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Pseudospherical surfaces on time scales: a geometric definition and the spectral approach. (English) Zbl 1138.53008

The paper consists mostly of extensions of known results about smooth and discrete surfaces to the case of surfaces defined over time scales. By a time scale \(\mathbb T\) one means an arbitrary closed subset of the reals (e.g., the discrete set of integers \(\mathbb Z,\) the Cantor set, etc.) and the surfaces here are defined over 2-dimensional time scales \(\mathbb T_1 \times \mathbb T_2.\) The author first reviews the requisite definitions from [M. Bohner and G. Guseinov, Dyn. Syst. Appl. 13, 351–379 (2004; Zbl 1090.26004)]. So, for example, the partial delta derivative \(D_jf:= \frac{\partial f (t)}{\Delta t_j}\) is defined as the limit of the quotient \(\frac{f(\sigma _j (t)) - f(s)}{\sigma (t_j) - s_j}\) as \(s_j \to t_j\), \(s_j \neq \sigma (t_j),\) where \(\sigma _j\) is the (forward) jump operator on the \(j\)th coordinate of \(t = (t_1, t_2) \in \mathbb T_1 \times \mathbb T_2.\)
A discrete surface \(\mathbf r: \varepsilon_1 \mathbb Z \times \varepsilon_2 \mathbb Z \to \mathbb R^3\) is defined as a map for which \(\Delta_1 \mathbf r\), \(\Delta_2 \mathbf r\) are linearly independent and \(\Delta_j f := \frac{T_j f - f}{\varepsilon_j}\), \(j = 1, 2\) \((T_j \) usual shift operators). In the discrete case \(D_j = \Delta_j.\) The author proves that for any discrete asymptotic weak Chebyshev net (discrete \(K\)-surface) the quantity (“the Gaussian curvature”)
\[ K := - \frac{(\Delta_1 \mathbf n \cdot \Delta_2 \mathbf r )(\Delta_2 \mathbf n \cdot \Delta_1 \mathbf r)}{(\Delta_1 \mathbf r)^2 (\Delta_2 \mathbf r)^2 - (\Delta_1 \mathbf r \cdot \Delta_2 \mathbf r)^2}, \]
is constant, where \(\mathbf n = \frac{\Delta_1 \mathbf r \times \Delta_2 \mathbf r}{| \Delta_1 \mathbf r \times \Delta_2 \mathbf r| }\) is the discrete version of the unit normal. The author obtains the same result for the asymptotic weak nets \(\mathbf r : \mathbb T_1 \times \mathbb T_2 \to \mathbb R^3\) on 2-dimensional time scales with \(\Delta_j'\)s in the above formula replaced with \(D_j'\)s, thus extending the notion of pseudospherical immersions to time scales. Also, a version of Darboux-Bäcklund transformation on time scales is considered.

MSC:

53A05 Surfaces in Euclidean and related spaces
53B25 Local submanifolds
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems

Citations:

Zbl 1090.26004
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