The purpose of this paper is to study the relations between quasiregular mappings on Riemannian manifolds and differential forms. Four classes of differential forms are introduced and it is shown that some differential expressions connected in a natural way to quasiregular mappings are members in these classes.
ABSTRACT. A tubular surface is an immersion u: Μ -> R" for which the section Π Π u(M) by an arbitrary hyperplane Π orthogonal to a fixed vector e e R" is a compact set.For tubular minimal surfaces in R" we prove that (a) if aim. Μ = 2 and u(M) lies in a half-space, then u(M) also lies in some hyperplane; and (b) if dim Μ > 3 , then a tubular minimal surface lies in the layer between two hyperplanes orthogonal to e .We obtain the corresponding results about the structure of the Gaussian image of two-dimensional tubular minimal surfaces.The case codimA/ = 1 was investigated earlier (Math. USSR Sb. 59 (1988), 237-245 (MR 88e:53009)).Bibliography: 19 titles.Let Μ be a /^-dimensional connected oriented noncompact manifold of class C 2 . Consider the surface (M, u) given by a C 2 -immersion u: Μ -• R" , where 2 < ρ < η -1. Everywhere below we assume that the immersion u is proper; that is, the inverse image of any compact set F cR" is a compact set in Μ.The surface (M, u) is said to be minimal if its mean curvature vector Η vanishes identically ([1], p. 34).Let Μ be a manifold without boundary. We shall say that the surface (Μ, u) is tubular if there exist a vector e 0 Ε R" and two numbers -00 < a < b < +00 such that for any hyperplane IL = {x € R": (x, e 0 ) = t} orthogonal to e 0 the section Σ £ (t) = u(M)nH t is not empty for any t e {a, b) and any portion included between two hyperplanes Π, and Π, with a < t x < t 2 < b is a compactum. In this case the interval {a, b) will be called the projection of the surface (M, u), and the finite or infinite difference b -a will be called its length.We shall say that the surface (Μ, u) is tubular in the large if it is tubular with projection (-00, +00).The simplest example of a minimal surface in R 3 that is tubular in the large is a catenoid. Numerous examples of two-dimensional immersed tubular minimal surfaces in R" can easily be constructed by starting from the representation for such surfaces given in [2].
A 6-dimensional grand unified theory with the compact space having the topology of a real projective plane, i.e., a 2-sphere with opposite points identified, is considered. The space is locally flat except for two conical singularities where the curvature is concentrated. One supersymmetry is preserved in the effective 4d theory. The unified gauge symmetry, for example SU(5) , is broken only by the non-trivial global topology. In contrast to the Hosotani mechanism, no adjoint Wilson-line modulus associated with this breaking appears. Since, locally, SU(5) remains a good symmetry everywhere, no UV-sensitive threshold corrections arise and SU(5)-violating local operators are forbidden. Doublettriplet splitting can be addressed in the context of a 6d N = 2 super Yang-Mills theory with gauge group SU(6). If this symmetry is first broken to SU(5) at a fixed point and then further reduced to the standard model group in the above non-local way, the two light Higgs doublets of the MSSM are predicted by the group-theoretical and geometrical structure of the model.
We consider the implications of the two-pocket Fermi surface for macroscopic quantum phenomena in cuprates. Superconductivity in this system can be described in terms of two coupled condensates. It results in a collective excitation corresponding to the relative phase oscillation-a phason. The energy of the phason is smaller than the maximum gap on the Fermi surface. We discuss the possibility of searching for this collective excitation in the dynamic resistance of a superconducting interference device (SQUID).
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