The duals of generic hypersurfaces.
The Euclidean plane kinematics
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.
The Euler characteristic of vector fields on Banach manifolds and the problem of Plateau
The Existence of Embedded Minimal Surfaces and the Problem of Uniqueness.
The exterior plateau problem.
The faces of the Grassmanian of three-planes in R7 (Calibrated geometries on R7).
The first bifurcation point for Delaunay nodoids.
The Fundamental Equations of Minimal Surfaces in C2.
The fundamental theorems of curves and hypersurfaces in centro-affine geometry.
The gap phenomenon in the dimension study of finite type systems
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.
The Gauss Map for Surfaces in 4-Space.
The Gauss map of a spacelike constant mean curvature hypersurface of Minkowski space.
The Gauss-Bonnet theorem for Cayley-Klein geometries of dimension two.
The generalized Holditch theorem for the homothetic motions on the planar kinematics
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...
The generalized regular functions over conformal quaternionic manifold
The geodesic curvature and geodesic torsion of the intersection curve of two surfaces.
The geometry and conformal structure of properly embedded minimal surfaces of finite topology in IR3.
The geometry of dented pentagram maps
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...
The Geometry of Periodic Minimal Surfaces.
The global structure of constant mean curvature surfaces.