Discrete Differential Geometry: Consistency as Integrability Yuri SURIS (TU München) Oberwolfach, March 6, 2006 Based on the ongoing textbook with A. Bobenko
Discrete Differential Geometry Differential geometry investigates smooth geometric shapes, such as curves and surfaces, with the help of mathematical analysis. Discrete geometry studies geometric shapes with finite number of elements, such as polyhedra, with an emphasis on their combinatoric properties. Discrete differential geometry develops discrete analogues and equivalents of notions and methods of the smooth theory. The aims are, on one hand, a better understanding of the nature and properties of the smooth objects, and, on the other hand, satisfying the needs and requirements of applications in modelling, computer graphics etc. 1
On the history of discrete differential geometry Robert Sauer, Munich: 1920-30s: Bending of discrete surfaces vs. isometric deformations of smooth surfaces; 1950: discrete pseudospheric surfaces; 1970: Differenzengeometrie, Springer-Verlag. Walter Wunderlich, Vienna, 1951: discrete pseudospheric surfaces. 2
More recent history of discrete differential geometry Discrete differential geometry meets theory of integrable systems: Alexander Bobenko, Ulrich Pinkall, Berlin, 1995: discrete pseudospheric surfaces, discrete isothermic surfaces and discrete minimal surfaces. Adam Doliwa, Paolo Santini, Rome, 1997: multi-dimensional discrete conjugate and orthogonal nets. The first book: Discrete Integrable Geometry and Physics, Clarendon Press: Oxford, 1999. The first textbook: Alexander Bobenko, Yuri Suris. Discrete Differential Geometry. Consistency as Integrability, 2006 (in preparation). The first Oberwolfach Workshop: 05-11.03.2006 (now and here). 3
Smooth and discrete conjugate nets Conjugate net: f : R m R n with i j f span( i f, j f). Equivalently: tangents f(u) + t i f(u) along each coordinate u j -curve build a developable surface, that is, the set of tangent lines to a curve (the edge of regression of the developable). Sauer s discretization: a quadrilateral net with planar elementary quadrilaterals (for m = 2 only!): f : Z m R n, four vertices f(u), f(u+e i ), f(u+e j ) and f(u+e i +e j ) of each elementary quadrilateral lie in a plane. 4
One sees and immediately understands a discrete developable surface with its discrete regression edge! The case m > 2 has not been considered by Sauer. 5
Jonas transformation of conjugate nets: f, f + : R m R n with f + f span( i f, i f + ). Equivalently: connecting lines ff + along each coordinate curve u i = const form a developable. Discretization (Doliwa, Santini, 1997): two layers of a (m + 1)-dimensional net with planar quadrilaterals: 6
Construction for m = 3: given 7 points f, f i, f ij with planar quadrilaterals (f, f i, f ij, f j ), find the 8th point f 123 so that the quadrilaterals (f i, f ij, f 123, f ik ) are planar as well. Solution: the planes in a threedimensional space intersect (generically) in exactly one point! f 23 f 123 f 23 f 123 f 3 f 13 f 3 f 13 f 2 f 12 f 2 f 12 f f 1 f f 1 Geometric construction. 3D system. 7
Multi-dimensional consistency property of a central importance: it guarantees existence of multi-dimensional nets. f 234 f 1234 f 134 f 34 f 23 f123 f 3 f 13 f 2 f 12 f f 1 f 24 f 124 f 4 f 14 8
Due to this property, discrete nets do not differ essentially from their transformations, and the transformations possess remarkable permutation properties. The theory becomes complicated only upon the smooth limit (break of symmetry between coordinate directions): 9
Smooth and discrete asymptotic nets Asymptotic net, or surface parametrized along asymptotic lines: f : R 2 R 3 with 2 1f, 2 2f span( 1 f, 2 f). Equivalently: osculating planes of both coordinate curves u 1 = const and u 2 = const through each point coincide with the tangent plane of the surface. Sauer s discretization: a quadrilateral net with planar vertices (for m = 2 only!): f : Z m R 3, all neighbor points f(u ± e i ) of f(u) lie in a plane P(u) through f(u). 10
For m = 2 this should hold for 4 neighbor points f(u ± e 1 ), f(u ± e 2 ): f 2 f 1 f f 2 f 1 Quadrilaterals are no more planar! 11
Construction for m = 3: given 7 pairs (f, P), (f i, P i ), (f ij, P ij ) with the property that each plane P contains the point f and its neighbors. Then also the 8th pair (f 123, P 123 ), with f 123 = P 12 P 13 P 23, P 123 = plane through f 12, f 13, f 23, has this property, i.e., P 123 contains f 123. f 23 P 123 P 23 f 123 P 3 f 13 f 3 P 13 P 2 f 12 f 2 P 12 f P 1 P f 1 12
This is a classical incidence theorem the Möbius theorem (1828) about pairs of tetrahedra, which are inscribed in one another. f 2 f 3 f f 1 f 13 f 123 f 12 f 23 See a combinatorial cube here? 13
Thus, the Möbius theorem is responsible for the multi-dimensional consistency of A-nets, i.e., for the existense of transformations with permutability properties. In the smooth limit these are Weingarten transformations of surfaces parametrized along asymptotic lines: two such surfaces f, f + : R 2 R 3 build a Weingarten pair, iff the connecting lines ff + are tangent to both surfaces in the corresponding points. 14
Smooth and discrete orthogonal nets Orthogonal net: conjugate net f : R m R n with an additional property: i f j f. In particular, for m = 2, n = 3 this is a surface parametrized along principal curvature lines. Discrete orthogonal net (Bobenko, Doliwa, Santini, 1997): quadrilateral net f : Z m R n with planar quadrilaterals and an additional property: each quadrilateral f(u), f(u + e i ), f(u + e i + e j ), f(u + e j ) is inscribed in a circle. 15
Construction for m = 3: given 7 points f, f i, f ij with circular quadrilaterals (f, f i, f ij, f j ), find the 8th point f 123 so that the quadrilaterals (f i, f ij, f 123, f ik ) are also circular. Solution: circularity constraint propagates in the construction of discrete conjugate nets! 16
A discrete 3D system for discrete orthogonal nets is a consequence of an incidence theorem of elementary geometry the Miquel theorem (1846). f 3 f 13 f 23 f 123 f f 2 12 f 1 See a combinatorial cube here? 17
This construction is also multidimensionally consistent. This yields transformations with permutability properties, which can be again interpreted as additional coordinate directions of the discrete nets. In the smooth limit they become classical Ribaucour transformations. 18
Smooth and discrete conjugate nets in quadrics Generalization (Doliwa, 1999): if seven points f, f i and f ij of a quadrilateral net with planar quadrilaterals lie on a quadric Q, then so does the 8th point f 123. In other words: constraint f Q propagates in the construction of nets with planar quadrilaterals! Again, this is a consequence of a classical incidence theorem of projective geometry (about associate points): for any seven points of CP 3 in general position there exists the eighth point, which belongs to any quadric through the original seven points. 19
Smooth and discrete Moutard nets Moutard net: f : R 2 R n with 1 2 f = qf. discrete Moutard net (Nimmo, Schief, 1997): f : Z m R n with f 12 + f = q(f 1 + f 2 ). multi-dimensionally consistent T-nets: f : Z m R n with f ij f = a ij (f i f j ). Thus, diagonals of elementary quadrilaterals are parallel. This constraint propagates through the construction of discrete nets with planar quadrilaterals! One more 3D system! 20
Projective characterization of T-nets (Doliwa): consider R n as the homogeneous coordinates space for RP n 1. Then the T-condition (parallel diagonals) for a net with m 3 is equivalent to planarity of tetrahedra of the even (or odd) sublattice: f 23 f 123 f 3 f 13 f 2 f 12 f f 1 Tetrahedra ff 12 f 13 f 23 and f 1 f 2 f 3 f 123 are planar. This condition is 4D consistent. 21
The planarity of the black tetrahedron yields the planarity of the white one a consequence of an incidence theorem of projective geometry: f 12 f f 23 f 13 f 3 f 123 f 1 f 2 See a combinatorial cube here? Recognize the picture for A-cube? 22
Other sorts of 3D systems Fields on faces: c jk τ i c jk 23
Examples of multi-dimensionally consistent systems: Discrete Darboux equations for discrete conjugate nets: τ i γ jk = γ jk + γ ji γ ik 1 γ ik γ ki. Star-triangle transformation, describes T-nets: a jk τ i a jk =. a ij a jk + a jk a ki + a ki a ij 24
Fields on edges: τ 2 τ 3 x 1 τ 3 x 2 τ 1 τ 3 x 2 τ 2 x 3 τ 3 x 1 τ 1 τ 2 x 3 x 3 τ 1 x 3 x 2 x 1 τ 2 x 1 τ 1 x 2 {Six black points} {Six white points} 25
Example (Schief): four points corresponding to four edges of any elementary quadrilateral are collinear. τ 1 τ 2 x 3 x 1 τ 1 x 2 τ 2 x 1 x 2 τ 1 x 3 τ 2 x 3 τ 1 τ 3 x 2 τ 3 x 1 τ 3 x 2 τ 2 τ 3 x 1 x 3 26
Smooth and discrete K-nets K-surface: surface f : R 2 R 3 parametrized along asymptotic lines with the (weak) Chebyshev property: 1 f is constant along u 2 -coordinate curves, and 2 f is constant along u 1 -coordinate curves. Bäcklund transformation: a Weingarten pair of K-surfaces f, f + : R 2 R 3 is a Bäcklund pair, iff the the distance f + f between the corresponding points is constant. Discrete K-nets (Sauer m = 2, discrete K-surfaces, Wunderlich m = 2, 3, K-surfaces and their transformations): quadrilateral nets f : Z m R 3 with planar vertices with all quadrilaterals being bent parallelograms: f ij f i = f j f. Equivalently: unit normals N : Z m S 2 form a T-net in a unit sphere. 27
Thus, imposed are two admissible reductions of conjugate nets simultaneously. The system becomes two-dimensional but inherits the property of being 3D consistent: N 23 N 123 N 3 N 13 N 2 N 12 N 2 N 12 N N 1 N N 1 28
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Discrete isothermic surfaces Discrete I-net (Bobenko, Pinkall, 1995): a circular net f : Z m R n with factorizable cross-ratios: q(f, f i, f ij, f j ) = α i α j. f j α i f ij f α j α i α j f i Equivalent (Möbius-geometric) characterization: T-nets in the light cone of the Minkowski space R n+1,1. A 3D consistent 2D system! 30
Other sorts of 2D systems: Fields on edges: x 2 y y 1 x R(x,y) = (x 2, y 1 ) Yang-Baxter property: R 23 R 13 R 12 = R 12 R 13 R 23. 31
y 13 R 23 x 23 R 13 z 1 z = y 13 R 12 x 23 y 3 x 3 z z 12 x 2 y 1 R 12 x y z 12 z 2 R 13 R 23 y x Example: discrete smoke-ring flow (Hoffmann): { x2 = (α β + x y)x(α β + x y) 1, y 1 = (α β + x y)y(α β + x y) 1, with x,y IH R 3, α, β R. 32
Differential geometry Discrete differential geometry Integrability surfaces discrete nets φ uv = sinφ integrable systems transformations of surfaces CONSISTENCY U v V u + [U, V ] = 0 Bäcklund-Darboux transformations 33
Differential geometry Discrete differential geometry Integrability Permutability a la Bianchi Multidimensional consistency integrable hierarchies Consistensy is the organizing principle of discrete differential geometry. 34
From integrable geometry to integrable equations The angle φ between asymptotic lines on a K-surface satisfies the integrable sine-gordon equation: x y φ = sinφ. The analogous quantity on a discrete K-surface satisfies (Bobenko-Pinkall, 1995) the discrete sine-gordon equation: sin 1 4 ( φ(x + ǫ,y + ǫ) φ(x + ǫ,y) φ(x,y + ǫ) + φ(x,y) ) = ǫ2 4 sin 1 4( φ(x + ǫ,y + ǫ) + φ(x + ǫ,y) + φ(x,y + ǫ) + φ(x,y) ). 35
Bäcklund transformations of smooth K-surfaces are described by the system: x φ (1) + x φ = 2 α sin φ(1) φ 2 For discrete K-surfaces: sin 1 4 sin 1 4, y φ (1) y φ = 2α sin φ(1) + φ. 2 ( φ (1) (x + ǫ,y) φ (1) (x, y) + φ(x + ǫ, y) φ(x,y) ) = ǫ 2α sin 1 ( φ (1) (x + ǫ, y) + φ (1) (x,y) φ(x + ǫ,y) φ(x,y) ), 4 ( φ (1) (x,y + ǫ) φ (1) (x, y) φ(x,y + ǫ) + φ(x,y) ) = ǫα 2 sin 1 4( φ (1) (x, y + ǫ) + φ (1) (x,y) + φ(x,y + ǫ) + φ(x,y) ). 36
In both cases for the Bäcklund transformations holds the permutability theorem of Bianchi: sin 1 4 ( φ (12) + φ (2) φ (1) φ ) = β α sin 1 4 ( φ (12) φ (2) + φ (1) φ ). Now note: in the discrete case all relevant equations have identical appearance: sin 1 4 ( φjk + φ k φ j φ ) = α k sin 1 ( φjk φ k + φ j φ ), α j 4 where the coordinate directions 1,2 (with α 1 = α 2 = ǫ/2) describe the surface itself, while the coordinate directions 3=(1), 4=(2) (with α 3 = α, α 4 = β) describe the Bäcklund transformations. The symmetry goes lost only upon the smooth limit. 37
With u = exp(iφ/2) we have an elegant discrete 2D equation which is 3D-consistent. u jk u = α ju j α k u k α j u k α k u j, u k u α k α j α j α k u jk u j u k u α j u jk α i u ijk α i u ik α k α k α k α k α j u j α i u i α j α i u ij α j 38
One can derive the zero-curvature representation of this equation from the fact of 3D consistency (Bobenko, Suris, 2002). Also, it is possible to classify integrable (3D consistent) 2D equations within certain classes (Adler, Bobenko, Suris, 2003). 39