Space Of Constant Sectional Curvature Research Articles

Umbilicity of constant mean curvature hypersurfaces into space forms

In this paper we establish conditions on the length of the traceless part of the second fundamental form of a complete constant mean curvature hypersurface immersed in a space of constant sectional curvature in order to show that it is totally umbilical.

Computationally Tractable Riemannian Manifolds for Graph Embeddings

Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases (e.g., hierarchical structures benefit from hyperbolic geometry). However, going beyond embedding spaces of constant sectional curvature, while potentially more representationally powerful, proves to be challenging as one can easily lose the appeal of computationally tractable tools such as geodesic distances or Riemannian gradients. Here, we explore two computationally efficient matrix manifolds, showcasing how to learn and optimize graph embeddings in these Riemannian spaces. Empirically, we demonstrate consistent improvements over Euclidean geometry while often outperforming hyperbolic and elliptical embeddings based on various metrics that capture different graph properties. Our results serve as new evidence for the benefits of non-Euclidean embeddings in machine learning pipelines.

Kissing number in non-Euclidean spaces of constant sectional curvature

This paper provides upper and lower bounds on the kissing number of congruent radius r > 0 r > 0 spheres in hyperbolic H n \mathbb {H}^n and spherical S n \mathbb {S}^n spaces, for n ≥ 2 n\geq 2 . For that purpose, the kissing number is replaced by the kissing function κ H ( n , r ) \kappa _H(n, r) , resp. κ S ( n , r ) \kappa _S(n, r) , which depends on the dimension n n and the radius r r . After we obtain some theoretical upper and lower bounds for κ H ( n , r ) \kappa _H(n, r) , we study their asymptotic behaviour and show, in particular, that κ H ( n , r ) ∼ ( n − 1 ) ⋅ d n − 1 ⋅ B ( n − 1 2 , 1 2 ) ⋅ e ( n − 1 ) r \kappa _H(n,r) \sim (n-1) \cdot d_{n-1} \cdot B(\frac {n-1}{2}, \frac {1}{2}) \cdot e^{(n-1) r} , where d n d_n is the sphere packing density in R n \mathbb {R}^n , and B B is the beta-function. Then we produce numeric upper bounds by solving a suitable semidefinite program, as well as lower bounds coming from concrete spherical codes. A similar approach allows us to locate the values of κ S ( n , r ) \kappa _S(n, r) , for n = 3 , 4 n= 3,\, 4 , over subintervals in [ 0 , π ] [0, \pi ] with relatively high accuracy.

Simplices modelled on spaces of constant curvature.

We give non-degeneracy criteria for Riemannian simplices based on simplices in spaces of constant sectional curvature. It extends previous work (SoCG15) on Riemannian simplices, where we developed Riemannian simplices with respect to Euclidean reference simplices. The criteria we give in this article are in terms of quality measures for spaces of constant curvature that we develop here. We see that simplices in spaces that have nearly constant curvature, are already non-degenerate under very weak quality demands. This is of importance because it allows for sampling of Riemannian manifolds based on anisotropy of the manifold and not (absolute) curvature.

Mean Curvature Flow Solitons in the Presence of Conformal Vector Fields

In this paper we introduce and study a notion of mean curvature flow soliton in Riemannian ambient spaces general enough to encompass target spaces of constant sectional curvature, Riemannian products or, in increasing generality, warped product spaces. As expected, our definition is motivated by the self-similarity of certain special solutions of the mean curvature flow with respect to the flow generated by a distinguished vector field on the target manifold. Our approach allows us to identify some natural geometric quantities that satisfy elliptic equations or differential inequalities in a simple and manageable form for which the machinery of weak maximum principles is valid. The latter is one of the main tools we apply to derive several new characterizations and rigidity results for mean curvature flow solitons that extend to our much more general setting known properties, for instance, in Euclidean space.

Trans-Sasakian 3-Manifolds with Reeb Flow Invariant Ricci Operator

Let M be a three-dimensional trans-Sasakian manifold of type ( α , β ) . In this paper, we obtain that the Ricci operator of M is invariant along Reeb flow if and only if M is an α -Sasakian manifold, cosymplectic manifold or a space of constant sectional curvature. Applying this, we give a new characterization of proper trans-Sasakian 3-manifolds.

Some examples of Dirac-harmonic maps

We discuss a method to construct Dirac-harmonic maps developed by Jost et al. (J Geom Phys 59(11):1512–1527, 2009). The method uses harmonic spinors and twistor spinors and mainly applies to Dirac-harmonic maps of codimension 1 with target spaces of constant sectional curvature. Before the present article, it remained unclear when the conditions of the theorems in Jost et al. (2009) were fulfilled. We show that for isometric immersions into space forms, these conditions are fulfilled only under special assumptions. In several cases, we show the existence of solutions.

On para-Kenmotsu manifolds

In this paper we study para-Kenmotsu manifolds. We characterize this manifolds by tensor equations and study their properties. We are devoted to a study of ?-Einstein manifolds. We show that a locally conformally flat para-Kenmotsu manifold is a space of constant negative sectional curvature -1 and we prove that if a para-Kenmotsu manifold is a space of constant ?-para-holomorphic sectional curvature H, then it is a space of constant sectional curvature and H = -1. Finally the object of the present paper is to study a 3-dimensional para-Kenmotsu manifold, satisfying certain curvature conditions. Among other, it is proved that any 3-dimensional para-Kenmotsu manifold with ?-parallel Ricci tensor is of constant scalar curvature and any 3-dimensional para-Kenmotsu manifold satisfying cyclic Ricci tensor is a manifold of constant negative sectional curvature -1.

Lagrangian submanifolds in the homogeneous nearly Kähler $${\mathbb {S}}^3 \times {\mathbb {S}}^3$$ S 3 × S 3

In this paper, we investigate Lagrangian submanifolds in the homogeneous nearly Kahler $$\mathbb {S}^3 \times \mathbb {S}^3$$ . We introduce and make use of a triplet of angle functions to describe the geometry of a Lagrangian submanifold in $$\mathbb {S}^3 \times \mathbb {S}^3$$ . We construct a new example of a flat Lagrangian torus and give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature. As a corollary of our main result, we obtain that the radius of a round Lagrangian sphere in the homogeneous nearly Kahler $$\mathbb {S}^3 \times \mathbb {S}^3$$ can only be $$\frac{2}{\sqrt{3}}$$ or $$\frac{4}{\sqrt{3}}$$ .

Emergent gravity in spaces of constant curvature

In physical theories where the energy (action) is localized near a submanifold of a constant curvature space, there is a universal expression for the energy (or the action). We derive a multipole expansion for the energy that has a finite number of terms, and depends on intrinsic geometric invariants of the submanifold and extrinsic invariants of the embedding of the submanifold. This is the second of a pair of articles in which we try to develop a theory of emergent gravity arising from the embedding of a submanifold into an ambient space equipped with a quantum field theory. Our theoretical method requires a generalization of a formula due to by Hermann Weyl. While the first paper discussed the framework in Euclidean (Minkowski) space, here we discuss how this framework generalizes to spaces of constant sectional curvature. We focus primarily on anti de Sitter space. We then discuss how such a theory can give rise to a cosmological constant and Planck mass that are within reasonable bounds of the experimental values.

Locally Symmetric CR-Integrable Almost Kenmotsu Manifolds

In this paper, it is proved that a locally symmetric almost Kenmotsu manifold of dimension 2n+1, n > 1, with CR-integrable structure is locally isometric to either the hyperbolic space of constant sectional curvature −1, or the Riemannian product of an (n + 1)-dimensional manifold of constant sectional curvature −4 and a flat n-dimensional manifold.

Conformally flat Lorentzian manifolds with special holonomy groups

We obtain a local classification of conformally flat Lorentzian manifolds with special holonomy groups. The corresponding local metrics are certain extensions of Riemannian spaces of constant sectional curvature to Walker metrics.Bibliography: 28 titles.

Volume formula for a ℤ2-symmetric spherical tetrahedron through its edge lengths

The present paper considers volume formulæ, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation of angle π in the middle points of a certain pair of its skew edges.

Curvatures of the tangent bundle with a special almost product metric

We study a class of Riemannian almost product metrics on the tangent bundle of a smooth manifold. This class includes the Sasaki and Cheeger-Gromoll metrics as special cases. For this class of metrics, we find the dependence of the scalar curvature of the tangent bundle on objects of the base manifold. For the case in which the base manifold is a space of constant sectional curvature, we obtain conditions on the metric and the dimension of the base under which the scalar curvature of the tangent bundle is constant. For special cases of metrics of the class considered, we find the intervals on which the scalar curvature of the tangent bundle treated as a function of the sectional curvature of the base has constant sign.

Spinor Green’s functions via spherical means on products of space forms

We explicitly compute the Green’s function of the spinor Klein–Gordon equation on the Riemannian and Lorentzian manifolds of the form M 0 × ⋯ × M N , with each factor being a space of constant sectional curvature. Our approach is based on an extension of the method of spherical means to the case of spinor fields and on the use of Riesz distributions.

Algebraic Properties of Curvature Operators in Lorentzian Manifolds with Large Isometry Groups

Together with spaces of constant sectional curvature and products of a real line with a manifold of constant curvature, the socalled Egorov spaces and $\varepsilon$-spaces exhaust the class of $n$-dimensional Lorentzian manifolds admitting a group of isometries of dimension at least ${1/2} n(n-1)+1$, for almost all values of $n$ [Patrangenaru V., Geom. Dedicata 102 (2003), 25-33]. We shall prove that the curvature tensor of these spaces satisfy several interesting algebraic properties. In particular, we will show that Egorov spaces are Ivanov-Petrova manifolds, curvature-Ricci commuting (indeed, semi-symmetric) and $\mathcal P$-spaces, and that $\varepsilon$-spaces are Ivanov-Petrova and curvature-curvature commuting manifolds.

On an invariant on isometric immersions into spaces of constant sectional curvature

In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups G with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of G is not symmetric, then there are no local isometric immersions of G into Q c 4.

η-EINSTEIN TANGENT SPHERE BUNDLES OF CONSTANT RADII

We study the geometric properties of the base manifold for the tangent sphere bundle of radius r satisfying the η-Einstein condition with the standard contact metric structure. One of the main theorems is that the tangent sphere bundle of the n(≥3)-dimensional locally symmetric space, equipped with the standard contact metric structure, is an η-Einstein manifold if and only if the base manifold is a space of constant sectional curvature [Formula: see text] or [Formula: see text].

When are the tangent sphere bundles of a Riemannian manifold η-Einstein?

We study the geometry of a tangent sphere bundle of a Riemannian manifold (M, g). Let M be an n-dimensional Riemannian manifold and TrM be the tangent bundle of M of constant radius r. The main theorem is that TrM equipped with the standard contact metric structure is η-Einstein if and only if M is a space of constant sectional curvature \({\frac{1}{r^2}}\) or \({\frac{n-2}{r^2}}\).

Remarks on η-Einstein unit tangent bundles

We study the geometric properties of the base manifold for the unit tangent bundle satisfying the η-Einstein condition with the canonical contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, is η-Einstein manifold if and only if the base manifold is the space of constant sectional curvature 1 or 2.