1 ) ⊂ supp(h n ) for each n ∈ N and h n (x) Since we have the pointwise convergence and |f (x)h n (x)| |f (x)| for each n ∈ N with f ∈ L 1 (supp(h 1 ), µ), by the dominated convergence theorem we obtain Let us consider δ > 0 and the Borel sets +∞)) and On the other hand we have Therefore µ(K) = 0 for each K ⊂ E + δ , then µ(E + δ ) = 0. Hence as δ → 0 we obtain µ(E + ) = 0, where The following result provides a useful characterization of non-regularity Theorem 5.8. The immersion Φ restricted toΩ ε is non-regular if and only if there exist a pointx 0 ∈ Σ 0 and a row vector field Λ(x 1 ,x 0 ) = 0 for all x 1 ∈ [0, ε] that solves the following system Proof. Assume that Φ restricted toΩ ε is non-regular, then by Proposition 5.6 there exist a pointx 0 ∈ Σ 0 and a row vector Γ = 0 such that Since Γ is a constant vector and D(x 1 ,x 0 ) is a regular matrix by Lemma 5.3 , Λ(x 1 ,x 0 ) := ΓD(x 1 ,x 0 ) solves the system (5.5) and Λ(x 1 ,x 0 ) = 0 for all x 1 ∈ [0, ε].

Integrability of admissible vector fields for a ruled regular submanifold
In this section we deduce the main result Theorem 6.6. As we pointed out in the Introduction we need that the space of simple m-vectors of degree grater than deg(M ) is quite simple. Therefore we give the following definition.
. . , n. When ι 0 = s − 1 the submanifold has maximum degree therefore all vector fields are admissible, thus there are no singular submanifold.
Keeping the previous notation we now consider the following spaces where the norm is given by (3) Λ(Σ 0 ) is the set of elements given by where z i ∈ C(Ω ε ) vanishing on Σ 0 . We denote by Π d the orthogonal projection over the space Λ(Σ 0 ), that is the bundle over the vector space of simple m-vectors of degree strictly grater than d, thanks to Remark 6.2. Then we set The open set O is defined in Section 3 and here exp denotes the Riemannian exponential map defined by means of the geodesic flow on T N induced by the Riemannian metric ·, · (see [18,Chapter 3]).
In equation (6.3) we consider E j for each j = 1, . . . , m as vector fields restricted to O (to be exact we should useẼ j following the notation introduced in Section 3) and dΓ(Y ) denotes the differential of Γ(Y ). Thanks to the diffeomorphism Ξ defined in Section 4 we can read the map F (Y ) and the variation Γ(Y ) in exponential coordinates of the second kind (x 1 , x 2 , . . . , x m ) where the open cylinder Ω ε lives.
Observe that now F (Y ) = 0 implies that the degree of the variation Γ(Y ) is less than or equal to d. Then where DF (0)Y is given by that is the right hand side of the equation (2.10). Therefore, following the computations developed in Section 4 and using the exponential coordinates of the second kind we have on Ω ε ⊂ Ξ(O), defined in (4.5). Observe that DF (0)Y = 0 if and only if Y is an admissible vector field, namely Y solves (4.6). Moreover, we have that A and B are bounded the supremum norm on Ω ε , since they are continuous on Ξ(O) and bounded on the compactΩ ε . Our objective now is to prove that the map DG(0, 0) is an isomorphism of Banach spaces. To show this, we shall need the following result.
Lemma 6.4. In the above conditions, assume that DF (0)(Y ) = Y 2 and Y H = Y 1 and Y (a) = 0. Then there exists a constant K such that Proof. We write Then Y v is a solution of the ODE given by where B(x), A(x) are defined after (4.7), F , G are defined in (4.7) and we set Since Y V (0,x) = 0 an Y V solves (6.5) in (0, ε), by Lemma 6.5 there exists a constant K such that where A(t, λ) is a d×d continuos matrix, bounded in the supremum norm on [0, ε]× E and c(t, λ) a continuos vector field bounded in the supremum norm on [0, ε] × E. We denote by u ′ the partial derivative ∂ t u . Then, there exists a constant K such that (6.8) Proof. We start from the case r = 1. By [ where we set C 1 = εe ε sup t∈[0,ε] sup λ∈E A(t,λ) .
Since u is a solution of (6.7) it follows (6.10) sup Hence by (6.9) and (6.10) we obtain Finally, we use the previous constructions to give a criterion for the integrability of admissible vector fields along a horizontal curve.  (4.4). Assume that Φ is regular on the compactΩ ε . Then every admissible vector field with compact support in Ω ε is integrable.
Proof. If ι 0 = s − 1 all vector fields are admissible, then all immersions are automatically regular. Each vector field V is integrable for instance by the exponential map Γ t = exp(tV ).
Let now s − 3 ι 0 s − 2. Let us take V vector field on Ω ε and {V i } ∞ i=1 vector fields equi-bounded in the supremum norm onΩ ε . Let l 1 (R) the Banach space of summable sequences. We consider the map where F is defined in (6.3). The mapG is continuous with respect to the product norms (on each factor we put the natural norm, the Euclidean one on the interval, the l 1 norm and || · || ∞ and || · || 1 in the spaces of vectors on Ω). Moreover G(0, 0, 0, 0) = (0, 0), since the immersion Φ has degree equal to d. Denoting by D Y the differential with respect to the last two variables ofG we have that is a linear isomorphism thanks to Proposition 6.3. We can apply the Implicit Function Theorem to obtain maps such thatG(τ, (τ i ), (Y 1 )(τ, τ i ), (Y 2 )(τ, τ i )) = (0, 0). We denote by l 1 (ε) the ball of radio ε in Banach space l 1 (R). This implies that (Y 1 )(τ, (τ i )) = 0 and that Hence the submanifolds have degree equal to or less than d. Now we assume that V is an admissible vector field compactly supported on Ω ε , and that V i are admissible vector fields such that V i V vanishing on Σ 0 . Then, differentiating (6.11), we obtain that the vertical vector fields on Ω ε are admissible. Since they are admissible and vertical vector fields vanishing at (0,x), they are identically 0.
Since the image of the holonomy map is dense and contains a normalized Schauder basis for V(Σ ε ), we choose on Ω ε such that {T Σε (V i V )} i∈N is a normalized Schauder basis for V(Σ ε ). Then we consider the map given by where C 0 (Σ ε , N ) is the set of continuous functions from Σ ε to N vanishing at infinity, that inherits its differential structure as submanifold of the Banach space C 0 (Σ ε , R 2n ), see [43,Section 5]. For s, (s i ) small, the image of this map is an infinite-dimensional submanifold of C 0 (Σ ε , N ) with tangent space at Φ| Σε given by the Banach space V(Σ ε ) (as T Σε (V ) = 0 and T Σε ( for each i ∈ N. Therefore the differential D 2 P(0, 0) : l 1 (R) → V(Σ ε ) defined by is injective, surjective and continuous. Then, by [6, Corollary 2.7] D 2 P(0, 0) is a Banach space isomorphism. Moreover, we have since V is compactly supported in Ω ε . Hence we can apply the Implicit Function Theorem to conclude that there exist ε ′ < ε and a family of smooth functions τ i (τ ), with i |τ i (τ )| < ε for all τ ∈ (−ε ′ , ε ′ ), so that takes the value Φ(p) for eachp ∈ Σ ε . Since the vector fields {V i } ∞ i=1 are equibounded in the supremum norm onΩ ε , the series i τ i (τ )V i is absolutely convergent onΩ ε .
Example 6.7. An Engel structure (E, H) is 4-dimensional Carnot manifold where H is a two dimensional distribution of step 3. A representation of the Engel group E, which is the tangent cone to each Engel structure, is given by R 4 endowed with the distribution H generated by The second layer is generated by and the third layer by X 4 = [X 1 , X 3 ] = ∂ x4 . A well-known example of horizontal singular curve, first discovered by Engel, is given by γ : R → R 4 , γ(t) = (0, t, 0, 0). R. Bryant and L. Hsu proved in [10] that γ is rigid in the C 1 topology therefore this curve γ does not satisfy any geodesic equation. However H. Sussman [47] proved that γ is the minimizer among all the curves whose endpoints belongs to the x 2 -axis.
Let Ω be an open set in R 2 and Φ : Ω → R 4 be the ruled immersion parametrized by Φ(u, v) = (0, u, 0, v) whose tangent vectors are (X 2 ) Φ(u,v) and (X 4 ) Φ(u,v) . Then we have that the degree deg(Φ(Ω)) is equal to four. Fix the left invariant metric g that makes X 1 , . . . , X 4 an orthonormal basis. Taking into account equation (4.2), we have that a normal vector field V = f 3 X 3 + g 1 X 1 is admissible if and only if ∂f 3 ∂u = g 1 , since b 313 = X 3 , [X 2 , X 3 ] = 0 and a 311 = X 3 , [X 2 , X 1 ] = −1. Therefore A(u, v) = (−1) for all (u, v) ∈ Ω, then A is linearly full in R. Thus, by Proposition 5.6 we gain that ruled immersion Φ is regular. Despite the immersion Φ is foliated by singular curves that are also rigid in the C 1 topology, Φ is a regular ruled immersion. Thus, by Theorem 6.6 we obtain that each admissible vector field is integrable. Therefore it possible to compute the first variation formula [14, Eq. (8.7), Section 8] and verify that Φ is a critical point for the area functional with respect to the left invariant metric g since its mean curvature vector H 4 of degree 4 vanishes. Hence this plane foliated by abnormal geodesics, that do not verify any geodesic equations, satisfies the mean curvature equations for surface of degree 4.
Here we show some applications of Theorem 6.6 to lifted surfaces immersed of codimension 2 in an Engel structure that model the visual cortex, taking into account orientation and curvature. Example 6.8. Let E = R 2 × S 1 × R be a smooth manifold with coordinates p = (x, y, θ, k). We set H = span{X 1 , X 2 }, where (6.13) X 1 = cos(θ)∂ x + sin(θ)∂ y + k∂ θ and X 2 = ∂ k .
Since the curvature is the derivative of orientation we gain that κ(x, y) = X 1 (θ(x, y)) and therefore the degree of these immersion is always equal to four. Then a tangent basis of T p Σ adapted to 2.7 is given by (6.16) E 1 = cos(θ)Φ x + sin(θ)Φ y = X 1 + X 1 (κ)X 2 , Therefore Σ is a FGT-(s−3) ruled submanifoldruled manifold foliated by horizontal curves. Adding V 3 = X 2 −X 1 (κ)X 1 and V 4 = X 3 we obtain a basis of T E. Choosing the metric g that makes E 1 , E 2 , V 3 , V 4 an orthonormal basis we gain that Therefore the admissibility system (4.2) on the chart Ω is given bȳ where V ⊥ = g 3 V 3 + f 4 V 4 and the projection of the vector field X 1 and X 4 onto Ω is given byX 1 = cos(θ(x, y))∂ x + sin(θ(x, y))∂ ȳ X 4 = − sin(θ(x, y))∂ x + cos(θ(x, y))∂ y .