Maps Of Manifolds Research Articles

A note on harmonic maps of statistical manifolds

We show an existence and uniqueness result for a class of maps from a flat statistical manifold into a Riemannian manifold in a given homotopy class when the target Riemannian manifold is of negative sectional curvature under a global topological non-triviality condition. We also show that due to dualistic structure of the domain manifold the result is still valid in dual coordinates.

Local metric properties of level surfaces on Carnot–Caratheodory spaces

For level sets of -mappings of Carnot manifolds to Carnot–Caratheodory spaces, we introduce an adequate local metric characteristic. Moreover, for mappings defined on Carnot groups, we construct a special adapted basis in the preimage such that it assigns a suitable local sub-Riemannian structure on a complement of a kernel of a sub-Riemannian differential to the initial sub-Riemannian structure in the image.

A note on F-planar mappings of manifolds with non-symmetric linear connection

Following the basic facts of [Formula: see text]-planar mappings introduced by Mikeš and Sinyukov and further developed by Hinterleitner and Mikeš, we consider the notions of an [Formula: see text]-planar curve and an [Formula: see text]-planar mapping, but in case of manifolds endowed with a non-symmetric linear connection and an affinor structure. Consequently, we investigate infinitesimal [Formula: see text]-planar and particularly geodesic transformations of manifolds with non-symmetric linear connection and find necessary and sufficient conditions for the existence of these transformations.

Admissible changes of variables for Sobolev functions on (sub-)Riemannian manifolds

We consider the properties of measurable maps of complete Riemannian manifolds which induce by composition isomorphisms of the Sobolev classes with generalized first variables whose exponent of integrability is distinct from the (Hausdorff) dimension of the manifold. We show that such maps can be re-defined on a null set so that they become quasi-isometries. Bibliography: 39 titles.

Albanese maps of projective manifolds with nef anticanonical bundles

Let $X$ be a projective manifold such that the anticanonical bundle $-K_X$ is nef. We prove that the Albanese map $p: X \rightarrow Y$ is locally isotrivial. In particular, $p$ is a submersion.

ELIMINATION OF DEFINITE FOLD. II

In this paper, we first give a new simple proof to the elimination theorem of definite fold by homotopy for generic smooth maps of manifolds of dimension strictly greater than $2$ into the $2$--sphere or into the real projective plane. Our new proof has the advantage that it is not only constructive, but is also algorithmic: the procedures enable us to construct various explicit examples. We also study simple stable maps of $3$--manifolds into the $2$--sphere without definite fold. Furthermore, we prove the non-existence of singular Legendre fibrations on $3$--manifolds, answering negatively to a question posed in our previous paper.

Geodesic mappings of spaces with affine connnection onto generalized Ricci symmetric spaces

The presented work is devoted to study of the geodesic mappings of spaces with affine connection onto generalized Ricci symmetric spaces. We obtained a fundamental system for this problem in a form of a system of Cauchy type equations in covariant derivatives depending on no more than 1/2 n2(n+1)+n real parameters. Analogous results are obtained for geodesic mappings of manifolds with affine connection onto equiaffine generalized Ricci symmetric spaces.

Aerodynamic inverse design using multifidelity models and manifold mapping

Aerodynamic inverse design is proposed using multifidelity models and the manifold mapping (MM) technique. Aerodynamic inverse design aims at achieving a target performance characteristic, such as a pressure coefficient distribution of an airfoil or local lift distribution of a wing. Due to the high computational cost of accurate aerodynamic models and the large number of design variables, the overall cost of inverse design can be prohibitive. The MM-based optimization algorithm leverages the speed of the low-fidelity model to accelerate the optimization process, but refers back to the high-fidelity model to ensure an accurate solution. In this work, the MM technique is applied to the characteristic distribution under consideration in each application. In particular, the pressure coefficient distribution is modeled with the MM technique in the case of airfoil inverse design, and the sectional lift distribution in the case of wing design. The proposed method is tested and evaluated on six airfoil inverse design cases and one rectangular wing inverse design case. In the two-dimensional cases, parameterized with eight design variables, direct aerodynamic inverse design using pattern search required 700 to 1200 high-fidelity model evaluations, which took 300 to 700 hours in total. The MM-based design algorithm required less than 20 high-fidelity simulations and 1000 to 2000 low-fidelity evaluations, which took 30 to 90 hours. In the three-dimensional case, parameterized with three design variables, direct aerodynamic inverse design took around 50 hours, whereas the MM-based design needed around six hours.

Tangent maps and tangent groupoid for Carnot manifolds

This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (i.e., maps that are compatible with the Carnot manifold structure). This differential is obtained as a group map between the corresponding tangent groups. We prove that, at every point, a Carnot manifold map is osculated in a very precise way by its Carnot differential at the point. We also show that, in the case of maps between nilpotent graded groups, the Carnot differential is given by the Pansu derivative. Therefore, the Carnot differential is the natural generalization of the Pansu derivative to maps between general Carnot manifolds. Another main result is a construction of an analogue for Carnot manifolds of Connes' tangent groupoid. Given any Carnot manifold (M,H) we get a smooth groupoid that encodes the smooth deformation of the pair M×M to the tangent group bundle GM. This shows that, at every point, the tangent group is the tangent space in a true differential-geometric fashion. Moreover, the very fact that we have a groupoid accounts for the group structure of the tangent group. Incidentally, this answers a well-known question of Bellaïche [11].

Exponentially harmonic maps of complete Riemannian manifolds

We note the existence of exponentially harmonic maps from complete Riemannian manifolds into arbitrary compact Riemannian manifolds. We also investigate the constancy of exponentially harmonic maps with finite exponential energy under some curvature assumptions.

A Method for the Design of Inhomogeneous Materials and Block Structures

A new method for the design of inhomogeneous materials and block structures with complex physicomechanical properties is stated. This approach based on the method of a block element can conveniently complement different numerical approaches used for these purposes and reveal the boundary problem solution properties, which either are complicated or cannot be analyzed by other methods. It is proven that a crucial role in the construction of this new approach is played by packed block elements as topological manifolds with a boundary, which can form similar new topological manifolds as a result of conjugation. It is demonstrated how homeomorphisms, which represent the mappings of topological manifolds of block elements onto real number spaces and further enable the construction of quotient topologies, which “glue” together both block elements and boundary problem solution fragments on block elements as carriers, are constructed for these purposes.

Multiplicity and asymptotic behavior of solutions for Kirchhoff type equations involving the Hardy\u2013Sobolev exponent and singular nonlinearity

In this paper, we study a class of critical elliptic problems of Kirchhoff type: \t\t\t[a+b(∫R3|∇u|2−μu2|x|2dx)2−α2](−Δu−μu|x|2)=|u|2∗(α)−2u|x|α+λf(x)|u|q−2u|x|β,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\biggl[a+b \\biggl( \\int_{\\mathbb{R}^{3}}\\vert \\nabla u\\vert ^{2}-\\mu \\frac{u^{2}}{\\vert x\\vert ^{2}}\\,dx \\biggr)^{\\frac{2-\\alpha }{2}} \\biggr]\\biggl(-\\Delta u- \\mu \\frac{u}{\\vert x\\vert ^{2}}\\biggr) = \\frac{\\vert u\\vert ^{2^{*}(\\alpha )-2}u }{\\vert x\\vert ^{\\alpha }}+\\lambda \\frac{f(x)\\vert u\\vert ^{q-2}u }{\\vert x\\vert ^{\\beta }}, $$\\end{document} where a,b>0, mu in [0,1/4), alpha , beta in [0,2), and qin (1,2) are constants and 2^{*}(alpha )=6-2alpha is the Hardy–Sobolev exponent in mathbb{R}^{3}. For a suitable function f(x), we establish the existence of multiple solutions by using the Nehari manifold and fibering maps. Moreover, we regard b>0 as a parameter to obtain the convergence property of solutions for the given problem as bsearrow 0^{+} by the mountain pass theorem and Ekeland’s variational principle.

Microformal Geometry and Homotopy Algebras

We extend the category of (super)manifolds and their smooth mappings by introducing a notion of microformal or "thick" morphisms. They are formal canonical relations of a special form, constructed with the help of formal power expansions in cotangent directions. The result is a formal category so that its composition law is also specified by a formal power series. A microformal morphism acts on functions by an operation of pullback, which is in general a nonlinear transformation. More precisely, it is a formal mapping of formal manifolds of even functions (bosonic fields), which has the property that its derivative for every function is a ring homomorphism. This suggests an abstract notion of a "nonlinear algebra homomorphism" and the corresponding extension of the classical "algebraic-functional" duality. There is a parallel fermionic version. The obtained formalism provides a general construction of $L_{\infty}$-morphisms for functions on homotopy Poisson ($P_{\infty}$-) or homotopy Schouten ($S_{\infty}$-) manifolds as pullbacks by Poisson microformal morphisms. We also show that the notion of the adjoint can be generalized to nonlinear operators as a microformal morphism. By applying this to $L_{\infty}$-algebroids, we show that an $L_{\infty}$-morphism of $L_{\infty}$-algebroids induces an $L_{\infty}$-morphism of the "homotopy Lie--Poisson" brackets for functions on the dual vector bundles. We apply this construction to higher Koszul brackets on differential forms and to triangular $L_{\infty}$-bialgebroids. We also develop a quantum version (for the bosonic case), whose relation with the classical version is like that of the Schr\"odinger equation with the Hamilton--Jacobi equation. We show that the nonlinear pullbacks by microformal morphisms are the limits at $\hbar\to 0$ of certain "quantum pullbacks", which are defined as special form Fourier integral operators.

A Nonparametric Deep Generative Model for Multimanifold Clustering.

Multimanifold clustering separates data points approximately lying on a union of submanifolds into several clusters. In this paper, we propose a new nonparametric Bayesian model to handle the manifold data structure. In our framework, we first model the manifold mapping function between Euclidean space and topological space by applying a deep neural network, and then construct the corresponding generation process of multiple manifold data. To solve the posterior approximation problem, in the optimization procedure, we apply a variational auto-encoder-based optimization algorithm. Especially, as the manifold algorithm has poor performance on the real dataset where nonmanifold and manifold clusters are appearing simultaneously, we expand our proposed manifold algorithm by integrating it with the original Dirichlet process mixture model. Experimental results have been carried out to demonstrate the state-of-the-art clustering performance.

О строении комплексов m–мерных плоскостей проективного пространства $P^n$, содержащих конечное число торсов

Статья посвящена дифференциальной геометрии подмногообразий многообразий G(m, n), m-мерных плоскостей проективного пространства $P^n$, содержащих конечное число торсов. Для исследования таких подмногообразий используется грассманово отображение многообразия G(m, n) m-мерных плоскостей проективного пространства $P^n$ на (m+1)(n−m)-мерное алгебраическое многообразие $\Omega(m, n)$ пространства $P^N$, где $N=\left(\begin{array}{c}m+1\\n+1\\\end{array}\right)-1$. Это отображение в сочетании с методом внешних форм Э. Картана и методом подвижного репера позволило определить зависимость строения изучаемых многообразий и конфигурации (m − 1)-мерных характеристических плоскостей и (m+ 1)-мерных касательных плоскостей торсов, принадлежащих рассматриваемым многообразиям. We consider the projective differential geometry of m-dimensional plane submanifolds of manifolds G(m, n) in projective space $P^n$ containing a finite number of developable surfaces. To study such submanifolds we use the Grassmann mapping of manifolds G(m, n) of m-dimensional planes in projective space $P^n$ to $(m + 1)(n-m)$-dimensional algebraic manifold $\Omega(m, n)$ of space $P^N$, where $N=\left(\begin{array}{c}m+1\\n+1\\\end{array}\right)-1$. This mapping combined with the method of external Cartan’s forms and moving frame method made it possible to determine the dependence of considered manifolds structure and the configuration of the (m − 1)-dimensional characteristic planes and (m + 1)-dimensional tangential planes of developable surfaces that belong to considered manifolds.

Comparison of three space mapping techniques on electromagnetic design optimization

PurposeThis paper aims to investigate three low-evaluation-budget optimization techniques: output space mapping (OSM), manifold mapping (MM) and Kriging-OSM. Kriging-OSM is an original approach having high-order mapping.Design/methodology/approachThe electromagnetic device to be optimally sized is a five-phase linear induction motor, represented through two levels of modeling: coarse (Kriging model) and fine.The optimization comparison of the three techniques on the five-phase linear induction motor is discussed.FindingsThe optimization results show that the OSM takes more time and iteration to converge the optimal solution compared to MM and Kriging-OSM. This is mainly because of the poor quality of the initial Kriging model. In the case of a high-quality coarse model, the OSM technique would show its domination over the other two techniques. In the case of poor quality of coarse model, MM and Kriging-OSM techniques are more efficient to converge to the accurate optimum.Originality/valueKriging-OSM is an original approach having high-order mapping. An advantage of this new technique consists in its capability of providing a sufficiently accurate model for each objective and constraint function and makes the coarse model converge toward the fine model more effectively.

Lump-loaded antenna optimization by manifold mapping algorithm with method of moments

We propose a new method of manifold mapping optimization method for the lump-loaded antennas to obtain a miniaturization. The surrogate model can be constructed using explicit formulas during optimization iterations. The coarse and fine models of manifold mapping method are evaluated with an adaptive precision method of moments (MoM) simulator. The computation precision is determined by compression thresholds in the adaptive cross approximation (ACA) for the MoM far coupling evaluations. Numerical optimization of the lump-loaded log-periodic dipole antenna is tested to validate the efficiency of the proposed method.

Harmonic maps of finite energy for Finsler manifolds

Abstract In this paper, we study some properties of harmonic maps for Finsler manifolds. Some Liouville theorems on harmonic maps for Finsler manifolds are given. Let M be a complete simply connected Riemannian manifold with non-negative Ricci curvature and M ¯ be a complete Berwald manifold with non-positive flag curvature. The main purpose of this paper is to prove that there exists no non-degenerate harmonic map ϕ from M to M ¯ with ∫ S M e ( ϕ ) d V S M ∞ , which generalizes the result of Schoen and Yau (1976) from Riemannian manifolds to Berwald manifolds.

Geodesic mappings of manifolds with affine connection onto the Ricci symmetric manifolds

In the present paper we investigate geodesic mappings of manifolds with affine connection onto Ricci symmetric manifolds which are characterized by the covariantly constant Ricci tensor. We obtained a fundamental system for this problem in a form of a system of Cauchy type equations in covariant derivatives depending on no more than n(n+1) real parameters. Analogous results are obtained for geodesic mappings of manifolds with affine connection onto symmetric manifolds.

Existence of multiple solutions for a p-Kirchhoff problem with the non-linear boundary condition

ABSTRACTIn this paper, using the Nehari manifold and fibering maps, we study the existence of at least two positive solutions for the following p-Kirchhoff equationwhere is a bounded domain, with a, b, , is the p-Laplacian operator, is a parameter.