Deformation of a submanifold in an Euclidean space with fixed Gauss image

In a previous paper [4] the present author studied a C mapping where M is an m-dimensional C manifold, I is some interval and for each tI the mapping is an immersion satisfying the following conditions, (i) The Gauss map associated with the immersion is regular. (ii) The Gauss image of the immersed submanifold is fixed against t for each point p of M. Such a mapping was called an admissible deformation. The purpose of the present paper is to give results obtained since then.     

Deformation of a submanifold in a Euclidean space with fixed gauss image: II

Deformation of an m-dimensional submanifold immersed in a Euclidean n-space is studied when the Gauss image does not move pointwise. The present paper is the third one and contains deformations where the tensor of deformation satisfies some differential equations, normal deformations and deformations of totally-real submanifolds in a complex space.     

Mapping a sphere into Euclidean space

Translated from Matematicheskie Zametki, Vol. 46, No. 3, pp. 3–8, September, 1989.     

Brownian Motion on a submanifold of Euclidean Space

A martingale characterization of Brownian motion on a submanifoldof Euclidean space is proved and the implications of the consequentmartingale representation are discussed.     

Unique determination of multidimensional submanifolds of Euclidean space from the Grassman image

Translated from Matematicheskie Zametki, Vol. 51, No. 1, pp. 8–15, January, 1992.     

Knaster's problem on continuous maps of a sphere into Euclidean space

A survey of known results and additional new ones on Knaster's problem: on the standard sphere S^n–1Rⁿ find configurations of points A₁, , A_k, such that for any continuous map fS^n–1R^m one can find a rotation a of the sphere S^n–1 such that f(a(A₁)==f(a(A_k)) and some problems closely connected with it. We study the connection of Knaster's problem with equivariant mappings, with Dvoretsky's theorem on the existence of an almost spherical section of a multidimensional convex body, and we also study the set {a S0(n)f(a(A₁))==f(a(A_k))} of solutions of Knaster's problem for a fixed configuration of points A₁, , A_kS^n–1 and a map fS^n–1R^m in general position. Unsolved problems are posed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 169–178, 1987.     

Transformations of R-surfaces of Euclidean space preserving their Grassman image

Translated from Matematicheskie Zametki, Vol. 45, No. 1, pp. 20–27, January, 1989.     

On the Grassmann image of a four-dimensional submanifold inE 6

The four-dimensional submanifold F⁴ in the six-dimensional Euclidean space E⁶ is considered. The relation between the curvature of Grassmannian manifold G_2,6 along the Grassmann image of a two-dimensional area element from the tangent space to F⁴ E⁶ and the projection of a four-dimensional section onto a three-dimensional space is clarified.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 125–132, 1992.     

On the boundary behavior of imbeddings of metric spaces into a Euclidean space

We investigate the boundary behavior of so-called Q-homeomorphisms with respect to a measure in some metric spaces. We formulate a series of conditions for the function Q(x) and the boundary of the domain under which any Q-homeomorphism with respect to a measure admits a continuous extension to a boundary point. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1068–1074, August, 2007.     

Isometric imbedding of a complete two-dimensional locally Euclidean manifold in a Euclidean space

In this article we consider the problem of isometric imbedding of a complete two-dimensional locally Euclidean manifold in a Euclidean space. For each of the possible topological types the corresponding minimal dimension of the extending Euclidean space is indicated. See [3].Translated from Matematicheskie Zametki, Vol. 13, No. 3, pp. 427–429, March, 1973.     

Constructing topologically distinct energy-critical curves in the path space of the Euclidean line

We construct topologically distinct global, non-embedding solutions to the Euler-Lagrange equation for a natural energy functional on the space of maps .     

An invariant of a surface in Euclidean space

We study the angle of deviationd of the normal from the normal plane a at given point A of a surface under an infinitesimal displacementds along the surface. We introduce the concept of normal deviation as the ratio of the angled to the displacementds. It is proved that the normal deviation of a surface at a given point and in a given direction is the torsion of the geodesic line emanating from the point in the given direction.Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 89–92.