Integrable discrete nets in Grassmannians
Abstract
We consider discrete nets in Grassmannians which generalize Qnets (maps with planar elementary quadrilaterals) and Darboux nets (valued maps defined on the edges of such that quadruples of points corresponding to elementary squares are all collinear). We give a geometric proof of integrability (multidimensional consistency) of these novel nets, and show that they are analytically described by the noncommutative discrete Darboux system.
Key words: discrete differential geometry, multidimensional consistency, Grassmannian, noncommutative Darboux system
Mathematics Subject Classification: 15A03, 37K25
1 Introduction
One of the central notions in discrete differential geometry constitute discrete nets, that is, maps specified by certain geometric properties. Their study was initiated by R. Sauer [14], while their appearance in the modern theory of integrable systems is connected with the work of A. Bobenko and U. Pinkall [1, 2] and of A. Doliwa and P. Santini [11]. A systematic exposition of discrete differential geometry, including detailed bibliographical and historical remarks, is given in the monograph [5] by two of the present authors. In many aspects, the discrete differential geometry of parametrized surfaces and coordinate systems turns out to be more transparent and fundamental than the classical (smooth) differential geometry, since the transformations of discrete surfaces possess the same geometric properties and therefore are described by the same equations as the surfaces themselves. This leads to the notion of multidimensional consistency which can be seen as the fundamental geometric definition of integrability in the discrete context, which yields standard integrability structures of both discrete and continuous systems, such as Bäcklund and Darboux transformations, zero curvature representations, hierarchies of commuting flows etc.
In this note we present a generalization of two classes of multidimensional nets, Qnets (or discrete conjugate nets) and Darboux nets, to the maps with values in the Grassmannian instead of . The basic idea underlying this work goes back to H. Grassmann and J. Plücker and consists in regarding more complicated objects than just points (like lines, spheres, multidimensional planes, contact elements etc.) as elementary objects of certain geometries. Such objects are then represented as points belonging to some auxiliary projective spaces or to certain varieties in these spaces. In the framework of discrete differential geometry, one can assign such objects to the sites of the lattice and impose certain geometric conditions to characterize interesting classes of multidimensional nets. Several such classes have been introduced in the literature, for instance:

discrete line congruences [12], which are nets in the set of lines in subject to the condition that any two neighboring lines intersect (are coplanar);

discrete Wcongruences [7], which are nets in the set of lines in such that four lines corresponding to the vertices of every elementary square of belong to a regulus. If one represents the lines in by points of the Plücker quadric in , then this condition is equivalent to the planarity of elementary quadrilaterals;

principal contact element nets [3], which are nets in the set of contact elements in such that any two neighboring contact elements share a common oriented sphere. In the framework of Lie geometry such nets are represented by isotropic line congruences.
Again, one can find detailed information and additional bibliographical notes about these nets in [5].
In the present work, we study two related classes of multidimensional nets in Grassmannians which generalize Qnets (nets in with planar elementary quadrilaterals) and the so called Darboux nets introduced in [15].
It turns out that Grassmannian Qnets can be analytically described by a noncommutative version of the so called discrete Darboux system which was introduced, without a geometric interpretation, in [6]. Our present investigations provide also a geometric meaning for the abstract Qnets in a projective space over a noncommutative ring, considered in [10]. More precisely, we demonstrate that equations of abstract Qnets in a projective space over the matrix ring can be interpreted as the analytical description of the Grassmannian Qnets in the suitable parametrization. The fact that the equations of Qnets are considered over a ring rather than over a field is not very essential in this context, since the very notion of Qnets is related to subspaces in general position, and an accident degeneration of some coefficients is treated as a singularity of the discrete mapping. A much more important circumstance is the noncommutativity of the matrix ring, which is equivalent to the absence of Pappus theorem in the geometries over this ring.
2 Multidimensional consistency of Grassmannian Qnets
Recall that the Grassmannian is defined as the variety of planes in . It can be also described as the variety of dimensional vector subspaces of the dimensional vector space . In the latter realization, the Grassmannian is alternatively denoted by . In what follows, the term “dimension” is used in the projective sense.
Definition 1.
(Grassmannian Qnet) A map , , , is called a dimensional Grassmannian Qnet of rank if for every elementary square of the four planes in corresponding to its vertices belong to some plane.
Note that three generic planes in span a plane. Therefore, the meaning of Definition 1 is that if any three of the planes corresponding to an elementary cell are chosen in general position, then the last one belongs to the plane spanned by the first three.
Example 1.
In the case of rank Definition 1 requires that four points corresponding to any elementary square of be coplanar. Thus we arrive at the notion of usual Qnets.
Example 2.
Qnets of rank are built of projective lines assigned to vertices of the lattice , and Definition 1 requires that four lines corresponding to any elementary square lie in a 5plane.
The main properties of usual Qnets which will be generalized now to the Grassmannian context are the following (see a detailed account in [5]):

Within an elementary cube of , the points assigned arbitrarily to any seven vertices determine the point assigned to the eighth vertex uniquely. This can be expressed by saying that Qnets are described by a discrete 3D system.

This 3D system can be imposed on all 3D faces of an elementary cube of any dimension . This property is called the multidimensional consistency of the corresponding 3D system and follows for any from the 4D consistency. The multidimensional consistency is treated as the integrability of the corresponding 3D system.
These properties are illustrated on Fig. 1. On this figure and everywhere else we use the notation for the shift of the th argument of a function on , that is, for . It is clear that the order of the subscripts does not matter, .
Theorem 1.
(Grassmannian Qnets are described by a discrete 3D system) Let seven planes , , , , , be given such that
for each pair of indices , but with no other degeneracies. Then there exists a unique plane such that the conditions
are fulfilled as well.
Proof.
The general position condition yields that the projective plane is of dimension . The assumptions of the theorem imply that the planes are contained in the corresponding planes , and therefore are also contained in . In the case of the general position, the planes spanned by are also dimensional. The plane , if exists, must lie in the intersection of three such planes. In the dimensional space , the dimension of a pairwise intersection is , and therefore the dimension of the triple intersection is , as required. ∎
Theorem 2.
(Multidimensional consistency of Grassmannian Qnets) The 3D system governing Grassmannian Qnets is 4Dconsistent and therefore dimensionally consistent for all .
Proof.
One has to show that the four planes,
and the three others obtained by cyclic shifts of indices, coincide. Thus, we have to prove that the six planes intersect along a plane. We assume that the ambient space has dimension . Then, in general position, the plane which contains all elements of our construction is of dimension . It is easy to understand that the plane is the intersection of two planes and . Indeed, the plane contains also , , and . Therefore, both and contain the three planes , and , which determine the plane . Now the intersection in question can be alternatively described as the intersection of the four planes , , , of one and the same dimensional space. This intersection is generically an plane. ∎
3 Analytical description: noncommutative Qnets
Here we give an analytical description of Grassmannian Qnets. In the case of ordinary Qnets (of rank ), the planarity condition is written in affine coordinates as
(1) 
where the scalar coefficients are naturally assigned to the corresponding elementary squares of (parallel to the coordinate plane ). Consistency of these equations around an elementary cube (Theorem 1) yields a mapping
This mapping can be rewritten in a rather nice form in terms of so called rotation coefficients. The same approach works in the case as well, with the only difference that now we have to assume that the coefficients are noncommutative.
In order to demonstrate this we use the interpretation of the Grassmannian as the variety of all dimensional subspaces of the vector space . One can represent an dimensional subspace of by a matrix whose rows contain vectors of some basis of . The change of basis of corresponds to a left multiplication of by an element of . Thus, one gets the isomorphism .
The condition that the dimensional vector subspace belongs to the dimensional vector space spanned by is now expressed by an equation
The set of coefficients is abundant since it contains parameters while . In order to get rid of this abundance we adopt an “affine” normalization of the representatives , analogous to the case . Namely, the representative of a generic subspace can be chosen, by applying the left multiplication by a suitable matrix, in the form
(2) 
with the unit matrix in the last columns. Under this normalization the coefficients in the equation obey the relation , and we come to the equation of the form (1).
The calculation of the consistency conditions of the equations (1) remains rather simple in the noncommutative setup. One of three ways of getting vector is
Note that after we substitute and from (1), the matrix enters the right hand side only once with the coefficient . Therefore, alternating of and yields the relation
(3) 
Analysis of relations (3) is based on the following statement.
Lemma 1.
(Integration of closed multiplicative matrixvalued oneform) Let the valued functions be defined on edges of parallel to the th coordinate axis (so that is assigned to the edge , where is the unit vector of the th coordinate axis). If satisfy
(4) 
then there exists a valued function defined on vertices of such that
(5) 
Proof.
Prescribe arbitrarily. In order to extend to any point of , connect it to by a lattice path , where the endpoint of any edge coincides with the initial point of the edge . Extend along the path according to (5). This extension does not depend on the choice of the path. Indeed, any two lattice paths connecting any two points can be transformed into one another by means of elementary flips exchanging two edges , to the two edges , . The value of at the common points and of such two paths remain unchanged under the flip, as follows from the “closedness condition” (4). ∎
Equations (3) together with Lemma 1 yield existence of matrices (assigned to edges of parallel to the th coordinate axis) such that . They are called the discrete Lamé coefficients. Equation (1) takes the form
Let us introduce the new variable (also assigned to edges parallel to the th coordinate axis) by the formula . Then , and, on the other hand, . This allows to rewrite the equation (1) finally as
(6) 
The matrices
(7) 
(assigned to the elementary squares parallel to the th coordinate plane) are called the discrete rotation coefficients. The compatibility conditions in terms of these coefficients are perfectly simple. We have
which leads to the coupled equations
These can be solved for to give an explicit map.
Theorem 3.
(Grassmannian Qnets are described by the noncommutative discrete Darboux system) Rotation coefficients of Qnets in the Grassmannian satisfy the noncommutative discrete Darboux system
(8) 
This map is multidimensionally consistent.
Consistency is a corollary of Theorem 2, but it is also not too difficult to prove it directly.
4 Grassmannian Darboux nets
Definition 2.
(Grassmannian Darboux net) A Grassmannian Darboux net (of rank ) is a map defined on edges of the regular square lattice, such that for every elementary quadrilateral of the four planes corresponding to its sides lie in a plane.
In particular, for one arrives at the notion of Darboux nets introduced in [15]: the four points corresponding to the sides of every elementary square are required to be collinear.
We will denote by the planes assigned to the edges of parallel to the th coordinate axis; the subscripts will be still reserved for the shift operation.
To find an analytical description of Grassmannian Darboux nets, we continue to work with the “affine” representatives from normalized as in (2). The defining property yields:
Hence
and therefore
Comparing with (3) and using Lemma 1, we conclude that . Set , then the linear problem takes the form (6) with the rotation coefficients
Thus, we come to the conclusion that Grassmannian Darboux nets are described by the same noncommutative discrete Darboux system (8) as Grassmannian Qnets, with rotation coefficients defined by the last formula. Of course, this is not a coincidence, since Qnets and Darboux nets are closely related. Indeed, considering an intersection of a Grassmannian Qnet in with some plane of codimension , one will find a Grassmannian Darboux net in . Conversely, any Grassmannian Darboux net can be extended (nonuniquely) to a Grassmannian Qnet. This is analogous to the case of ordinary nets (of rank ) explained in [5, p.76].
Acknowledgements.
The research of V.A. is supported by the DFG Research Unit “Polyhedral Surfaces” and by RFBR grants 080100453 and NSh3472.2008.2. The research of A.B. is partially supported by the DFG Research Unit “Polyhedral Surfaces”.
References
 [1] Bobenko, A.I., Pinkall, U.: Discrete surfaces with constant negative Gaussian curvature and the Hirota equation, J. Diff. Geom. 43 (1996) 527–611.
 [2] Bobenko, A.I., Pinkall, U.: Discrete isothermic surfaces, J. Reine Angew. Math. 475 (1996) 187–208.
 [3] Bobenko, A.I., Suris, Yu.B.: On organizing principles of discrete differential geometry. Geometry of spheres, Russian Math. Surveys 62:1 (2007) 1–43 (English translation of Uspekhi Mat. Nauk 62:1 (2007) 3–50).
 [4] Bobenko, A.I., Suris, Yu.B.: Discrete Koenigs nets and discrete isothermic surfaces, arXiv:0709.3408v1 [math.DG]
 [5] Bobenko, A.I., Suris, Yu.B.: Discrete differential geometry. Integrable structure. Graduate Studies in Mathematics, vol. 98. Providence: AMS, 2008, xxiv+404 pp.
 [6] Bogdanov, L.V., Konopelchenko, B.G.: Lattice and difference DarbouxZakharovManakov systems via dressing method, J. Phys. A 28:5 L173–178 (1995)
 [7] Doliwa, A.: Discrete asymptotic nets and congruences in Plücker line geometry, J. of Geometry and Physics 39, 9–29 (2001)
 [8] Doliwa, A.: The Ribaucour congruences of spheres within Lie sphere geometry, In: Bäcklund and Darboux transformations. The geometry of solitons (Halifax, NS, 1999), CRM Proc. Lecture Notes, vol. 29, Amer. Math. Soc., Providence, RI, pp. 159–166.
 [9] Doliwa, A.: The C(symmetric) quadrilateral lattice, its transformations and the algebrogeometric construction, arXiv: 0710.5820 [nlin.SI]
 [10] Doliwa, A.: Geometric algebra and quadrilateral lattices, arXiv: 0801.0512 [nlin.SI]
 [11] Doliwa, A., Santini, P.M.: Multidimensional quadrilateral lattices are integrable. Phys. Lett A 233:4–6, 365–372 (1997)
 [12] Doliwa, A., Santini, P.M., Mañas, M.: Transformations of quadrilateral lattices, J. Math. Phys. 41, 944–990 (2000)
 [13] A.D. King, W.K. Schief. Application of an incidence theorem for conics: Cauchy problem and integrability of the dCKP equation. J. Phys. A 39:8 (2006) 1899–1913.
 [14] Sauer, R.: Projektive Liniengeometrie, Berlin: W. de Gruyter & Co., 1937, 194 pp.
 [15] Schief, W.K.: Lattice geometry of the discrete Drboux, KP, BKP and CKP equations. Menelaus’ and Carnot’s theorems, J. Nonlin. Math. Phys. 10, Suppl. 2, 194–208 (2003).