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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
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