# zbMATH — the first resource for mathematics

Discrete Laplace cycles of period four. (English) Zbl 1247.51010
Within the framework of discrete differential geometry, the author studies discrete conjugate nets, their Laplace transforms, discrete $$W$$-congruences, and discrete asymptotic nets.
The key definitions in this field of geometry are as follows:
{(1)} A discrete conjugate net is a map $$f: \mathbb Z^2\to\mathbb P^3$$, $$(i,j)\mapsto f(i,j)=f_i^j$$, such that every elementary quadrangle $$f_i^j$$, $$f_{i+1}^j$$, $$f_{i+1}^{j+1}$$, $$f_i^{j+1}$$ is planar.
{(2)} A discrete conjugate net is regular if the four points $$f_i^j$$, $$f_{i+1}^j$$, $$f_{i+1}^{j+1}$$, $$f_i^{j+1}$$ are pairwise different, are not collinear, and the osculating planes $$f_{i-1}^j\vee f_i^j\vee f_{i+1}^j$$ and $$f_i^{j-1}\vee f_i^j\vee f_i^{j+1}$$ are of projective dimension two. Here the symbol $$\vee$$ stands for the span of projective subspaces.
{(3)} The first and second Laplace transforms of a regular discrete conjugate net are the discrete conjugate nets $\mathcal{L}_1 f:\mathbb Z^2\to\mathbb P^3, \quad (i,j)\mapsto (f_i^j\vee f_{i+1}^j)\cap (f_i^{j+1}\vee f_{i+1}^{j+1}),$
$\mathcal{L}_2 f:\mathbb Z^2\to\mathbb P^3, \quad (i,j)\mapsto (f_i^j\vee f_i^{j+1})\cap (f_{i+1}^j\vee f_{i+1}^{j+1}).$
{(4)} The $$n$$th Laplace sequence to a discrete conjugate net $$f$$ is the sequence $$n\mapsto \mathcal{L}_i^nf$$ where $$\mathcal{L}_i^nf$$ is recursively defined by $$\mathcal{L}_i^nf =\mathcal{L}_i\mathcal{L}_i^{n-1}f$$ and $$\mathcal{L}_i^0f=f.$$
{(5)} The Laplace sequences of a discrete conjugate net $$f$$ are called a Laplace cycle of period four, if $$\mathcal{L}_1^1f = \mathcal{L}_2^2f$$.
{(6)} Denote the Grassmannian of lines in $$\mathbb P^3$$ by $$\mathbb L^3$$. A map $$L:\mathbb Z^2\to\mathbb L^3$$ is called a discrete $$W$$-congruence if its Klein image on the Plücker quadric is a conjugate net.
For discrete conjugate nets whose Laplace sequence is of period four, the author shows that the connecting lines of corresponding points form a discrete $$W$$-congruence. He derives some properties of discrete Laplace cycles of period four and describes two explicit methods for their construction.
The author explores discrete versions of some results pertaining to asymptotic transforms, $$W$$-congruences, conjugate nets, and Laplace cycles of period four established in the 1930’s by H. Jonas [Math. Ann. 114, 237–274 (1937; Zbl 0016.18101) and ibid., 749–780 (1937; Zbl 0017.18702)] and R. Sauer [Projektive Liniengeometrie. Berlin, Leipzig: Walter de Gruyter (1937; Zbl 0016.21804)].
##### MSC:
 51K10 Synthetic differential geometry 53A20 Projective differential geometry 51A20 Configuration theorems in linear incidence geometry 53A25 Differential line geometry 53A05 Surfaces in Euclidean and related spaces 51M30 Line geometries and their generalizations
Full Text:
##### References:
  Barner M., Über geschlossene Laplace-Ketten, Arch. Math. (Basel), 1958, 9(5), 366-377 · Zbl 0089.17103  Bobenko A.I., Suris Y.B., Discrete Differential Geometry, Grad. Stud. Math., 98, American Mathematical Society, Providence, 2008 http://dx.doi.org/10.1007/978-3-7643-8621-4 · Zbl 1158.53001  Degen W., Über geschlossene Laplace-Ketten ungerader Periodenzahl, Math. Z., 1960, 73(2), 95-120 http://dx.doi.org/10.1007/BF01162471 · Zbl 0101.39504  Doliwa A., Geometric discretisation of the Toda system, Phys. Lett. A, 1997, 234(3), 187-192 http://dx.doi.org/10.1016/S0375-9601(97)00477-5 · Zbl 1044.37527  Doliwa A., Asymptotic lattices and W-congruences in integrable discrete geometry, J. Nonlinear Math. Phys., 2001, 8(suppl.), 88-92 http://dx.doi.org/10.2991/jnmp.2001.8.s.16 · Zbl 0990.37055  Doliwa A., Discrete asymptotic nets and W-congruences in Plücker line geometry, J. Geom. Phys., 2001, 39(1), 9-29 http://dx.doi.org/10.1016/S0393-0440(00)00070-X · Zbl 0990.37056  Doliwa A., Santini P.M., Mañas M., Transformations of quadrilateral lattices, J. Math. Phys., 2000, 41(2), 944-990 http://dx.doi.org/10.1063/1.533175 · Zbl 0987.37070  Edge W.L., The net of quadric surfaces associated with a pair of Möbius tetrads, Proc. Lond. Math. Soc., 1936, 41(1), 337-360 http://dx.doi.org/10.1112/plms/s2-41.5.337 · JFM 62.0744.02  Hammond E.S., Periodic conjugate nets, Ann. of Math, 1921, 22(4), 238-261 http://dx.doi.org/10.2307/1967906 · JFM 48.0844.03  Hu H.S., Laplace sequences of surfaces in projective space and two-dimensional Toda equations, Lett. Math. Phys., 2001, 57(1), 19-32 http://dx.doi.org/10.1023/A:1017923708829  Jonas H., Allgemeine Transformationstheorie der konjugierten Systeme mit viergliedrigen Laplaceschen Zyklen, Math. Ann., 1937, 114(1), 749-780 http://dx.doi.org/10.1007/BF01594207 · JFM 63.0665.01  Jonas H., Ein allgemeiner Satz über W-Kongruenzen mit Anwendungen auf Laplacesche Zyklen, Biegungsflächen des einschaligen Hyperboloids und schiefe Weingarten Systeme, Math. Ann., 1937, 114(1), 237-274 http://dx.doi.org/10.1007/BF01594175 · JFM 63.0664.05  Möbius A.F., Kann von zwei dreiseitigen Pyramiden eine jede in Bezug auf die andere um- und eingeschrieben zugleich heißen?, J. Reine Angew. Math., 1828, 3, 273-278 http://dx.doi.org/10.1515/crll.1828.3.273  Nieszporski M., On a discretization of asymptotic nets, J. Geom. Phys., 2002, 40(3-4), 259-276 http://dx.doi.org/10.1016/S0393-0440(01)00038-9 · Zbl 0997.35080  Pottmann H., Wallner J., Computational Line Geometry, Math. Vis., Springer, Berlin, 2010 http://dx.doi.org/10.1007/978-3-642-04018-4  Sasaki T., Line congruence and transformation of projective surfaces, Kyushu J. Math., 2006, 60(1), 101-243 http://dx.doi.org/10.2206/kyushujm.60.101 · Zbl 1104.53008  Sauer R., Projektive Liniengeometrie, Göschens Lehrbücherei I: Reine und Angewandte Mathematik, 23, de Gruyter, Berlin, 1937  Veblen O., Young J.W., Projective Geometry. I, Ginn and Company, Boston-London, 1910  Wilczynski E.J., The general theory of congruences, Trans. Amer. Math. Soc., 1915, 16(3), 311-327 http://dx.doi.org/10.1090/S0002-9947-1915-1501014-2 · JFM 45.0927.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.