Riemannian Manifold Of Constant Curvature Research Articles

Separation of Variables and Superintegrability on Riemannian Coverings

We introduce Stäckel separable coordinates on the covering manifolds $M_k$, where $k$ is a rational parameter, of certain constant-curvature Riemannian manifolds with the structure of warped manifold. These covering manifolds appear implicitly in literature as connected with superintegrable systems with polynomial in the momenta first integrals of arbitrarily high degree, such as the Tremblay-Turbiner-Winternitz system. We study here for the first time multiseparability and superintegrability of natural Hamiltonian systems on these manifolds and see how these properties depend on the parameter $k$.

Results of Hyperbolic Ricci Solitons

We obtain some properties of a hyperbolic Ricci soliton with certain types of potential vector fields, and we point out some conditions when it reduces to a trivial Ricci soliton. We also study those soliton submanifolds whose vector fields are the tangential components of a concurrent vector field on the ambient manifold, and in particular, we show that a totally umbilical hyperbolic Ricci soliton is an Einstein manifold. We prove that if the hyperbolic Ricci soliton hypersurface of a Riemannian manifold of constant curvature and endowed with a concurrent vector field has a parallel shape operator, then it is a metallic-shaped hypersurface, and we determine some conditions for it to be minimal. Moreover, we show that it is also a pseudosymmetric hypersurface.

Identifying the latent space geometry of network models through analysis of curvature

Abstract A common approach to modelling networks assigns each node to a position on a low-dimensional manifold where distance is inversely proportional to connection likelihood. More positive manifold curvature encourages more and tighter communities; negative curvature induces repulsion. We consistently estimate manifold type, dimension, and curvature from simply connected, complete Riemannian manifolds of constant curvature. We represent the graph as a noisy distance matrix based on the ties between cliques, then develop hypothesis tests to determine whether the observed distances could plausibly be embedded isometrically in each of the candidate geometries. We apply our approach to datasets from economics and neuroscience.

On the Differential Equation Governing Torqued Vector Fields on a Riemannian Manifold

In this article, we show that the presence of a torqued vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian manifolds of constant curvature. More precisely, we show that there is no torqued vector field on n-sphere Sn(c). A nontrivial example of torqued vector field is constructed on an open subset of the Euclidean space En whose torqued function and torqued form are nowhere zero. It is shown that owing to topology of the Euclidean space En, this type of torqued vector fields could not be extended globally to En. Finally, we find a necessary and sufficient condition for a torqued vector field on a compact Riemannian manifold to be a concircular vector field.

Riemannian manifolds dual to static spacetimes

We establish a one-to-one correspondence between static spacetimes and Riemannian manifolds that maps causal geodesics to geodesics, as suggested by L. C. Epstein. We then explore constant curvature spacetimes - such as the de Sitter and the anti-de Sitter universes - and find that they map to constant curvature Riemannian manifolds, namely the Euclidean space, the sphere and the hyperbolic space. By imposing the conditions required to map to the sphere, we obtain the spherically symmetric metrics for which there is radial oscillatory motion with a period independent of the amplitude. We then consider the case of a perfect fluid and an Einstein cluster and determine the pressure and density profiles required to find this type of motion. Finally, we give examples of surfaces corresponding to certain types of motion for metrics that do not exhibit constant curvature, such as the Schwarzschild, Schwarzschild de Sitter and Schwarzschild anti-de Sitter solutions, and even for a simplified model of a wormhole.

Magnetic trajectories on tangent sphere bundle with g-natural metrics

We study magnetic trajectories in the unit tangent sphere bundle with pseudo-Riemannian g-natural metrics of a Riemannian manifold. A high interest is dedicated to the case when the base manifold is a space form and when the metric is of Kaluza–Klein type. Slant curves are obtained when a certain conservation law is satisfied. We give a complete classification of slant magnetic curves (respectively, geodesics) on $$T_1M$$ , when M is a two-dimensional Riemannian manifold of constant curvature.

Identifying the Latent Space Geometry of Network Models Through Analysis of Curvature

Statistically modeling networks, across numerous disciplines and contexts, is fundamentally challenging because of (often high-order) dependence between connections. A common approach assigns each person in the graph to a position on a low-dimensional manifold. Distance between individuals in this (latent) space is inversely proportional to the likelihood of forming a connection. The choice of the latent geometry (the manifold class, dimension, and curvature) has consequential impacts on the substantive conclusions of the model. More positive curvature in the manifold, for example, encourages more and tighter communities; negative curvature induces repulsion among nodes. Currently, however, the choice of the latent geometry is an a priori modeling assumption and there is limited guidance about how to make these choices in a data-driven way. In this work, we present a method to consistently estimate the manifold type, dimension, and curvature from an empirically relevant class of latent spaces: simply connected, complete Riemannian manifolds of constant curvature. Our core insight comes by representing the graph as a noisy distance matrix based on the ties between cliques. Leveraging results from statistical geometry, we develop hypothesis tests to determine whether the observed distances could plausibly be embedded isometrically in each of the candidate geometries. We explore the accuracy of our approach with simulations and then apply our approach to data-sets from economics and sociology as well as neuroscience.

Change Detection in Graph Streams by Learning Graph Embeddings on Constant-Curvature Manifolds.

The space of graphs is often characterized by a nontrivial geometry, which complicates learning and inference in practical applications. A common approach is to use embedding techniques to represent graphs as points in a conventional Euclidean space, but non-Euclidean spaces have often been shown to be better suited for embedding graphs. Among these, constant-curvature Riemannian manifolds (CCMs) offer embedding spaces suitable for studying the statistical properties of a graph distribution, as they provide ways to easily compute metric geodesic distances. In this paper, we focus on the problem of detecting changes in stationarity in a stream of attributed graphs. To this end, we introduce a novel change detection framework based on neural networks and CCMs, which takes into account the non-Euclidean nature of graphs. Our contribution in this paper is twofold. First, via a novel approach based on adversarial learning, we compute graph embeddings by training an autoencoder to represent graphs on CCMs. Second, we introduce two novel change detection tests operating on CCMs. We perform experiments on synthetic data, as well as two real-world application scenarios: the detection of epileptic seizures using functional connectivity brain networks and the detection of hostility between two subjects, using human skeletal graphs. Results show that the proposed methods are able to detect even small changes in a graph-generating process, consistently outperforming approaches based on Euclidean embeddings.

Adversarial autoencoders with constant-curvature latent manifolds

Constant-curvature Riemannian manifolds (CCMs) have been shown to be ideal embedding spaces in many application domains, as their non-Euclidean geometry can naturally account for some relevant properties of data, like hierarchy and circularity. In this work, we introduce the CCM adversarial autoencoder (CCM-AAE), a probabilistic generative model trained to represent a data distribution on a CCM. Our method works by matching the aggregated posterior of the CCM-AAE with a probability distribution defined on a CCM, so that the encoder implicitly learns to represent data on the CCM to fool the discriminator network. The geometric constraint is also explicitly imposed by jointly training the CCM-AAE to maximise the membership degree of the embeddings to the CCM. While a few works in recent literature make use of either hyperspherical or hyperbolic manifolds for different learning tasks, ours is the first unified framework to seamlessly deal with CCMs of different curvatures. We show the effectiveness of our model on three different datasets characterised by non-trivial geometry: semi-supervised classification on MNIST, link prediction on two popular citation datasets, and graph-based molecule generation using the QM9 chemical database. Results show that our method improves upon other autoencoders based on Euclidean and non-Euclidean geometries on all tasks taken into account.

Characterization of Curves that Lie on a Geodesic Sphere or on a Totally Geodesic Hypersurface in a Hyperbolic Space or in a Sphere

The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion. In this work, we extend these investigations to characterize curves that lie on a geodesic sphere or totally geodesic hypersurface in a Riemannian manifold of constant curvature. Using that geodesic spherical curves are normal curves, i.e., they are the image of an Euclidean spherical curve under the exponential map, we are able to characterize geodesic spherical curves in hyperbolic spaces and spheres through a non-homogeneous linear equation. Finally, we also show that curves on totally geodesic hypersurfaces, which play the role of hyperplanes in Riemannian geometry, should be characterized by a homogeneous linear equation. In short, our results give interesting and significant similarities between hyperbolic, spherical, and Euclidean geometries.

Integral formulae for codimension-one foliated Randers spaces

Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. This paper continues our recent study and presents new integral formulae and their applications for codimension-one foliated Randers spaces. The goal is a generalization of Reeb's formula (that the total mean curvature of the leaves is zero) and its companion (that twice total second mean curvature of the leaves equals to the total Ricci curvature in the normal direction). We also extend results by Brito, Langevin and Rosenberg (that total mean curvatures of arbitrary order for a codimension-one foliated Riemannian manifold of constant curvature don't depend on a foliation). All of that is done by a comparison of extrinsic and intrinsic curvatures of the two Riemannian structures which arise in a natural way from a given Randers structure.

Affine analogues of the Sasaki-Shchepetilov connection

AbstractFor two-dimensional manifold M with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle TM ⊕ E, where E is a trivial bundle. The metrizable case, when M is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: 36 (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on TM ⊕ (ℝ × M) and two on TM ⊕ (ℝ2 × M) are constructed. It is shown that two of those connections – one from each pair – may be identified with the standard flat connection in ℝN, after suitable local affine embedding of (M,∇) into ℝN.

On a Classification of 4-d Gradient Ricci Solitons with Harmonic Weyl Curvature

We study a characterization of 4-dimensional (not necessarily complete) gradient Ricci solitons $(M, g, f)$ which have harmonic Weyl curvature, i.e. $\delta W=0$. Roughly speaking, we prove that the soliton metric $g$ is locally isometric to one of the following four types: an Einstein metric, the product $ \mathbb{R}^2 \times N_{\lambda}$ of the Euclidean metric and a 2-d Riemannian manifold of constant curvature ${\lambda} \neq 0$, a certain singular metric and a locally conformally flat metric. The method here is motivated by Cao-Chen's works \cite{CC1, CC2} and Derdzi\'{n}ski's study on Codazzi tensors \cite{De}. Combined with the previous results on locally conformally flat solitons, our characterization yields a new classification of 4-d complete steady solitons with $\delta W=0$. For shrinking case, it reproves the rigidity result \cite{FG, MS} in 4-d. It also helps to understand the expanding case; we now understand all 4-d non-conformally-flat ones with $\delta W=0$. We also characterize {\it locally} 4-d (not necessarily complete) gradient Ricci solitons with harmonic curvature.

Dimensions of projections of sets on Riemannian surfaces of constant curvature

We apply the theory of Peres and Schlag to obtain generic lower bounds for Hausdorff dimension of images of sets by orthogonal projections on simply connected two-dimensional Riemannian manifolds of constant curvature. As a conclusion we obtain appropriate versions of Marstrand’s theorem, Kaufman’s theorem, and Falconer’s theorem in the above geometrical settings.

Hessian structures on deformed exponential families and their conformal structures

An exponential family is an important class of statistical models in statistical sciences. In information geometry, it is known that an exponential family naturally has dualistic Hessian structures. A deformed exponential family is a statistical model which is a generalization of exponential families. A deformed exponential family naturally has two kinds of dualistic Hessian structures. In this paper, such Hessian geometries are summarized.In addition, a deformed exponential family has a generalized conformal structure of statistical manifolds. In the case of q-exponential family, which is a special class of deformed exponential families, the family naturally has two kinds of different Riemannian metrics which are obtained from conformal transformations of Hessian metrics. Then it is showed that a q-exponential family is a Riemannian manifold of constant curvature.

Levi Harmonic Maps of Contact Riemannian Manifolds

We study Levi harmonic maps, i.e., C∞ solutions f:M→M′ to \(\tau_{\mathcal{H}} (f) \equiv \operatorname{trace}_{g} ( \varPi_{\mathcal{H}}\beta_{f} ) = 0\), where (M,η,g) is an (almost) contact (semi) Riemannian manifold, M′ is a (semi) Riemannian manifold, βf is the second fundamental form of f, and \(\varPi_{\mathcal{H}} \beta_{f}\) is the restriction of βf to the Levi distribution \({\mathcal{H}} = \operatorname{Ker}(\eta)\). Many examples are exhibited, e.g., the Hopf vector field on the unit sphere S2n+1, immersions of Brieskorn spheres, and the geodesic flow of the tangent sphere bundle over a Riemannian manifold of constant curvature 1 are Levi harmonic maps. A CR map f of contact (semi) Riemannian manifolds (with spacelike Reeb fields) is pseudoharmonic if and only if f is Levi harmonic. We give a variational interpretation of Levi harmonicity. Any Levi harmonic morphism is shown to be a Levi harmonic map.

BIMINIMAL CURVES IN 2-DIMENSIONAL SPACE FORMS

Abstract. We study biminimalcurves in 2-dimensional Riemannian man-ifolds of constant curvature. IntroductionElastic curves provide examples of classically known geometric variationalproblem. A plane curve is said to be an elastic curve if it is a critical point ofthe elastic energy, or equivalently a critical point of the total squared curvature[9].In this paper, we study another geometric variational problem of curves inRiemannian 2-manifolds of constant curvature. The Euler-Lagrange equationstudied in this paper is derived from the theory of biharmonic maps in Rie-mannian geometry.A smooth map φ : (M,g) → (N,h) between Riemannian manifolds is saidto be biharmonic if it is a critical point of the bienergy functional:E 2 (φ) =Z M |τ(φ)| 2 dv g ,where τ(φ) = tr ∇dφ is the tension field of φ. Clearly, if φ is harmonic, i.e.,τ(φ) = 0, then φ is biharmonic. A biharmonic map is said to be proper if it isnot harmonic.Chen and Ishikawa [3] studied biharmonic curves and surfaces in semi-Euclidean space (see also [6]). Caddeo, Montaldo and Piu [1] studied bihar-monic curves on Riemannian 2-manifolds. They showed that biharmonic curveson Riemannian 2-manifolds of non-positive curvature are geodesics. Proper bi-harmonic curves on the unit 2-sphere are small circles of radius 1/√2.Next, Loubeau and Montaldo introduced the notion of biminimal immersion[10].An isometric immersion φ : (M,g) → (N,h) is said to be biminimal if it is acritical point of the bienergy functional under all normal variations. Thus thebiminimality is weaker than biharmonicity for isometric immersions, in general.

Complete hypersurfaces with [formula omitted] in a unit sphere

By investigating hypersurfaces M n in the unit sphere S n + 1 ( 1 ) with H k = 0 and with two distinct principal curvatures, we give a characterization of torus S 1 ( k / n ) × S n − 1 ( ( n − k ) / n ) . We extend recent results of T. Hasanis et al. [T. Hasanis, A. Savas-Halilaj, T. Vlachos, Complete Minimal hypersurfaces in a sphere, Monatsh. Math. 145 (2005) 301–305] and T. Otsuki [T. Otsuki, Minimal hypersurfaces in a Riemannian manifold of constant curvature, Amer. J. Math. 92 (1970) 145–173].

Patterson-Sullivan measures and quasi-conformal deformations

In this paper we relate the ergodic action of a Kleinian group on the space of line elements to the conformal action of the group on the sphere at infinity. In particular, we show that for a pair of geometrically isomorphic convex co-compact Kleinian groups, the ratio of the length of the Patterson-Sullivan measure on line element space to the length of its push-forward is bounded below by the ratio of the Hausdorff dimensions of the limit sets. Our primary techniques come from ergodic theory and Patterson-Sullivan theory. 1 Basics and Statement of Results Let Isom+(H) n ≥ 2 be the space of orientation-preserving isometries of Hn. As is well-known, this space of isometries can be given the topology induced by uniform convergence on compact sets. A Kleinian group Γ is a discrete subgroup of Isom+(H). As such, Γ acts discontinuously on Hn, and because we make a standing assumption that the action is torsion-free, the quotient manifoldN = Hn/Γ is a complete Riemannian manifold of constant curvature −1. A Kleinian group Γ also acts as a discrete subgroup of conformal automorphisms of the sphere at infinity Sn−1 ∞ ; this action partitions S n−1 ∞ into two disjoint sets. The regular set ΩΓ is the largest open set in Sn−1 ∞ on which Γ acts properly discontinuously, and the limit set LΓ is its complement. In the case that LΓ contains more than 2 points, the limit set is characterized as being the smallest closed Γ-invariant subset of Sn−1 ∞ . Define the convex hull CH(LΓ) of the limit set LΓ to be the smallest convex subset of Hn so that all geodesics with both limit points in LΓ are contained in CH(LΓ). We can take the quotient of CH(LΓ) by Γ (denoted by C(Γ)); this is

Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension

Let f:\Sigma_1 --> \Sigma_2 be a map between compact Riemannian manifolds of constant curvature. This article considers the evolution of the graph of f in the product of \Sigma_1 and \Sigma_2 by the mean curvature flow. Under suitable conditions on the curvature of \Sigma_1 and \Sigma_2 and the differential of the initial map, we show that the flow exists smoothly for all time. At each instant t, the flow remains the graph of a map f_t and f_t converges to a constant map as t approaches infinity. This also provides a regularity estimate for Lipschtz initial data.