Riemannian Manifold Research Articles

Tensor Tomography on Negatively Curved Manifolds of Low Regularity

We prove solenoidal injectivity for the geodesic X-ray transform of tensor fields on simple Riemannian manifolds with C1,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,1}$$\\end{document} metrics and non-positive sectional curvature. The proof of the result rests on Pestov energy estimates for a transport equation on the non-smooth unit sphere bundle of the manifold. Our low regularity setting requires keeping track of regularity and making use of many functions on the sphere bundle having more vertical than horizontal regularity. Some of the methods, such as boundary determination up to gauge and regularity estimates for the integral function, have to be changed substantially from the smooth proof. The natural differential operators such as covariant derivatives are not smooth.

Small spheres with prescribed nonconstant mean curvature in Riemannian manifolds

Given a function f on a smooth Riemannian manifold without boundary, we prove that if p∈M is a non-degenerate critical point of f, then a neighborhood of p contains a foliation by spheres with mean curvature proportional to f. This foliation is essentially unique. The nondegeneracy assumption can be substantially relaxed, at the expense of losing the property that the family of spheres with prescribed mean curvature defines a foliation.

Estimates for low Steklov eigenvalues of surfaces with several boundary components

R\'esum\'eIn this article, we give computable lower bounds for the first non-zero Steklov eigenvalue $$\sigma _1$$ σ 1 of a compact connected 2-dimensional Riemannian manifold M with several cylindrical boundary components. These estimates show how the geometry of M away from the boundary affects this eigenvalue. They involve geometric quantities specific to manifolds with boundary such as the extrinsic diameter of the boundary. In a second part, we give lower and upper estimates for the low Steklov eigenvalues of a hyperbolic surface with a geodesic boundary in terms of the length of some families of geodesics. This result is similar to a well known result of Schoen, Wolpert and Yau for Laplace eigenvalues on a closed hyperbolic surface.

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

Considering an n-dimensional Riemannian manifold M whose sectional curvature is bounded above by κ and Ricci curvature is bounded below by (n−1)K, we obtain an upper bound for the harmonic mean of the first (n−1) non-zero Neumann eigenvalues of the Laplacian for domains contained in M. This can be viewed as certain isoperimetric inequality and generalizes the results for domains in space forms [5,15].

The Covariance Metric in the Blaschke Locus

We prove that the Blaschke locus has the structure of a finite dimensional smooth manifold away from the Teichmüller space and study its Riemannian manifold structure with respect to the covariance metric introduced by Guillarmou, Knieper and Lefeuvre in Guillarmou et al. in (Ergod Theory Dyn Syst 43:974–1022, 2021). We also identify some families of geodesics in the Blaschke locus arising from Hitchin representations for orbifolds and show that they have infinite length with respect to the covariance metric.

A new approach from the calculus of variations to the Lichnerowicz theorem

The well-known Lichnerowicz theorem states that on an n-dimensional compact Riemannian manifold without boundary with the assumptionRicg≥(n−1)g on the Ricci curvature, the first eigenvalue λ of the Laplacian on the manifold satisfiesλ≥n. This theorem is proved using a Bochner formula.In this paper we give a new approach from the calculus of variations to the Lichnerowicz theorem. We use the energy Econf() of conformality, introduced in [3]. For a smooth map f between Riemannian manifolds, the map f is a weakly conformal map (almost everywhere) if and only if Econf(f) = 0 (the minimum value).Our approach is as follows:(1)The identity map on M is a conformal map, hence a minimizer of the energy Econf(). Especially the identity map is stable, i.e., the second variation is non-negative for the identity map.(2)For the first eigenfunction φ, we take the variation vector field corresponding to dφ, and the non-negativity of the second variation for the variation vector field at the identity map implies the Lichnerowicz theorem.

Motif-Aware Riemannian Graph Neural Network with Generative-Contrastive Learning

Graphs are typical non-Euclidean data of complex structures. In recent years, Riemannian graph representation learning has emerged as an exciting alternative to Euclidean ones. However, Riemannian methods are still in an early stage: most of them present a single curvature (radius) regardless of structural complexity, suffer from numerical instability due to the exponential/logarithmic map, and lack the ability to capture motif regularity. In light of the issues above, we propose the problem of Motif-aware Riemannian Graph Representation Learning, seeking a numerically stable encoder to capture motif regularity in a diverse-curvature manifold without labels. To this end, we present a novel Motif-aware Riemannian model with Generative-Contrastive learning (MotifRGC), which conducts a minmax game in Riemannian manifold in a self-supervised manner. First, we propose a new type of Riemannian GCN (D-GCN), in which we construct a diverse-curvature manifold by a product layer with the diversified factor, and replace the exponential/logarithmic map by a stable kernel layer. Second, we introduce a motif-aware Riemannian generative-contrastive learning to capture motif regularity in the constructed manifold and learn motif-aware node representation without external labels. Empirical results show the superiority of MofitRGC.

Distributed Manifold Hashing for Image Set Classification and Retrieval

Conventional image set methods typically learn from image sets stored in one location. However, in real-world applications, image sets are often distributed or collected across different positions. Learning from such distributed image sets presents a challenge that has not been studied thus far. Moreover, efficiency is seldom addressed in large-scale image set applications. To fulfill these gaps, this paper proposes Distributed Manifold Hashing (DMH), which models distributed image sets as a connected graph. DMH employs Riemannian manifold to effectively represent each image set and further suggests learning hash code for each image set to achieve efficient computation and storage. DMH is formally formulated as a distributed learning problem with local consistency constraint on global variables among neighbor nodes, and can be optimized in parallel. Extensive experiments on three benchmark datasets demonstrate that DMH achieves highly competitive accuracies in a distributed setting and provides faster classification and retrieval than state-of-the-arts.

Unified Framework for Diffusion Generative Models in SO(3): Applications in Computer Vision and Astrophysics

Diffusion-based generative models represent the current state-of-the-art for image generation. However, standard diffusion models are based on Euclidean geometry and do not translate directly to manifold-valued data. In this work, we develop extensions of both score-based generative models (SGMs) and Denoising Diffusion Probabilistic Models (DDPMs) to the Lie group of 3D rotations, SO(3). SO(3) is of particular interest in many disciplines such as robotics, biochemistry and astronomy/cosmology science. Contrary to more general Riemannian manifolds, SO(3) admits a tractable solution to heat diffusion, and allows us to implement efficient training of diffusion models. We apply both SO(3) DDPMs and SGMs to synthetic densities on SO(3) and demonstrate state-of-the-art results. Additionally, we demonstrate the practicality of our model on pose estimation tasks and in predicting correlated galaxy orientations for astrophysics/cosmology.

The Log-Sobolev Inequality for a Submanifold in Manifolds With Asymptotic Non-Negative Intermediate Ricci Curvature

We prove a sharp Log-Sobolev inequality for submanifolds of a complete non-compact Riemannian manifold with asymptotic non-negative intermediate Ricci curvature and Euclidean volume growth. Our work extends a result of Dong et al. (Acta Math. Sci. Ser. B (Engl. Ed.), 44(1):189–194 (2024)) which already generalizes Yi and Zheng (To appear in Chin. Ann. Math., (2023)) and Brendle (Comm. Pure Appl. Math. 75(3), 449–454 (2022)).

Some inequalities for bi-slant Riemannian submersions in complex space forms

The goal of this paper is to analyze sharp-type inequalities including the scalar and Ricci curvatures of bi-slant Riemannian submersions in complex space forms. Then, for bi-slant Riemannian submersion between a complex space form and a Riemannian manifold, we give inequalities involving the Casorati curvature of the space ker [Formula: see text]. Also, we mention some examples.

Phase-only beampattern synthesis for maximizing mainlobe gain via Riemannian Newton method

Beampattern synthesis with phase-only weights is extensively utilized in phased-array systems due to the virtue of low-cost and high power efficiency. It is always a non-convex optimization problem having constant modulus constraints (CMCs) and the performance significantly depends on the employed solvers. This paper discusses the phase-only beampattern synthesis for maximizing the mainlobe gain with the constraints on the mainlobe ripple and the relative sidelobe level. As CMC is equivalent to a complex circle manifold, the proposed beampattern synthesis is considered as a manifold optimization problem, and then decomposed into several subproblems under the framework of manifold alternating direction method of multipliers (ADMM). In particular, the subproblem for updating phase-only weights is efficiently tackled by a second-order Riemannian Newton method, which has faster convergence speed and no requirement on line search compared to widely-used first-order Riemannian manifold methods. The explicit convergence condition is derived and the computational complexity is also analyzed. Numerical results show that the proposed method has better performance on the mainlobe gain and sidelobe control accuracy, or less computational cost compared to several representative approaches, such as convex relaxation, standard ADMM, Riemannian gradient descent-based ADMM and Riemannian conjugate gradient methods.

The Kastler–Kalau–Walze-type theorems about J-Witten deformation

In this paper, we obtain a Lichnerowicz-type formula for [Formula: see text]-Witten deformation and give the proof of the Kastler–Kalau–Walze-type theorems associated with [Formula: see text]-Witten deformation on four-dimensional and six-dimensional almost product Riemannian manifold with (respectively, without) boundary. We give an explanation of the Einstein–Hilbert action for [Formula: see text]-Witten deformation on four-dimensional manifold with boundary.

Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension

We consider second order (maximally) conformally superintegrable systems and explain how the definition of such a system on a (pseudo-)Riemannian manifold gives rise to a conformally invariant interpretation of superintegrability. Conformal equivalence in this context is a natural extension of the classical (linear) Stäckel transform, originating from the Maupertuis-Jacobi principle. We extend our recently developed algebraic geometric approach for the classification of second order superintegrable systems in arbitrarily high dimension to conformally superintegrable systems, which are presented via conformal scale choices of second order superintegrable systems defined within a conformal geometry. For superintegrable systems on constant curvature spaces, we find that the conformal scales of Stäckel equivalent systems arise from eigenfunctions of the Laplacian and that their equivalence is characterised by a conformal density of weight two. Our approach yields an algebraic equation that governs the classification under conformal equivalence for a prolific class of second order conformally superintegrable systems. This class contains all non-degenerate examples known to date, and is given by a simple algebraic constraint of degree two on a general harmonic cubic form. In this way the yet unsolved classification problem is put into the reach of algebraic geometry and geometric invariant theory. In particular, no obstruction exists in dimension three, and thus the known classification of conformally superintegrable systems is reobtained in the guise of an unrestricted univariate sextic. In higher dimensions, the obstruction is new and has never been revealed by traditional approaches.

Infinite-time blowing-up solutions to small perturbations of the Yamabe flow

In this paper, we examine a PDE aspect of the Yamabe flow as an energy-critical parabolic equation of the fast-diffusion type. It is well-known that under the validity of the positive mass theorem, the Yamabe flow on a smooth closed Riemannian manifold M exists for all time t and uniformly converges to a solution to the Yamabe problem on M as t→∞. We show that such results no longer hold if some arbitrarily small and smooth perturbation is imposed on it, by constructing solutions to the perturbed flow that blow up at multiple points on M in the infinite time. We also examine the stability of the blow-up phenomena under a negativity condition on the Ricci curvature at blow-up points.

Cherrier–Escobar problem for the elliptic Schrödinger-to-Neumann map

In this paper, we study a Cherrier–Escobar problem for the extended problem corresponding to the elliptic Schrödinger-to-Neumann map on a compact 3-dimensional Riemannian manifold with boundary. Using the algebraic topological argument of Bahri and Coron (1988), we show solvability under the assumption that the extended problem corresponding to the elliptic Schrödinger-to-Neumann map has a positive first eigenvalue and a positive Green’s function.

A non-Newtonian magnetic curves in multiplicative Riemann manifolds

The aim of this study is to rearrange magnetic curves and their main properties with the help of multiplicative calculi. Magnetic curves have been examined in many spaces with the tools of traditional (Newtonian) analysis and their characterizations have been obtained. The innovation brought by this study; magnetic curves and many other getometric and physical expressions were studied for the first time with non-Newtonian arguments in multiplicative space. In the study, the advantages of purely multiplicative operations and multiplicative calculation are used. Moreover, it unveils the distinctions (angle, norm, distance, line vb.) between the multiplicative Euclidean space and the conventional Euclidean space, offering a novel perspective on geometrically magnetic curves. As a result, the concept of multiplicative magnetic curves (t − magnetic, n − magnetic and b − magnetic) are introduced to the academic discourse, and the essential characterizations are established. The study also provides illustrative examples to facilitate a better is understood of the subject matter and employs Geogebra to generate visual representations of new concepts.

Pseudo-Riemannian Geodesic Orbit Nilmanifolds of Signature (n-2,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{(n-2,2)}$$\\end{document}

The geodesic orbit property is useful and interesting in itself, and it plays a key role in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here we extend Riemannian and Lorentz results to trans-Lorentz nilmanifolds. Those are the geodesic orbit pseudo Riemannian manifolds M=G/H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M = G/H$$\\end{document} of signature (n-2,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-2,2)$$\\end{document} such that a nilpotent analytic subgroup of G is transitive on M. For that we suppose that there is a reductive decomposition g=h⊕n(vector space direct sum) with[h,n]⊂n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {g}}= {\\mathfrak {h}}\\oplus {\\mathfrak {n}}\ ext { (vector space direct sum) with } [{\\mathfrak {h}},{\\mathfrak {n}}] \\subset {\\mathfrak {n}}$$\\end{document} and n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {n}}$$\\end{document} nilpotent. When the metric is nondegenerate on [n,n]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[{\\mathfrak {n}},{\\mathfrak {n}}]$$\\end{document} we show that n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {n}}$$\\end{document} is abelian or 2-step nilpotent. That is the same result as for geodesic orbit Riemannian and Lorentz nilmanifolds. When the metric is degenerate on [n,n]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[{\\mathfrak {n}},{\\mathfrak {n}}]$$\\end{document} we show that n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {n}}$$\\end{document} is a double extension of a geodesic orbit nilmanifold of either Riemannian or Lorentz signature.

Riemannian‐geometric regime‐switching covariance hedging

AbstractThis study develops a regime‐switching Riemannian‐geometric covariance framework for futures hedging. The covariance of conventional regime‐switching BEKK (Baba, Engle, Kraft and Kroner) (RSBEKK) evolves on flat spaces that exclude a prior the possibility of inherent geometric covariance dynamic. A Riemannian‐geometric regime‐switching BEKK (RG‐RSBEKK) is proposed such that the covariance moves along a trajectory on Riemannian manifolds. RG‐RSBEKK is applied to China Securities Index 300 futures for hedging the stock sector exposures. Empirical results reveal that specifying covariance dynamic on curved spaces enhances hedging effectiveness based on the model confidence set with loss measures of variance, utility, value‐at‐risk, and Frobenius distance.

Liouville theorem for exponentially harmonic functions on Riemannian manifolds with compact boundary

Liouville theorem for exponentially harmonic functions on Riemannian manifolds with compact boundary