Restricting his considerations to the Euclidean plane, the author shows a method leading to the solution of the equivalence problem for all Lie groups of motions. Further, he presents all transitive one-parametric system of motions in the Euclidean plane.
Several examples of gaps (lacunes) between dimensions of maximal and submaximal symmetric models are considered, which include investigation of number of independent linear and quadratic integrals of metrics and counting the symmetries of geometric structures and differential equations. A general result clarifying this effect in the case when the structure is associated to a vector distribution, is proposed.
W. Blaschke and H. R. Müller [4, p. 142] have given the following theorem as a generalization of the classic Holditch Theorem: Let be a 1-parameter closed planar Euclidean motion with the rotation number and the period . Under the motion , let two points , trace the curves and let be their orbit areas, respectively. If is the orbit area of the orbit curve of the point which is collinear with points and then In this paper, under the 1-parameter closed planar homothetic motion...
We propose a new family of natural generalizations of the pentagram map from 2D to higher dimensions and prove their integrability on generic twisted and closed polygons. In dimension there are such generalizations called dented pentagram maps, and we describe their geometry, continuous limit, and Lax representations with a spectral parameter. We prove algebraic-geometric integrability of the dented pentagram maps in the 3D case and compare the dimensions of invariant tori for the dented maps...