Dimensional Manifold Research Articles

Morse – Smale 3-Diffeomorphisms with Saddles of the Same Unstable Manifold Dimension

In this paper, we consider a class of Morse – Smale diffeomorphisms defined on a closed 3-manifold (not necessarily orientable) under the assumption that all their saddle points have the same dimension of the unstable manifolds. The simplest example of such diffeomorphisms is the well-known “source-sink” or “north pole – south pole” diffeomorphism, whose non-wandering set consists of exactly one source and one sink. As Reeb showed back in 1946, such systems can only be realized on the sphere. We generalize his result, namely, we show that diffeomorphisms from the considered class also can be defined only on the 3-sphere.

On some important characterizations of Lorentz para-Kenmotsu manifolds on some special curvature tensors

In this paper, some properties of Lorentz para-Kenmotsu manifolds are studied using specified curvature tensors. The Lorentz para-Kenmotsu manifold is investigated in terms of the curvature tensors [Formula: see text] and [Formula: see text]. Initially, the tensor-based characterization of semisymmetric Lorentz para-Kenmotsu manifolds is studied. Subsequently, we consider the Lorentzian para-Kenmotsu manifold, which admits almost [Formula: see text]-Ricci solitons via these curvature tensors. According to the [Formula: see text] and [Formula: see text] curvature tensors, Ricci pseudosymmetry notions of Lorentzian para-Kenmotsu manifolds accepting [Formula: see text]-Ricci soliton have been developed. Following that, required conditions for the Lorentzian para-Kenmotsu manifold, admitting [Formula: see text]-Ricci soliton to be Ricci semisymmetric, are presented based on the curvature tensors chosen. Further, various characterizations are provided, and classifications are made under certain conditions. Finally, the characterizations of the invariant submanifolds of Lorentz para-Kenmotsu manifold on the [Formula: see text] and [Formula: see text] curvature tensors are investigated. We obtained the necessary and sufficient conditions for an invariant submanifold of a para-Kenmotsu to be [Formula: see text] and [Formula: see text] pseudoparallel.

On metric and cohomological properties of Oeljeklaus-Toma manifolds

On metric and cohomological properties of Oeljeklaus-Toma manifolds

Lorentzian Para-Sasakian Manifolds and *-Ricci Solitons

We study the properties of Lorentzian para-Sasakian manifolds endowed with ∗-Ricci solitons and gradient ∗-Ricci solitons. Finally, the existence of ∗-Ricci soliton on a 4-dimensional Lorentzian para-Sasakian manifold is proved by constructing a non-trivial example

Теоремы о дифференцируемых сферах для многообразий с ограниченными сверху кривизнами Риччи

In the present paper, we prove that if is an -dimensional compact Riemannian manifold and if where , and are the sectional and Ricci curvatures of respectively, then is diffeomorphic to a spherical space form where is a finite group of isometries acting freely. In particular, if is simply connected, then it is diffeo­mor­phic to the Euclidian sphere

Large-scale geometry of the Universe

The large scale geometry of the late Universe can be decomposed as ℝ × Σ3, where ℝ stands for cosmic time and Σ3 is the three dimensional spatial manifold. We conjecture that the geometry of the Universe's spatial section Σ3 conforms with the Thurston-Perelman theorem, according to which the geometry of Σ3 is either one of the eight geometries from the Thurston geometrization conjecture, or a combination of Thurston geometries smoothly sewn together. We assume that topology of individual geometries plays no observational role, i.e. the size of individual geometries is much larger than the Hubble radius today. We investigate the dynamics of each of the individual geometries by making use of the simplifying assumption that our local Hubble patch consists of only one such geometry, which is approximately homogeneous on very large scales, but spatial isotropy is generally violated.Spatial anisotropies grow in time in decelerating universes, but they decay in accelerating universes. The thus-created anisotropy problem can be solved by a period of primordial inflation, akin to how the flatness problem is solved. Therefore, as regards Universe's large scale geometry, any of the Thurston's geometries should be considered on a par with Friedmann's geometries.We consider two observational methods that can be used to test our conjecture: one based on luminosity distance and one on angular diameter distance measurements, but leave for the future their detailed forecasting implementations.

Certain results on tangent bundle endowed with generalized Tanaka Webster connection (GTWC) on Kenmotsu manifolds

<p>This work studies the complete lifts of Kenmotsu manifolds associated with the generalized Tanaka-Webster connection (GTWC) in the tangent bundle. Using the GTWC, this study explores the complete lifts of various curvature tensors and geometric structures from Kenmotsu manifolds to their tangent bundles. Specifically, it examines the complete lifts of Ricci semi-symmetry, the projective curvature tensor, $ \Phi $-projectively semi-symmetric structures, the conharmonic curvature tensor, the concircular curvature tensor, and the Weyl conformal curvature tensor. Additionally, the research delves into the complete lifts of Ricci solitons on Kenmotsu manifolds with the GTWC within the tangent bundle framework, providing new insights into their geometric properties and symmetries in the lifted space. The data on the complete lifts of the Ricci soliton in Kenmotsu manifolds associated with the GTWC in the tangent bundle are also investigated. An example of the complete lifts of a $ 5 $-dimensional Kenmotsu manifold is also included.</p>

On the differential geometry of some classes of infinite dimensional manifolds

On the differential geometry of some classes of infinite dimensional manifolds

Some Curvature Conditions on 3-Dimensional Quasi-Sasakian Manifolds Admitting Conformal Ricci Soliton

In this paper, we examine 3-dimensional quasi-Sasakian manifold admitting conformal Ricci soliton. We give some theorems for $W_{0}^{*}$ flat, $\xi-W_{0}^{*}$ flat and $\phi-W_{0}^{*}$ semisymmetric 3-dimensional quasi-Sasakian manifold admitting conformal Ricci soliton. Also we study conformal Ricci soliton on a 3-dimensional quasi-Sasakian manifold satisfying the conditions $W_{0}^{*}(\xi,X).S=0$ and $R(\xi,X).W_{0}^{*}=0$.

Geometric phases of nonlinear elastic [formula omitted]-rotors via Cartan’s moving frames

We study the geometric phases of nonlinear elastic N-rotors with continuous rotational symmetry. In the Hamiltonian framework, the geometric structure of the phase space is a principal fiber bundle, i.e., a base, or shape manifold B, and fibers F along the symmetry direction attached to it. The symplectic structure of the Hamiltonian dynamics determines the connection and curvature forms of the shape manifold. Using Cartan’s structural equations with zero torsion we find an intrinsic (pseudo) Riemannian metric for the shape manifold. Without lose of generality, we show that one has the freedom to define the rotation sign of the total angular momentum A of the elastic rotors as either positive or negative, e.g., counterclockwise or clockwise, respectively, or viceversa. This endows the base manifold B with two distinct metrics both compatible with the geometric phase. In particular, the metric is pseudo-Riemannian if A<0, and the shape manifold is a 2D Robertson-Walker spacetime with positive curvature. For A>0, the shape manifold is the hyperbolic plane H2 with negative curvature. We then generalize our results to free elastic N-rotors. We show that the associated shape manifold B is reducible to the product manifold of N−1 hyperbolic planes H2 (A>0), or 2D Robertson-Walker spacetimes (A<0) depending on the convection used to define the rotation sign of the total angular momentum. We then consider elastic N-rotors subject to time-dependent self-equilibrated moments. The N-dimensional shape manifold of the extended autonomous system has a structure similar to that of the (N−1)-dimensional shape manifold of free elastic rotors. The Riemannian structure of the shape manifold provides an intrinsic measure of the closeness of one shape to another in terms of curvature, or induced geometric phase.

Multiplicity results for nonhomogenous elliptic equation involving the generalized Paneitz-Branson operator

"Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3 without boundary ∂M, we consider the multiplicity result of solutions of the following nonhomogenous fourth order elliptic equation involving the generalized Paneitz-Branson operator, Pg (u) = f (x) |u|2 −2 u + h(x). Under some conditions and using critical points theory, we prove the existence of two distinct solutions of the above equation. At the end, we give a geometric example when the equation has negative and positive solutions. Keywords: Riemannian manifold, multiplicity result, nonhomogeneous, Paneitz-Branson operator, critical points theory."

The upper bound of the harmonic mean of the Steklov eigenvalues in curved spaces

AbstractConsidering an ‐dimensional Riemannian manifold whose sectional curvature is bounded above by and Ricci curvature is bounded below by , we obtain an upper bound of the harmonic mean of the first nonzero Steklov eigenvalues for domains contained in . This can be viewed as certain isoperimetric inequality and generalizes the result on comparing the first nonzero Steklov eigenvalues for such domains (Li–Wang–Wu, 2020, arXiv:2003.03093).

Estimates of Bergman kernels and Bergman metric on compact Picard surfaces

Let Γ⊂SU((2,1),C) be a torsion-free cocompact subgroup. Let B2 denote the 2-dimensional complex ball endowed with the hyperbolic metric μhyp, and let XΓ:=Γ﹨B2 denote the quotient space, which is a compact complex manifold of dimension 2. Let Λ:=ΩXΓ2 denote the line bundle on XΓ, whose sections are holomorphic (2,0)-forms. For any k≥1, the hyperbolic metric induces a point-wise metric on H0(XΓ,Λ⊗k), which we denote by |⋅|hyp. For any k≥1, let BΛk denote the Bergman kernel of the complex vector space H0(XΓ,Λ⊗k). For any k≥3, and z,w∈XΓ, the first main result of the article is the following off-diagonal estimate of the Bergman kernel BΛk|BΛk(z,w)|hyp=OXΓ(k2cosh3k−8⁡(dhyp(z,w)/2)), where dhyp(z,w) denotes the geodesic distance between the points z and w on XΓ, and the implied constant depends only on the Picard surface XΓ.For any k≥1, let μberk(z):=−i2π∂z∂z‾log⁡|BΛk(z,z)|hyp denote the Bergman metric associated the line bundle Λ⊗k, and let μberk,vol denote the associated volume form. For k≫1 sufficiently large, and ϵ>0, the second main result of the article is the following estimatesupz∈XΓ⁡|μberk,vol(z)μhypvol(z)|=OXΓ,ϵ(k4+ϵ), where μhypvol denotes the volume form associated to the hyperbolic metric μhyp, and the implied constant depends on the Picard surface XΓ, and on the choice of ϵ>0.

SPHERES AND CIRCLES WITH RESPECT TO AN INDEFINITE METRIC ON A RIEMANNIAN MANIFOLD WITH A SKEW-CIRCULANT STRUCTURE

We study hyper-spheres, spheres and circles, with respect to an indefinite metric, in a single tangent space on a 4-dimensional differentiable manifold. The manifold is equipped with a positive definite metric and an additional tensor structure of type (1, 1). The fourth power of the additional structure is minus identity and its components form a skew-circulant matrix in some local coordinate system. The both structures are compatible and they determine an associated indefinite metric on the manifold.

PROPAGATION OF POLARIZED LIGHT AND ELECTROMAGNETIC CURVES IN THE OPTICAL FIBER IN WALKER 3-MANIFOLDS

In the present paper, we define the three cases of the geometric phase equations associated with a monochromatic linearly polarized light wave traveling along an optical fiber in three dimensional Walker manifold (M, g^ε_f). Walker manifolds have many applications in mathematics and theoretical physics. We are working in the context of a pseudo-Riemannian manifold (i.e. a manifold equipped with a non-degenerate arbitrary signature metric tensor). That is, we generalize the motion of the light wave in the optical fiber and the associated electromagnetic curves that describe the motion of a charged particle under the influence of an electromagnetic field over a Walker space defined as a pseudo-Riemannian manifold with a light-like distribution, parallel to the Levi-Civita junction. These manifolds (especially Lorentzian) are important in physics because of their applications in general relativity. Then, we obtain the Rytov curves related to the cases of geometric phase models. Moreover, we give some examples and visualize the evolution of the electric field along the optical fiber in (M, g^ε_f ) via MAPLE program.

A DIFFERENTIABLE STRUCTURE ON A FINITE DIMENSIONAL REAL VECTOR SPACE AS A MANIFOLD

There are three conditions for a topological space to be said a topological manifold of dimension : Hausdorff space, second-countable, and the existence of homeomorphism of a neighborhood of each point to an open subset of or -dimensional locally Euclidean. The differentiable structure is given if the intersection of two charts is an empty chart or its transition map is differentiable. In this article, we study a differentiable manifold on finite dimensional real vector spaces. The aim is to prove that any finite-dimensional vector space is a differentiable manifold. First of all, it is proved that a finite dimensional vector space is a topological manifold by constructing a norm as its topology. Given a metric which is induced by a norm. Two norms on a finite dimensional vector space are always equivalent and they are determine the same topology. Secondly, it is proved that the transition map in the finite dimensional vector space is differentiable. As conclusion, we have that any finite dimensional vector space with independent norm topology choice is a differentiable manifold. As a matter of discussion, it can be studied that the vector space of all linear operators of a finite dimensional vector space has a differentiable manifold structure as well.

Existence and uniqueness results for a singular Kirchhoff type equation on a closed manifold

Using the variational methods and the critical points theory, we prove the existence and the uniqueness of a positive solution for a singular Kirchhoff type equation on a closed Riemannian manifold of dimension N≥3. At the end, we give a geometric application involving the conformal Laplacian.

On the Pontrjagin classes of spray manifolds

Abstract The characterization of projectively flat Finsler metrics on an open subset in $R^n$ is the Hilbert’s fourth problem in the regular case. Locally projectively flat Finsler manifolds form an important class of Finsler manifolds. Every Finsler metric induces a spray on the manifold via geodesics. Therefore, it is a natural problem to investigate the geometric and topological properties of manifolds equipped with a spray. In this paper, we study the Pontrjagin classes of a manifold equipped with a locally projectively flat spray and show that such manifold must have zero Pontrjagin classes.

Coarse-Graining and Forecasting Atomic Material Simulations with Descriptors.

Atomic simulations of materials require significant resources to generate, store, and analyze. Here, descriptor functions are proposed as a general, metric latent space for atomic structures, ideal for use in large-scale simulations. Descriptors can regress a broad range of properties, including character-dependent dislocation densities, stress states, or radial distribution functions. A vector autoregressive model can generate trajectories over yield points, resample from new initial conditions and forecast trajectory futures. A forecast confidence, essential for practical application, is derived by propagating forecasts through the Mahalanobis outlier distance, providing a powerful tool to assess coarse-grained models. Application to nanoparticles and yielding of nanoscale dislocation networks confirms low uncertainty forecasts are accurate and resampling allows for the propagation of smooth property distributions. Yielding is associated with a collapse in the intrinsic dimension of the descriptor manifold, which is discussed in relation to the yield surface.

Lorentzian manifolds: A characterization with a type of semi-symmetric non-metric connection

This paper characterizes the Lorentzian manifolds endowed with a semi-symmetric non-metric [Formula: see text]-connection (briefly, ssnmρc). First, the existence of semi-symmetric non-metric connection (ssnmc) on Lorentzian manifold is established, and it is shown that an [Formula: see text]-dimensional Lorentzian manifold equipped with an ssnmρc is a generalized Robertson–Walker spacetime. We also establish the condition for a Lorentzian manifold together with an ssnmρc to be a Robertson–Walker spacetime. In this series, the properties of Ricci semi-symmetric Lorentzian manifold endowed with an ssnmρc, almost Ricci solitons and gradient Ricci solitons are explored. Finally, a non-trivial example of Lorentzian manifold admits an ssnmρc, which is constructed to verify some of our results.