N-dimensional Manifold Research Articles

Ricci flow on surfaces of revolution: an extrinsic view

A Ricci flow (M, g(t)) on an n-dimensional Riemannian manifold M is an intrinsic geometric flow. A family of smoothly embedded submanifolds \((S(t), g_E)\) of a fixed Euclidean space \(E = \mathbb {R}^{n+k}\) is called an extrinsic representation in \(\mathbb {R}^{n+k}\) of (M, g(t)) if there exists a smooth one-parameter family of isometries \((S(t), g_E) \rightarrow (M, g(t))\). When does such a representation exist? We formulate a new way of framing this question for Ricci flows on surfaces of revolution immersed in \(\mathbb {R}^3\). This framework allows us to construct extrinsic representations for the Ricci flow initialized by any compact surface of revolution immersed in \(\mathbb {R}^3\). In particular, we exhibit the first explicit extrinsic representations in \(\mathbb {R}^4\) of the Ricci flows initialized by toroidal surfaces of revolution immersed in \(\mathbb {R}^3\).

The Han-Li conjecture in constant scalar curvature and constant boundary mean curvature problem on compact manifolds

The Han-Li conjecture states that: Let (M,g0) be an n-dimensional (n≥3) smooth compact Riemannian manifold with boundary having positive (generalized) Yamabe constant and c be any real number, then there exists a conformal metric of g0 with scalar curvature 1 and boundary mean curvature c. Combining with Z.C. Han and Y.Y. Li's results, we answer this conjecture affirmatively except for the case that n≥8, the boundary is umbilic, the Weyl tensor of M vanishes on the boundary and has an interior non-zero point.

Horosphere slab separation theorems in manifolds without conjugate points

Let mathcal {W}^{n} be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let Win mathcal {W}^{n} and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in Win mathcal {W} are considered.

Inertial manifolds for the 3D modified-Leray-α model

In this paper, we investigate the critical modified-Leray-α model in T3 and prove the existence of an N-dimensional inertial manifold for this problem.

There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with n Complete Light-Like Geodesics

In the present paper, we study conformal mappings between a connected n-dimension pseudo-Riemannian Einstein manifolds. Let g be a pseudo-Riemannian Einstein metric of indefinite signature on a connected n-dimensional manifold M. Further assume that there is a point at which not all sectional curvatures are equal and through which in linearly independent directions pass n complete null (light-like) geodesics. If, for the function ψ the metric ψ − 2 g is also Einstein, then ψ is a constant, and conformal mapping is homothetic. Note that Kiosak and Matveev previously assumed that all light-lines were complete. If the Einstein manifold is closed, the completeness assumption can be omitted (the latter result is due to Mikeš and Kühnel).

A Black Hole at the Center of Earth Plays the Role of the Biggest System of Telecommunication for Connecting DNAs, Dark DNAs and Molecules of Water on 4+N- Dimensional Manifold.

Recently, some scientists from NASA have claimed that there may be a black hole like structure at the centre of the earth. We show that the existence of life on the earth may be a reason that this black hole like object is a black brane that has been formed from biological materials like DNA. Size of this DNA black brane is 109 times longer than the size of the earth’s core and compacted interior it. By compacting this long object, a curved space-time emerges, and some properties of black holes emerge. This structure is the main cause of the emergence of the large temperature of the core, magnetic field around the earth and gravitational field for moving around the sun. Also, this structure produces some waves which act like topoisomerase in biology and read the information on DNAs. However, on the four-dimensional manifold, DNAs are contracted at least four times around various axis’s and waves of earth couldn’t read their information. While, by adding extra dimensions on 4 +n-dimensional manifold, the separation distance between particles increases and all of the information could be recovered by waves. For this reason, each DNA has two parts which one can be seen on the four-dimensional universe, and another one has existed in extra dimensions, and only it’s e_ects is observed. This dark part of DNA called as a dark DNA in an extra dimension. These dark DNAs not only exchange information with DNAs but also are connected with some of the molecules of water and helps them to store information and have memory. Thus, the earth is the biggest system of telecommunication which connects DNAs, dark DNAs and molecules of water.

Bôcher-type theorem on n-dimensional manifolds with conical metric

We generalize the Bôcher-type theorem and give a sharp characterization of the behavior at the isolated singularities of a solution bounded on one side for the equation Δ g u = 0 \Delta _g u =0 on singular manifolds with conical metrics. Furthermore, we also obtain a Liouville-type result which demonstrates that the fundamental solution is the unique nontrivial solution of div ⁡ ( | x | θ ∇ u ) = 0 \operatorname {div}(|x|^\theta \nabla u)=0 in R n ∖ { 0 } \mathbb {R}^n\setminus \{0\} that is bounded on one side in both a neighborhood of the origin as well as at infinity.

Klein–Gordon–Maxwell–Proca type systems in the electro-magneto-static case: the high dimensional case

We investigate Klein–Gordon–Maxwell–Proca type systems in the context of closed n-dimensional manifolds with $$n \ge 4$$ . We prove existence of solutions and compactness of the system both in the subcritical and in the critical case.

Riesz transforms on a class of non-doubling manifolds

We consider a class of manifolds obtained by taking the connected sum of a finite number of N-dimensional Riemannian manifolds of the form where is a compact manifold, with the product metric. The case of greatest interest is when the Euclidean dimensions ni are not all equal. This means that the ends have different ‘asymptotic dimension’, and implies that the Riemannian manifold is not a doubling space. We completely describe the range of exponents p for which the Riesz transform on is a bounded operator on Namely, under the assumption that each ni is at least 3, we show that Riesz transform is of weak type (1, 1), is continuous on Lp for all and is unbounded on Lp otherwise. This generalizes results of the first-named author with Carron and Coulhon devoted to the doubling case of the connected sum of several copies of Euclidean space and of Carron concerning the Riesz transform on connected sums.

Formal and Non-Archimedean Structures of Dynamic Systems on Manifolds

New results are presented and a brief review is given for new methods of the theory of dynamic systems on manifolds over local fields and formal groups over local rings. For the analysis of n-dimensional manifolds and dynamic systems on such manifolds, formal structures are used, in particular, n-dimensional formal groups. Infinitesimal deformations are presented in terms of formal groups. The well-known one-dimensional case is extended and n-dimensional (n ≥ 1) analytic mappings of an open p-adic polydisc (n-disk) \( {D}_p^n \) are considered. The n-dimensional analogs of modules arising in formal and non-Archimedean dynamic systems are introduced and investigated and their formal-algebraic structure is presented. Rigid structures, objects, and methods are outlined. From the point of view of systems analysis, new, namely formal and non-Archimedean, faces and structures of systems, mappings and iterations of mappings between these faces and structures are introduced and investigated.

Geometric and Functional Inequalities on Finsler Manifolds

In this paper, we give an (integrated) \(p(>1)\)-Bochner–Weitzenbock formula and the p-Reilly type formula on Finsler manifolds. As applications, we obtain the p-Poincare inequality on an n-dimensional compact Finsler manifold without boundary or with convex boundary under the assumption that Ricci\(_N\ge K\) for \(N\in [n, \infty ]\) and \(K\in \mathbb {R}\). In fact, we have the sharp p-Poincare inequality on such a manifold. Based on this, we further prove the existence of two types of optimal \((\mathfrak p, \mathfrak q)\)-Sobolev inequalities on compact Finsler manifolds with Ric\(_N\ge K\) for \(N\in [n, \infty )\) and \(K\in \mathbb {R}\). In particular, when \(K>0\), we establish the sharp \((\mathfrak p, 2)\)-Sobolev inequality for \(2\le \mathfrak p\le 2N/(N-2)\). With this, we obtain the sharp lower bound estimate for the first eigenvalue \(\lambda _1\) of the Finsler Laplacian and characterize the manifold on which \(\lambda _1\) reaches its extremum, generalizing the well known Lichnerowicz–Obata’s results (in: Geometrie des groupes de transformations, Dunod, Paris, 1958; J Math Soc Jpn 14:333–340, 1962).

On Compact Riemannian Manifolds with Harmonic Weyl Curvature

We give some rigidity theorems for an n-dimensional ( $$n\ge 4$$ ) compact Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and positive constant $$\sigma _2$$ curvature. Moreover, we prove that a 4-dimensional compact locally conformally flat Riemannian manifold with positive scalar curvature and positive constant $$\sigma _2$$ curvature is isometric to a quotient of the round $$\mathbb {S}^4$$ .

Semi-concurrent vector fields in Finsler geometry

In the present paper, we give an answer to a question which is closely related to doubly warped product of Finsler metrics: ‘‘For each n, is there an n-dimensional Finsler manifold (M,F), admitting a non-constant smooth function f on M such that ∂f∂xi∂gij∂yk=0?”. We relate the preceding mentioned condition to different concepts appeared and studied in Finsler geometry. We introduce and investigate the notion of a semi concurrent vector field on a Finsler manifold. We show that some special Finsler manifolds admitting such vector fields turn out to be Riemannian. We prove that Tachibana's characterization of Finsler manifolds admitting a concurrent vector field leads to Riemannian metrics. Various examples for conic Finsler spaces that admit semi-concurrent vector field are presented.

Elliptic equation with critical and negative exponents involving the GJMS operator on compact Riemannian manifolds

In this paper we consider an elliptic equation with critical and negative exponents involving the GJMS (Graham–Jenne–Mason–Sparling) operator of order k on n-dimensional compact Riemannian manifolds with 2k<n. Multiplicity and nonexistence results are established.

New geometric aspects of Moser–Trudinger inequalities on Riemannian manifolds: the non-compact case

In the first part of the paper we investigate some geometric features of Moser–Trudinger inequalities on complete non-compact Riemannian manifolds. By exploring rearrangement arguments, isoperimetric estimates, and gluing local uniform estimates via Gromov's covering lemma, we provide a Coulhon, Saloff-Coste and Varopoulos type characterization concerning the validity of Moser–Trudinger inequalities on complete non-compact n-dimensional Riemannian manifolds (n≥2) with Ricci curvature bounded from below. Some sharp consequences are also presented both for non-negatively and non-positively curved Riemannian manifolds, respectively. In the second part, by combining variational arguments and a Lions type symmetrization-compactness principle, we guarantee the existence of a non-zero isometry-invariant solution for an elliptic problem involving the n-Laplace–Beltrami operator and a critical nonlinearity on n-dimensional homogeneous Hadamard manifolds. Our results complement in several directions those of Y. Yang [J. Funct. Anal., 2012].

On a Class of P-Kenmotsu Manifolds Admitting Weyl-projective Curvature Tensor of Type (1, 3)

We study a class of para-Kenmotsu manifolds admitting Weyl-projective curvature tensor of type (1, 3). At the end, it is shown that an n-dimensional (n > 2) P-Kenmotsu manifold is Ricci semisymmetric if and only if it is an Einstein manifold.

Contact metric manifolds with large automorphism group and (κ, µ)-spaces

AbstractWe discuss the classifiation of simply connected, complete (κ, µ)-spaces from the point of view of homogeneous spaces. In particular, we exhibit new models of (κ, µ)-spaces having Boeckx invariant -1. Finally, we prove that the number ${{(n + 1)(n + 2)} \over 2}$ is the maximum dimension of the automorphism group of a contact metric manifold of dimension 2n +1, n ≥ 2, whose symmetric operator h has rank at least 3 at some point; if this dimension is attained, and the dimension of the manifold is not 7, it must be a (κ, µ)-space. The same conclusion holds also in dimension 7 provided the almost CR structure of the contact metric manifold under consideration is integrable.

О дифференциальных уравнениях тензоров кривизны фундаментально-групповой и аффинной связностей

The principal bundle is considered, the base of which is an n-dimensional smooth manifold, and the typical fiber is an r-fold Lie group. Structure equations for the forms of the fundamental group and affine connections are given, each of which contains the corresponding components of the curvature tensor. For each connection, an approach is shown that allows to find the differential equations for the components of the curvature tensor of the corresponding connection in a faster way than by differentiating the expressions of these objects in terms of the connection objects and their Pfaffian derivatives. The method consists in successively solving cubic equations, first by Laptev’s lemma, then by Cartan’s lemma. Taking into account the comparisons modulo basic forms, we obtain already known results (see [3]). Thus, differential equations are derived for the components of the curvature tensor of the first-order fundamentalgroup connection, as well as for the components of the curvature tensor of the affine connection.

Variation of the first eigenvalue of p-Laplacian on evolving geometry and applications

Let (M,g)be an n-dimensional compact Riemannian manifold whose metric g(t)evolves by the generalised abstractgeometric flow. This paper discusses the variation formulas, monotonicity and differentiability for the first eigenvalue of thep-Laplacian on (M,g(t)). It is shown that the first nonzero eigenvalue is monotonically nondecreasing along the flow undercertain geometric conditions and that it is differentiable almost everywhere. These results provide a unified approach to the study of eigenvalue variations and applications under many geometric flows

Some Remarks on Pohozaev-Type Identities

In this note we present some Pohozaev-type identities that have been recently established in a joint work with Paul Laurain and Tristan Riviere in the framework of half-harmonic maps defined either on the real line or on the unit circle with values into a closed n-dimensional manifold. Weak half-harmonic maps are defined as critical points of the so-called half Dirichlet energy. By using the invariance of the half Dirichlet energy with respect to the trace of the Mobius transformations we derive a countable family of relations involving the Fourier coefficients of weak half-harmonic maps. We also present a short overview of Pohozaev formulas in 2-D in connection with Noether's theorem.