Related papers: B\"acklund transformations for transparent connect…
We give a relatively simple proof that a translation surface in Euclidean space that satisfies a relation of type $aH+bK=c$, for some real numbers $a,b,c$, where $H$ and $K$ are the mean curvature and the Gauss curvature of the surface,…
We show that one can achieve transversality for lifts of holomorphic disks to a projectivized vector bundle by locally enlarging the structure group and considering the action of gauge transformations on the almost complex structure, which…
All the connections, pure toward the nilpotent structure, are found. Examples of manifolds, for which the curvature tensor is pure or hybrid, are given. For a manifold of B-type a necessary and sufficient condition for purity of the…
We begin by considering several properties commonly (but not universally) possessed by B\"acklund transformations between hyperbolic Monge-Amp\`ere equations: wavelike nature of the underlying equations, preservation of independent…
In this paper, we study translation surfaces in the Euclidean space endowed with a canonical semi-symmetric non-metric connection. We completely classify the translation surfaces of constant sectional curvature with respect to this…
Binary symmetry constraints are applied to constructing B\"acklund transformations of soliton systems, both continuous and discrete. Construction of solutions to soliton systems is split into finding solutions to lower-dimensional Liouville…
Proper lattices for the discrete BKP and the discrete DKP equaitons are determined. Linear B\"acklund transformation equations for the discrete BKP and the DKP equations are constructed, which possesses the lattice symmetries and generate…
Treating an integrable quad-equation along with its two generalised symmetries as a compatible system allows one to construct an auto-B\"acklund transformation for solutions of the related NLS-type system. A fixed periodic reduction of the…
We construct explicit solutions to the discrete motion of discrete plane curves that has been introduced by one of the authors recently. Explicit formulas in terms the $\tau$ function are presented. Transformation theory of the motions of…
We give a new mechanism for constructing Backlund transformations by using symmetry reduction of differential systems. We then characterize a family of Backlund transformations between Darboux integrable systems where the Backlund…
Harmonic maps from $\BR^2$ or one-connected domain ${\O}\subset \BR^2$ into $GL(m, \BC)$ and $U(m)$ are treated. The GBDT version of the B\"acklund-Darboux transformation is applied to the case of the harmonic maps. A new general formula on…
Consider an orientable compact surface in three dimensional Euclidean space with minimum total absolute curvature. If the Gaussian curvature changes sign to finite order and satisfies a nondegeneracy condition along closed asymptotic…
We study, theoretically and experimentally, a 1-parameter family of transformations and their limiting vector field on the space of plane polygons. These transformations are discrete analogs of completely integrable transformation on closed…
Pre-geodesics of an affine connection are the curves that are geodesics after a reparametrization (the analogous concept in K\"ahler geometry is known as J-planar curves). Similarly, dual-geodesics on a Riemannian manifold are curves along…
Reflection at an interface separating two different media is a rather universal phenomenon which arises because of wave mismatching at the interface. By means of supersymmetric quantum mechanics methods, it is shown that a fully transparent…
We approach the construction of Backlund transformations for Darboux integrable hyperbolic partial differential equations in the plane through the reduction of exterior differential systems. For example it is shown that all the Backlund…
We first describe the action of the fundamental group of a closed surface of variable negative curvature on the oriented geodesics in its universal covering in terms of a naturally-defined flat connection whose holonomy lies in the group of…
The first examples of complete projective connections are uncovered: normal projective connections on surfaces whose geodesics are all closed and embedded are complete, as are normal projective connections induced from complete affine…
The new integrable mapping with a simple geometric interpretation is presented. This mapping arise from the nonlinear superposition principle for the B\"acklund transformations of some vector evolution equation.
There are three complete plane geometries of constant curvature: spherical, Euclidean and hyperbolic geometry. We explain how a closed oriented surface can carry a geometry which locally looks like one of these. Focussing on the hyperbolic…