Mappings of uniform spaces that have perfect sections and preserve completeness

Mathematical Notes -     

Isometric immersions of a Euclidean plane in Lobachevskii space

It is proved that any regular isometric immersion of a Euclidean plane in (three-dimensional) Lobachevskii space is either a homeomorphism onto an orisphere or a covering of the surface formed by the rotation of an equidistant about its base.Translated from Matematicheskie Zametki, Vol. 10, No. 3, pp. 327–332, September, 1971.     

Bending of infinite convex surfaces of Lobachevskii space

We solve the problem of the isometric immersion of a complete Riemannian metric g_ij, prescribed on a plane, with curvature K₄>−1, in a three-dimensional Lobachevskii space (with curvature-1). We assume here that the metric g_ij is close to Euclidean: It deviates from zero only in some bounded domain and certain of its integral characteristics are small. We show that isometric immersions exist and, moreover, the second form of the desired immersion can be arbitrarily prescribed at infinity (with only the Gauss equation taken into account). Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 45, pp. 56–62, 1974.     

Self-dual cones in Hilbert space

Self-dual orderings of Hilbert spaces are defined and a structure theory is developed. As an application it is shown that a projection π is in the commuting algebra of a “nice” topological group iff π and I — π leave the cone of L² positive definite functions invariant.     

Geometric methods of constructing discrete groups of motions of Lobachevskii space

We present five methods that make it possible to obtain discrete groups of motions of Lobachevskii space by means of the synthetic geometry of this space: the method of variation of one parameter, the method of variation of several parameters, the method of truncation of ideal faces, the method of gluing, and the method of buffer polyhedra. The action of the methods is illustrated by examples.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 15, pp. 3–59, 1983.     

Quantum channels that preserve entanglement

Let M and N be full matrix algebras. A unital completely positive (UCP) map is said to preserve entanglement if its inflation has the following property: for every maximally entangled pure state ρ of , is an entangled state of . We show that there is a dichotomy in that every UCP map that is not entanglement breaking in the sense of Horodecki–Shor–Ruskai must preserve entanglement, and that entanglement preserving maps of every possible rank exist in abundance. We also show that with probability 1, all UCP maps of relatively small rank preserve entanglement, but that this is not so for UCP maps of maximum rank.