Narrow Results

An algebraic formulation is given for the embedded noncommutative spaces over the Moyal algebra developed in a geometric framework in [8]. We explicitly construct the projective modules corresponding to the tangent bundles of the embedded noncommutative spaces, and recover from this algebraic formulation the metric, Levi–Civita connection and related curvatures, which were introduced geometrically in [8]. Transformation rules for connections and curvatures under general coordinate changes are given. A bar involution on the Moyal algebra is discovered, and its consequences on the noncommutative differential geometry are described.

We show that the indefinite complex space form ${ℂ}^{r,s}$ is not a relative to the indefinite complex space form $ℂ{\mathbb{P}}_l^N(b)$ or $ℂ{ℍ}_m^N(b)$. We further study whether two Fubini–Study spaces are relatives or not.

We study the possibility to explain the mystery of the dark matter (DM) through the transition from General Relativity to embedding gravity. This modification of gravity, which was proposed by Regge and Teitelboim, is based on a simple string-inspired geometrical principle: our spacetime is considered here as a four-dimensional surface in a flat bulk. We show that among the solutions of embedding gravity, there is a class of solutions equivalent to solutions of GR with an additional contribution of non-relativistic embedding matter, which can serve as cold DM. We prove the stability of such type of solutions and obtain an explicit form of the equations of motion of embedding matter in the non-relativistic limit. According to them, embedding matter turns out to have a certain self-interaction, which could be useful in the context of solving the core-cusp problem that appears in the $\mathrm{\Lambda }$CDM model.

The Fisher–Rao metric on the parameter space for the set of lines in a two-dimensional convex image is approximated under the assumption that the errors in the measurements are small. The volume of the parameter space under the approximating metric is proportional to the area of the image under the Euclidean metric. In the case of a rectangular image, expressions for the approximating metric are obtained and an algorithm is given for sampling the parameter space. The sample points are used in an algorithm for detecting lines in a rectangular image. Experimental results are reported. In the case of a disc shaped image the parameter space for lines embeds isometrically, under the approximating metric, into three-dimensional Euclidean space.

We study the embedding theory being a formulation of the gravitation theory where the independent variable is the embedding function for the four-dimensional spacetime in a flat ambient space. We do not impose additional constraints which are usually used to remove from the theory the extra solutions not being the solutions of Einstein equations. In order to show the possibility of automatic removal of these extra solutions we analyze the equations of the theory, assuming an inflation period during the expansion of the universe. In the framework of Friedmann–Robertson–Walker (FRW) symmetry we study the initial conditions for the inflation, and we show that after its termination the Einstein equations begin to satisfy with a very high precision. The properties of the theory equations allow us to suppose with confidence that the Einstein equations will satisfy with enough precision out of the framework of FRW symmetry as well. Thus the embedding theory can be considered as a theory of gravity which explains observed facts without any additional modification of it and we can use this theory in a flat space when we try to develop a quantum theory of gravity.

We study the procedure that allows to rewrite the actions of some modified gravity theories like mimetic gravity and Regge–Teitelboim embedding theory as Einstein–Hilbert actions with additional matter contributions. It turns out that this procedure can be applied to brane action, which leads to a polynomial brane action. We also examine the connection between this procedure and Polyakov trick in string theory.

The present paper is concerned with the existence of golbal smooth solutions for the homogeneous Dirichlet boundary value problem of the Darboux equation and the case degenerate on the boundary is contained. As some applications the smooth isometric embeddings of positively and nonnegatively curved disks into R³ are constructed.

In this paper we consider piecewise linear (pl) isometric embeddings of Euclidean polyhedra into Euclidean space. A Euclidean polyhedron is just a metric space which admits a triangulation such that each n-dimensional simplex of is affinely isometric to a simplex in 𝔼ⁿ. We prove that any 1-Lipschitz map from an n-dimensional Euclidean polyhedron into 𝔼³ⁿ is ϵ-close to a pl isometric embedding for any ϵ > 0. If we remove the condition that the map be pl, then any 1-Lipschitz map into 𝔼^{2n + 1} can be approximated by a (continuous) isometric embedding. These results are extended to isometric embedding theorems of spherical and hyperbolic polyhedra into Euclidean space by the use of the Nash–Kuiper C¹ isometric embedding theorem ([9] and [13]).

We prove that every proper $n$-dimensional length metric space admits an “approximate isometric embedding” into Lorentzian space ${ℝ}^{3n+6,1}$. By an “approximate isometric embedding” we mean an embedding which preserves the energy functional on a prescribed set of geodesics connecting a dense set of points.