Algebraic identities for the Nijenhuis tensors

The general algebraic identities are discovered for the Nijenhuis and Haantjes tensors on an arbitrary manifold Mn. For n=3, the special algebraic identities involving the symmetric bilinear form H(u,v) are derived.     

The existence of riemann invariants in the one-dimensional equations of the non-linear theory of elasticity

The conditions for the existence of Riemann invariants of a one-dimensional system of equations of the non-linear theory of elasticity are investigated. Haantjes' diagonalization criterion is used to determine the form of the elastic potential for which the system has six Riemann invariants or three Riemann invariants (for waves which propagate in one direction). In particular, it is shown that the Haantjes criterion is satisfied and there are three Riemann invariants in the case of the elastic potential for slightly-non-linear weakly-anisotropic elastic media [1–3]. A procedure for computing Riemann invariants is described. The Riemann invariants are computed approximately for a form of elastic potential which satisfies the Haantjes criterion.     

Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

The integrability of an m-component system of hydrodynamic type, u _t = V(u)u _x , by the generalized hodograph method requires the diagonalizability of the m ×  m matrix V(u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains—infinite-component systems of hydrodynamic type for which the ∞ ×  ∞ matrix V(u) is ‘sufficiently sparse’. For such systems the Haantjes tensor is well-defined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor. We prove that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability.     

Isometries,submetries and distance coordinates on Finsler manifolds

This paper considers fundamental issues related to Finslerian isometries, submetries, distance and geodesics. It is shown that at each point of a Finsler manifold there is a distance coordinate system. Using distance coordinates, a simple proof is given for the Finslerian version of the Myers–Steenrod theorem and for the differentiability of Finslerian submetries.     

Introduction of local coordinates for a countable system of differential equations in the neighborhood of an invariant manifold

We establish conditions of the existence of local coordinates for a countable system of differential equations in the neighborhood of an invariant manifold and present the form of this system in these coordinates.     

Manifold-theoretic compactifications of configuration spaces

We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to Fulton–MacPherson and Axelrod–Singer in the setting of smooth manifolds, as well as a simplicial variant of this compactification initiated by Kontsevich. Our constructions are elementary and give simple global coordinates for the compactified configuration space of a general manifold embedded in Euclidean space. We stratify the canonical compactification, identifying the difieomorphism types of the strata in terms of spaces of configurations in the tangent bundle, and give completely explicit local coordinates around the strata as needed to define a manifold with corners. We analyze the quotient map from the canonical to the simplicial compactification, showing it is a homotopy equivalence. Using global coordinates we define projection maps and diagonal maps, which for the simplicial variant satisfy cosimplicial identities.     

Flat coordinates for Saito Frobenius manifolds and string theory

We investigate the connection between the models of topological conformal theory and noncritical string theory with Saito Frobenius manifolds. For this, we propose a new direct way to calculate the flat coordinates using the integral representation for solutions of the Gauss–Manin system connected with a given Saito Frobenius manifold. We present explicit calculations in the case of a singularity of type A _n . We also discuss a possible generalization of our proposed approach to SU(N) _k /(SU(N) _k+1 × U(1)) Kazama–Suzuki theories. We prove a theorem that the potential connected with these models is an isolated singularity, which is a condition for the Frobenius manifold structure to emerge on its deformation manifold. This fact allows using the Dijkgraaf–Verlinde–Verlinde approach to solve similar Kazama–Suzuki models.     

On the Construction of Geometric Integrators in the RKMK Class

We consider the construction of geometric integrators in the class of RKMK methods. Any differential equation in the form of an infinitesimal generator on a homogeneous space is shown to be locally equivalent to a differential equation on the Lie algebra corresponding to the Lie group acting on the homogeneous space. This way we obtain a distinction between the coordinate-free phrasing of the differential equation and the local coordinates used. In this paper we study methods based on arbitrary local coordinates on the Lie group manifold. By choosing the coordinates to be canonical coordinates of the first kind we obtain the original method of Munthe-Kaas [16]. Methods similar to the RKMK method are developed based on the different coordinatizations of the Lie group manifold, given by the Cayley transform, diagonal Padé approximants of the exponential map, canonical coordinates of the second kind, etc. Some numerical experiments are also given.     

Haantjes Manifolds with Symmetry

We introduce the definition of Haantjes manifolds with symmetry and explain why these manifolds appear in the theory of integrable systems of hydrodynamic type and in topological field theories.     

Invariance of the cone algebra without asymptotics

Let B be a manifold with conical singularities, and denote by the smooth bounded manifold with cylindrical ends obtained by blowing up near the singularities.B.-W. Schulze has developed a framework for a pseudodifferential calculus on B by defining various classes of distribution spaces and operator algebras, working in fixed coordinates on the manifold . I am showing here that the Mellin Sobolev spaces without asymptotics, the cone algebra without asymptotics, and its ideal of smoothing operators are independent of the choice of coordinates and therefore may be considered intrinsic objects for manifolds with conical singularities.     

Manifold construction based on local distance invariance

This paper presents a distance invariant manifold that preserves the neighborhood relations among data patterns. All patterns have their corresponding cells in the manifold space. The constellation of neighborhood cells closely resembles that of patterns. The manifold is invariant under the translation, rotation and scale of the pattern coordinates. The neighborhood relations among cells are adjusted and improved in each iteration according to the reduction of the distance preservation energy.     

Variational method for adaptive mesh generation

A variational method is suggested for generating adaptive grids composed of hexahedral cells. The method is based on the minimization of a functional written on a manifold in a space whose variables are usual spatial coordinates in a physical domain and the components of a monitor vector function. A grid is constructed in the manifold, and its projection onto the physical domain yields an adaptive grid. Examples of adaptive grid generation are given.     

The affine connection in the normal coordinates

We propose a method to calculate the derivatives of the Christoffel symbols in the normal coordinates on a smooth manifold with symmetric affine connection. As an application, we construct an algorithm for computing the Taylor series of the double exponential map.     

Transition Manifolds of Complex Metastable Systems

We consider complex dynamical systems showing metastable behavior, but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.     

Harmonic analysis for star graphs and the spherical coordinate trapezoidal rule

Novel ideas in harmonic analysis are used to analyze the trapezoidal rule integration for two spheres. Sampling in spherical coordinates links three levels of harmonic analysis. Eigenfunctions of a nonstandard manifold Laplacian descend by restriction, first to a differential graph Laplacian, and then to difference operators. Trapezoidal rule integration with appropriate sampling is exact on eigenspaces of the manifold Laplacian, a fact which leads to trapezoidal rule error estimates on Sobolev-style spaces of functions. Singular functions with accurate trapezoidal rule integrals are identified, and a simplified analysis of smooth function numerical integration is provided.     

Two integrable systems with integrals of motion of degree four

We discuss the possibility of using second-order Killing tensors to construct Liouville-integrable Hamiltonian systems that are not Nijenhuis integrable. As an example, we consider two Killing tensors with a nonzero Haantjes torsion that satisfy weaker geometric conditions and also three-dimensional systems corresponding to them that are integrable in Euclidean space and have two quadratic integrals of motion and one fourth-order integral in momenta.     

Criteria for existence of a Hamiltonian structure

The necessary and sufficient conditions are derived for the existence of a Hamiltonian structure for 3-component non-diagonalizable systems of hydrodynamic type. The conditions are formulated in terms of tensor invariants defined by the metric h _ij (u) constructed from the Haantjes (1,2)-tensor.     

Bifurcation in a Quasilinear Variational Problem on a One-Dimensional Manifold

Necessary and sufficient conditions for bifurcation in a variational problem on a one-dimensional closed manifold are obtained. In local coordinates, this problem corresponds to a semilinear 4th-order differential equation and can be regarded as a model equation of problems in the theory of shells. Bibliography: 5 titles.     

黎曼流形上的法坐標

<正> 1952年P.Hartman和A.Wintner建立了下述定理:設連續函數E(u,ν),F(u,ν),G(u,ν)在點(0,0)的鄰域內满足條件