Riemannian Manifold Research Articles

Inequalities for eigenvalues of the bi-drifting Laplacian on bounded domains in complete noncompact Riemannian manifolds and related results

Inequalities for eigenvalues of the bi-drifting Laplacian on bounded domains in complete noncompact Riemannian manifolds and related results

Semi-supervised metric learning incorporating weighted triplet constraint and Riemannian manifold optimization for classification

Semi-supervised metric learning incorporating weighted triplet constraint and Riemannian manifold optimization for classification

Parabolic Schrödinger networks

A Schrödinger network Nq={X,ζ,q} is an infinite graph with a set of vertices X and edges that are countably infinite, a transition function ζ(a,b)≥0 and a function q(a) on X. A perturbed Laplace operator Δq is defined as Δql(a)=Δl(a)−q(a)l(a). A solution h(a) to the equation Δqh(a)=0 is termed a q-average function, and a q-superaverage function v(a) satisfies Δqv(a)≤0. The network is called q-hyperbolic if there exists a function s(a)>0, q-superaverage on X; otherwise, it is a q-parabolic network, also mentioned as a parabolic Schrödinger network. In this paper, we develop a theory of function on parabolic Schrödinger networks by using potential-theoretic techniques associated with q-superaverage and q-average functions, which are similar to the function theory on parabolic Riemannian manifolds.

Substructures of the Weyl group and their physical applications

We study substructures of the Weyl group of conformal transformations of the metric of (pseudo)Riemannian manifolds. These substructures are identified by differential constraints on the conformal factors of the transformations which are chosen such that their composition is associative. Mathematically, apart from rare exceptions, they are partial associative groupoids, not groups, so they do not have an algebra of infinitesimal transformations, but this limitation can be partially circumvented using some of their properties cleverly. We classify and discuss the substructures with two-derivatives differential constraints, the most famous of which being known as the harmonic or restricted Weyl group in the physics literature, but we also show the existence of a lightcone constraint which realizes a proper subgroup of the Weyl group. We then show the physical implications that come from invariance under the two most important substructures, concentrating on classical properties of the energy-momentum tensor and a generalization of the quantum trace anomaly. We also elaborate further on the harmonic substructure, which can be interpreted as partial gauge fixing of full Weyl invariance using BRST methods. Finally, we discuss how to construct differential constraints of arbitrary higher-derivative order and present, as examples, generalizations involving scalar constraints with four and six derivatives.

Asymptotics as s → 0+ of the fractional perimeter on Riemannian manifolds

Asymptotics as s → 0+ of the fractional perimeter on Riemannian manifolds

Invariant Gibbs measure for a Schrödinger equation with exponential nonlinearity

We investigate the invariance of the Gibbs measure for the fractional Schrödinger equation of exponential type (expNLS) i∂tu+(−Δ)α2u=2γβeβ|u|2u on d-dimensional compact Riemannian manifolds Md, for a dispersion parameter α>d, some coupling constant β>0, and γ≠0. (i) We first study the construction of the Gibbs measure for (expNLS). We prove that in the defocusing case γ>0, the measure is well-defined in the whole regime α>d and β>0 (Theorem 1.1 (i)), while in the focusing case γ<0 its partition function is always infinite for any α>d and β>0, even with a mass cut-off of arbitrary small size (Theorem 1.1 (ii)). (ii) We then study the dynamics (expNLS) with random initial data of low regularity. We first use a compactness argument to prove weak invariance of the Gibbs measure, in the defocusing case γ>0, in the whole regime α>d and β>0 (Theorem 1.4 (i)). In the large dispersion regime α>2d, we can improve this result by constructing a local deterministic flow for (expNLS) for any β>0 and γ≠0. Using the Gibbs measure, we prove that solutions are almost surely global for any β,γ>0, and that the Gibbs measure is invariant (Theorem 1.4 (ii)). (iii) Finally, in the particular case d=1 and M=T, we are able to exploit some probabilistic multilinear smoothing effects to build a global probabilistic flow for (expNLS) for 1+22<α≤2 for any β,γ>0 (Theorem 1.6).

Joint Probability Densities on Riemannian Manifolds are Symmetric Tensor Densities

This paper presents the tensor properties of joint probability densities on a Riemannian manifold. Initially, we develop a binary data matrix to record the values of a large number of particles confining in a closed system at a certain time in order to retrieve the joint probability densities of related variables. By introducing the particle-oriented coordinate and the generalized inner product as a multi-linear operation on the basis of this coordinate, we extract the set of joint probabilities and prove them to meet covariant tensor properties on a general Riemannian space of variables. Based on the Taylor expansion of scalar fields in Riemannian manifolds, it has been shown that the symmetrized iterative covariant derivatives of the cumulative probability function defined on Riemannian manifolds also give the set of related joint probability densities equivalent to the aforementioned multi-linear method. We show these covariant tensors reduce to classical ordinary partial derivatives in ordinary Euclidean space with Cartesian coordinates and give the formal definition of joint probabilities by partial derivatives of the cumulative distribution function. The equivalence between the symmetrized covariant derivative and the generalized inner product has been concluded. Some examples of well-known physical tensors clarify that many deterministic physical variables are presented as tensor densities with an interpretation similar to probability densities.

On the Potential Vector Fields of Soliton-Type Equations

We highlight some properties of a class of distinguished vector fields associated to a (1,1)-tensor field and to an affine connection on a Riemannian manifold, with a special view towards the Ricci vector fields, and we characterize them with respect to statistical, almost Kähler, and locally product structures. In particular, we provide conditions for these vector fields to be closed, Killing, parallel, or semi-torse forming. In the gradient case, we give a characterization of the Euclidean sphere. Among these vector fields, the Ricci and torse-forming-like vector fields are particular cases.

Tube formulas for valuations in complex space forms

AbstractGiven an isometry invariant valuation on a complex space form we compute its value on the tubes of sufficiently small radii around a set of positive reach. This generalizes classical formulas of Weyl, Gray and others about the volume of tubes. We also develop a general framework on tube formulas for valuations in Riemannian manifolds.

Analytical calculation of Kerr and Kerr-Ads black holes in f(R) theory

In this paper, we extend Chandrasekhar’s method of calculating rotating black holes into f(R) theory. We show that the solution with constant Ricci scalar always exists in a general f(R) gravity and derive the Kerr and Kerr-Ads metric by using the analytical mathematical method. Suppose that the spacetime is a 4-dimensional Riemannian manifold with a general stationary axisymmetric metric, we calculate Cartan’s equation of structure and derive the Einstein tensor. In order to reduce the solving difficulty, we fix the gauge freedom to transform the metric into a more symmetric form. We solve the field equations in the two cases of the Ricci scalar R=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R=0$$\\end{document} and R≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R\ e 0$$\\end{document}. In the case of R=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R=0$$\\end{document}, the Ernst’s equations are derived. We give the elementary solution of Ernst’s equations and show the way to obtain more solutions including Kerr metric. In the case of R≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R\ e 0$$\\end{document}, we reasonably assume that the solution to the equations consists of two parts: the first is Kerr part and the second is introduced by the Ricci scalar. Giving solution to the second part and combining the two parts, we obtain the Kerr-Ads metric. The calculations are carried out in a general f(R) theory. Furthermore, the whole solving process can be treated as a standard calculation procedure to obtain rotating black holes, which can be applied to other modified gravities.

Scalar curvature for metric spaces: Defining curvature for quantum gravity without coordinates

Geometrical properties of spacetime are difficult to study in nonperturbative approaches to quantum gravity like causal dynamical triangulations (CDT), where one uses simplicial manifolds to define the gravitational path integral, instead of Riemannian manifolds. In particular, in CDT one only relies on two mathematical tools, a distance measure and a volume measure. In this paper, we define a notion of scalar curvature, for metric spaces endowed with a volume measure or a random walk, without assuming nor using notions of tensor calculus. Furthermore, we directly define the Ricci scalar, without the need of defining and computing the Riemann or the Ricci tensor . For this, we make use of quantities, like the surface of a geodesic sphere, or the return probability of scalar diffusion processes, that can be computed in these metric spaces, as in a Riemannian manifold, where they receive scalar curvature contributions. Our definitions recover the classical results of scalar curvature when the sets are Riemannian manifolds. We propose two methods to compute the scalar curvature in these spaces, and we compare their features in natural implementations in discrete spaces. The defined generalized scalar curvatures are easily implemented on discrete spaces, like graphs. We present the results of our definitions on random triangulations of a 2D sphere and plane. Additionally, we show the results of our generalized scalar curvatures on the quantum geometries of 2D CDT, where we find that all our definitions indicate a flat ground state of the gravitational path integral. Published by the American Physical Society 2024

Conformal trajectories in three-dimensional space forms

We introduce the notion of conformal trajectories in three-dimensional Riemannian manifolds [Formula: see text]. Given a conformal vector field [Formula: see text], a conformal trajectory of [Formula: see text] is a regular curve [Formula: see text] in [Formula: see text] satisfying [Formula: see text], for some fixed nonzero constant [Formula: see text]. In this paper, we study the conformal trajectories in the space forms [Formula: see text], [Formula: see text] and [Formula: see text]. For (non-Killing) conformal vector fields in [Formula: see text] (respectively, in [Formula: see text]), we prove that conformal trajectories have constant curvature and its torsion is a linear combination of trigonometric (respectively, hyperbolic) functions on the arc-length parameter. In the case of Euclidean space [Formula: see text], we obtain the same result for the radial vector field and characterizing all conformal trajectories.

Existence of Optimal Pairs for Optimal Control Problems with States Constrained to Riemannian Manifolds

Existence of Optimal Pairs for Optimal Control Problems with States Constrained to Riemannian Manifolds

The 𝐿𝑝 restriction bounds for Neumann data on surface

Abstract Let { u λ } \{u_{\lambda}\} be a sequence of L 2 L^{2} -normalized Laplacian eigenfunctions on a compact two-dimensional smooth Riemanniann manifold ( M , g ) (M,g) . We seek to get an L p L^{p} restriction bound of the Neumann data λ − 1 ⁢ ∂ ν u λ | γ \lambda^{-1}\partial_{\nu}u_{\lambda}|_{\gamma} along a unit geodesic 𝛾. Using the 𝑇- T ∗ T^{*} argument, one can transfer the problem to an estimate of the norm of a Fourier integral operator and show that such bound is O ⁢ ( λ − 1 p + 3 2 ) O(\lambda^{-\frac{1}{p}+\frac{3}{2}}) . The stationary phase theorem plays the crucial role in our proof. Moreover, this upper bound is shown to be optimal.

A rescaled metric entropy of C1-vector field with singularities

For a C1-vector field on a closed Riemannian manifold, we add the flow speed control to define Bowen ball, and the new Bowen balls are given after considering reparametrization. That use the idea of rescaling the size of the neighborhoods of a regular point. Then we give new definitions of metric-entropy by using the minimum number of these new types of Bowen balls covering a set and the measure of this set is more than 1−δ. We prove that these new definitions are equivalent, and we call them rescaled metric entropy. Then we prove that for a C1-vector field, the rescaled metric entropy is equal to the metric entropy of the time-1 map of the flow providing that ln⁡‖X(x)‖ is integrable.

Functional Principal Component Analysis for Multiple Variables on Different Riemannian Manifolds

Functional Principal Component Analysis for Multiple Variables on Different Riemannian Manifolds

Quarter-Symmetric Non-Metric Connection of Non-Integrable Distributions

In this paper, we focus on non-integrable distributions with a quarter-symmetric non-metric connection (QSNMC) in generalized Riemannian manifold. First, by studying a quarter-symmetric connection on the generalized Riemannian manifold, we obtain the condition that the connection is non-metric. Then, the Gauss, Codazzi and Ricci equations are proved for non-integrable distributions with respect to a quarter-symmetric non-metric connection in generalized Riemannian manifold. Furthermore, we deduce Chen’s inequalities for non-integrable distributions of real space forms with a quarter-symmetric non-metric connection in generalized Riemannian manifold as applications. After that, we give some examples of non-integrable distributions in Riemannian manifold with quarter-symmetric non-metric connection.

Exploring Conformal Soliton Structures in Tangent Bundles with Ricci-Quarter Symmetric Metric Connections

In this study, we investigate the tangent bundle TM of an n-dimensional (pseudo-)Riemannian manifold M equipped with a Ricci-quarter symmetric metric connection ∇˜. Our primary goal is to establish the necessary and sufficient conditions for TM to exhibit characteristics of various solitons, specifically conformal Yamabe solitons, gradient conformal Yamabe solitons, conformal Ricci solitons, and gradient conformal Ricci solitons. We determine that for TM to be a conformal Yamabe soliton, the potential vector field must satisfy certain conditions when lifted vertically, horizontally, or completely from M to TM, alongside specific constraints on the conformal factor λ and the geometric properties of M. For gradient conformal Yamabe solitons, the conditions involve λ and the Hessian of the potential function. Similarly, for TM to be a conformal Ricci soliton, we identify conditions involving the lift of the potential vector field, the value of λ, and the curvature properties of M. For gradient conformal Ricci solitons, the criteria include the Hessian of the potential function and the Ricci curvature of M. These results enhance the understanding of the geometric properties of tangent bundles under Ricci-quarter symmetric metric connections and provide insights into their transition into various soliton states, contributing significantly to the field of differential geometry.

Universal inequalities for Dirichlet eigenvalues on discrete groups

We prove universal inequalities for Laplacian eigenvalues with Dirichlet boundary condition on subsets of certain discrete groups. The study of universal inequalities on Riemannian manifolds was initiated by Weyl, Pólya, Yau, and others. Here we focus on a version by Cheng and Yang. Specifically, we prove Yang-type universal inequalities for Cayley graphs of finitely generated amenable groups, as well as for the d -regular tree (simple random walk on the free group).

Curves as slant submanifolds of an almost product Riemannian manifold

Curves as slant submanifolds of an almost product Riemannian manifold