Notion Of Manifold Research Articles

Reshetnyak Rigidity for Riemannian Manifolds

We prove two rigidity theorems for maps between Riemannian manifolds. First, we prove that a Lipschitz map $f:M\to N$ between two oriented Riemannian manifolds, whose differential is almost everywhere an orientation-preserving isometry, is an isometric immersion. This theorem was previously proved using regularity theory for conformal maps; we give a new, simple proof, by generalizing the Piola identity for the cofactor operator. Second, we prove that if there exists a sequence of mapping $f_n:M\to N$, whose differentials converge in $L^p$ to the set of orientation-preserving isometries, then there exists a subsequence converging to an isometric immersion. These results are generalizations of celebrated rigidity theorems by Liouville (1850) and Reshetnyak (1967) from Euclidean to Riemannian settings. Finally, we describe applications of these theorems to non-Euclidean elasticity and to convergence notions of manifolds.

A generalization of manifolds with corners

In conventional Differential Geometry one studies manifolds, locally modelled on Rn, manifolds with boundary, locally modelled on [0,∞)×Rn−1, and manifolds with corners, locally modelled on [0,∞)k×Rn−k. They form categories Man⊂Manb⊂Manc. Manifolds with corners X have boundaries ∂X, also manifolds with corners, with dim∂X=dimX−1.We introduce a new notion of manifolds with generalized corners, or manifolds with g-corners, extending manifolds with corners, which form a category Mangc with Man⊂Manb⊂Manc⊂Mangc. Manifolds with g-corners are locally modelled on XP=HomMon(P,[0,∞)) for P a weakly toric monoid, where XP≅[0,∞)k×Rn−k for P=Nk×Zn−k.Most differential geometry of manifolds with corners extends nicely to manifolds with g-corners, including well-behaved boundaries ∂X. In some ways manifolds with g-corners have better properties than manifolds with corners; in particular, transverse fibre products in Mangc exist under much weaker conditions than in Manc.This paper was motivated by future applications in symplectic geometry, in which some moduli spaces of J-holomorphic curves can be manifolds or Kuranishi spaces with g-corners rather than ordinary corners.Our manifolds with g-corners are related to the ‘interior binomial varieties’ of Kottke and Melrose [20], and the ‘positive log differentiable spaces’ of Gillam and Molcho [6].

On topological filtrations of groups

The (weak) geometric simple connectivity and the quasi-simple filtration are topological notions of manifolds, which may be defined for discrete groups too. It turns out that they are equivalent for finitely presented groups, but the main problem is the absence of examples of groups which do not satisfy them. In this note we study some algebraic classes of groups with respect to these properties.

A high-order numerical manifold method with nine-node triangular meshes

The numerical manifold method (NMM) is a unified framework that is used to describe continuous and discontinuous problems. The NMM is derived based on the finite cover approximation theory and gains its name after the mathematical notion of manifold. It is also a method based on the partition of unity (PU) and it introduces two cover systems: the mathematical cover (MC) and the physical cover (PC). There are two approaches for constructing high-order approximations. The first approach involves a non-constant PU function and non-constant local approximations. This results in the linear dependence (LD) problem and leads to the singularity in a global matrix. The second approach involves a higher PU function and constant local approximations. The increase in the order of approximations should go along with the increase in star but the LD problem can be avoided completely in theory. In this paper, a new high-order NMM with nine-node triangular meshes is proposed. The upgrade from first-order NMM to high-order NMM is illustrated in detail. Moreover, the initial stress matrix is analyzed in detail. The effectiveness and accuracy of the proposed high-order NMM are validated using several typical examples. The proposed high-order NMM supplements the existing family of non-LD high- and low-order NMM under MC with triangular meshes.

Asymptotic Multiphase Solitonlike Solutions of the Cauchy Problem for a Singularly Perturbed Korteweg–de-Vries Equation with Variable Coefficients

UDC 517.9 We describe the set of initial conditions under which the Cauchy problem for a singularly perturbed Korteweg–de-Vries equation with variable coefficients has an asymptotic multiphase solitonlike solution. The notion of manifold of initial values for which the above-mentioned solution exists is proposed for the analyzed Cauchy problem. The statements on the estimation of the difference between the exact and constructed asymptotic solutions are proved for the Cauchy problem.

Lagrangian submanifolds in 3-dimensional complex space forms with isotropic cubic tensor

In this paper, we introduce the notion of manifolds with isotropic cubic tensor. We give a complete classification of the Lagrangian submanifolds in 3-dimensional complex space forms with isotropic cubic tensor.

Rings and spirals in barred galaxies - I. Building blocks

In this paper we present building blocks which can explain the formation and properties both of spirals and of inner and outer rings in barred galaxies. We first briefly summarise the main results of the full theoretical description we have given elsewhere, presenting them in a more physical way, aimed to an understanding without the requirement of extended knowledge of dynamical systems or of orbital structure. We introduce in this manner the notion of manifolds, which can be thought of as tubes guiding the orbits. The dynamics of these manifolds can govern the properties of spirals and of inner and outer rings in barred galaxies. We find that the bar strength affects how unstable the L1 and L2 Lagrangian points are, the motion within the 5A5A5Amanifold tubes and the time necessary for particles in a manifold to make a complete turn around the galactic centre. We also show that the strength of the bar, or, to be more precise, of the non-axisymmetric forcing at and somewhat beyond the corotation region, determines the resulting morphology. Thus, less strong bars give rise to R1 rings or pseudorings, while stronger bars drive R2, R1R2 and spiral morphologies. We examine the morphology as a function of the main parameters of the bar and present descriptive two dimensional plots to that avail. We also derive how the manifold morphologies and properties are modified if the L1 and L2 Lagrangian points become stable. Finally, we discuss how dissipation affects the manifold properties and compare the manifolds in gas-like and in stellar cases. Comparison with observations, as well as clear predictions to be tested by observations will be given in an accompanying paper.