Spherical Space Forms Research Articles

Four-dimensional steady gradient Ricci solitons with 3-cylindrical tangent flows at infinity

In this paper we consider 4-dimensional steady soliton singularity models, i.e., complete steady gradient Ricci solitons that arise as the rescaled limit of a finite time singular solution of the Ricci flow on a closed 4-manifold. In particular, we study the geometry at infinity of such Ricci solitons under the assumption that their tangent flow at infinity is the product of R with a 3-dimensional spherical space form. We also classify the tangent flows at infinity of 4-dimensional steady soliton singularity models in general.

Enhanced Bounds for rho-invariants for both general and spherical 3-manifolds

We establish enhanced bounds on Cheeger–Gromov [Formula: see text]-invariants for general 3-manifolds and yet stronger bounds for special classes of 3-manifold. As key ingredients, we construct chain null-homotopies whose complexity is linearly bounded by its boundary. This result can be regarded as an algebraic topological analogue of Gromov’s conjecture for quantitative topology. The author hopes for applications to various fields including the smooth knot concordance group, quantitative topology and complexity theory.

Cellularization for exceptional spherical space forms and the flag manifold of [formula omitted

We construct an explicit equivariant cellular decomposition of the (4n−1)-sphere with respect to binary polyhedral groups, and describe the associated cellular chain complex. As a corollary of the binary octahedral case, we deduce an S3-equivariant decomposition of the flag manifold of SL3(R).

On positive scalar curvature cobordisms and the conformal Laplacian on end-periodic manifolds

We show that the periodic $\eta$-invariants introduced by Mrowka--Ruberman--Saveliev~\cite{MRS3} provide obstructions to the existence of cobordisms with positive scalar curvature metrics between manifolds of dimensions $4$ and $6$. The proof combines a relative version of the Schoen--Yau minimal surface technique with an end-periodic index theorem for the Dirac operator. As a result, we show that the bordism groups $\Omega^{spin,+}_{n+1}(S^1 \times BG)$ are infinite for any non-trivial group $G$ which is the fundamental group of a spin spherical space form of dimension $n=3$ or $5$.

Bundles with even-dimensional spherical space form as fibers and fiberwise quarter pinched Riemannian metrics

Let E E be a smooth bundle with fiber an n n -dimensional real projective space R P n \mathbb {R}P^n . We show that, if every fiber carries a positively curved pointwise strongly 1 / 4 1/4 -pinched Riemannian metric that varies continuously with respect to its base point, then the structure group of the bundle reduces to the isometry group of the standard round metric on R P n \mathbb {R}P^n .

Hairy black holes in disformal scalar-tensor gravity theories

We show that the no-hair theorem for scalar-tensor theories with bimetric structure can be evaded. We find that hairy black hole solutions in the presence of an electric charge admit anti-de Sitter (AdS), flat or de Sitter asymptotics with spherical, flat, or hyperbolic base manifolds. Spherically symmetric, asymptotically flat black holes and asymptotically AdS configurations with any horizon topology are compatible with a regular scalar field on and outside the event horizon. The latter presents a rich thermodynamic behavior induced by the disformal factor that enters as a coupling parameter in the theory. In the grand canonical ensemble, there is an interplay of stability and first-order phase transitions between thermal AdS, the hairy black hole, and the Reissner-Nordstr\"om-AdS black hole, whose thermodynamic phase space resembles a solid-liquid-gas system, with an electric potential playing the role of pressure. In close analogy, there is a triple point where the three phases coexist, being equally probable.

Regularity of optimal transport maps on locally nearly spherical manifolds

Given a compact connected n-dimensional Riemannian manifold, we investigate the smoothness of the optimal transport map between the smooth densities with respect to the squared Riemannian distance cost. The optimal map is characterized by exp(gradu), where the potential function u satisfies a Monge–Ampère type equation. Delanoë [7] showed the smoothness of u on the Riemannian surfaces when the scalar curvature is close to 1 in C 2 norm. In this work, we study the regularity issue on Riemannian manifolds with curvature sufficiently close to curvature of round sphere in C 2 norm in all dimensions and prove that the 𝒞-curvature on such Riemannian manifolds satisfies an improved Ma-Trudinger-Wang condition and the Jacobian of the exponential map is positive. As a consequence, we imply the smoothness of the optimal transport map by the continuity method.

Geometric graph manifolds with non‐negative scalar curvature

We classify $n$-dimensional geometric graph manifolds with nonnegative scalar curvature, and first show that if $n>3$, the universal cover splits off a codimension 3 Euclidean factor. We then proceed with the classification of the 3-dimensional case by showing that such a manifold is either a lens space or a prism manifold with a very rigid metric. This allows us to also classify the moduli space of such metrics: it has infinitely many connected components for lens spaces, while it is connected for prism manifolds.

Application of the Generalized Bochner Technique to the Study of Conformally Flat Riemannian Manifolds

In this article, we discuss the global aspects of the geometry of locally conformally flat (complete and compact) Riemannian manifolds. In particular, the article reviews and improves some results (e.g., the conditions of compactness and degeneration into spherical or flat space forms) on the geometry “in the large" of locally conformally flat Riemannian manifolds. The results presented here were obtained using the generalized and classical Bochner technique, as well as the Ricci flow.

An Upper Bound on Pachner Moves Relating Geometric Triangulations

We show that any two geometric triangulations of a closed hyperbolic, spherical, or Euclidean manifold are related by a sequence of Pachner moves and barycentric subdivisions of bounded length. This bound is in terms of the dimension of the manifold, the number of top dimensional simplexes, and bound on the lengths of edges of the triangulation. This leads to an algorithm to check from the combinatorics of the triangulation and bounds on lengths of edges, if two geometrically triangulated closed hyperbolic or low dimensional spherical manifolds are isometric or not.

Cartography as Spatial Representation: A new assessment of the competing advantages and drawbacks across fields of science

The history of cartography has been marked by the endless search for the perfect form for the representation of the information on a spherical surface manifold into the flat planar format of the printed page or computer screen. Dozens of cartographic formats have been proposed over the centuries from ancient Greek times to the present. This is an issue not just for the mapping of the globe, but in all fields of science where spherical entities are found. The perceptual and representational advantages and drawbacks of many of these formats are considered, particularly in the tension between a unified representation, which is always distorted in some dimension, and a minimally distorted representation, which can only be obtained by segmentation into sectorial patches. The use of these same formats for the mapping of spherical manifolds are evaluated, from quantum physics through the mapping of the brain to the large-scale representation of the cosmos.

Single Parameter Model for Cosmic Scale Photon Redshift in a Closed Universe

A successful single parameter model has been formulated to match the observations of photons from type 1a supernovae which were previously used to corroborate the standard &#120556 cold dark matter model. The new single parameter model extrapolates all the way back to the cosmic background radiation (CMB) without requiring a separate model to describe inflation of the space dimensions after the Big Bang. This single parameter model assumes that spacetime forms a finite symmetrical manifold with positive curvature. For the spacetime manifold to be finite, the time dimension must also have positive curvature. This model was formulated to consider whether the curvature of the time dimension may be related to the curvature of the space dimensions. This possibility is not considered in the more complex models previously used to fit the available redshift data. The geometry for the finite spacetime manifold was selected to be compatible with the Friedmann equation with positive curvature. The manifold shape was motivated by an assumption that there exists a matter hemisphere (when considering time together with a single space dimension) and an antimatter hemisphere to give a symmetrical and spherical overall spacetime manifold. Hence, the space dimension expands from a pole to the equator, at a maximum value for the time dimension. This is analogous to the expansion of a circle of latitude on a globe from a pole to the equator. The three space dimensions are identical so that any arbitrary single space direction may be selected. The initial intention was to modify the assumed geometry for the spacetime manifold to account for the presence of matter. It was surprisingly found that, within the error of the reported measurements, no further modification was necessary to fit the data. The Friedmann equation reduces to the Schwarzschild equation at the equator so this can be used to predict the total amount of mass in the Universe. The resulting prediction is of the order of 1051 kg. The corresponding density of matter at the current time is approximately 1.6 × 10-28 kg·m-3.

Monoids of self-maps of topological spherical space forms

A topological spherical space form is the quotient of a sphere by a free action of a finite group. In general, their homotopy types depend on specific actions of a group. We show that the monoid of homotopy classes of self-maps of a topological spherical space form is determined by the acting group and the dimension of the sphere, not depending on a specific action.

On the Einstein condition for Lorentzian 3-manifolds

It is well known that in Lorentzian geometry there are no compact spherical space forms; in dimension 3, this means there are no closed Einstein 3-manifolds with positive Einstein constant. We generalize this fact here, by proving that there are also no closed Lorentzian 3-manifolds (M,g) whose Ricci tensor satisfiesRic=fg+(f−λ)T♭⊗T♭, for any unit timelike vector field T, any positive constant λ, and any nontrivial function f that never takes the value λ. (Observe that this reduces to the positive Einstein case when f=λ.) We show that there is no such obstruction if λ is negative. Finally, the “borderline” case λ=0 is also examined: we show that if λ=0, then (M,g) must be isometric to (S1×N,−dt2⊕h) with (N,h) a Riemannian manifold.

Tracking With CNN Based Correlation Filters on Spherical Manifolds

The correlation filters tracker allows features from multiple channels. The fusion of features by simply summing over them in an Euclidean space would destroy the inherent geometry among multiple features, resulting in the lose of their phase information which is crucial to tracking task. To provide a better fusion of features from multiple layers of a convolutional neural network (CNN) in the classical CNNs based correlation filters algorithm, we introduce spherical manifolds and computing intrinsic mean on spherical manifolds in the article, so that fusion of features and online update of filter kernels can be implemented over a spherical manifolds. In addition, we introduce a random projection method, imposed on CNN features before feature fusion to compress features for the sake of reducing computational complexity and modeling complexity. Extensive experiments on OTB-50 dataset demonstrate that the proposed algorithm outperforms state-of-the-art methods with respect to both precision and success rate.

On the classification of -dimensional spherical Sasakian manifolds

In this article we regard spherical hypersurfaces in with a fixed Reeb vector field as -dimensional Sasakian manifolds. We establish a correspondence between three different sets of parameters, namely, those arising from representing the Reeb vector field as an automorphism of the Heisenberg sphere, those used in Stanton’s description of rigid spheres, and those arising from the rigid normal forms. We also describe geometrically the moduli space for rigid spheres and provide a geometric distinction between Stanton hypersurfaces and those found in [1]. Finally, we determine the Sasakian automorphism groups of rigid spheres and detect the homogeneous Sasakian manifolds among them.

Geometric bistellar moves relate geometric triangulations

A geometric triangulation of a Riemannian manifold is a triangulation where the interior of each simplex is totally geodesic. Bistellar moves are local changes to the triangulation which are higher dimensional versions of the flip operation of triangulations in a plane. We show that geometric triangulations of a compact hyperbolic, spherical or Euclidean manifold are connected by geometric bistellar moves (possibly adding or removing vertices), after taking sufficiently many derived subdivisions. For dimensions 2 and 3, we show that geometric triangulations of such manifolds are directly related by geometric bistellar moves (without having to take derived subdivision).

Statistical learning theory of structured data

The traditional approach of statistical physics to supervised learning routinely assumes unrealistic generative models for the data: Usually inputs are independent random variables, uncorrelated with their labels. Only recently, statistical physicists started to explore more complex forms of data, such as equally labeled points lying on (possibly low-dimensional) object manifolds. Here we provide a bridge between this recently established research area and the framework of statistical learning theory, a branch of mathematics devoted to inference in machine learning. The overarching motivation is the inadequacy of the classic rigorous results in explaining the remarkable generalization properties of deep learning. We propose a way to integrate physical models of data into statistical learning theory and address, with both combinatorial and statistical mechanics methods, the computation of the Vapnik-Chervonenkis entropy, which counts the number of different binary classifications compatible with the loss class. As a proof of concept, we focus on kernel machines and on two simple realizations of data structure introduced in recent physics literature: k-dimensional simplexes with prescribed geometric relations and spherical manifolds (equivalent to margin classification). Entropy, contrary to what happens for unstructured data, is nonmonotonic in the sample size, in contrast with the rigorous bounds. Moreover, data structure induces a transition beyond the storage capacity, which we advocate as a proxy of the nonmonotonicity, and ultimately a cue of low generalization error. The identification of a synaptic volume vanishing at the transition allows a quantification of the impact of data structure within replica theory, applicable in cases where combinatorial methods are not available, as we demonstrate for margin learning.

Spherical 3-Manifolds Bounding Rational Homology Balls

We give a complete classification of the spherical 3-manifolds that bound smooth rational homology 4-balls. Furthermore, we determine the order of spherical 3-manifolds in the rational homology cobordism group of rational homology 3-spheres. To this end, we use constraints for 3-manifolds to bound rational homology balls induced from Donaldson’s diagonalization theorem and Heegaard Floer correction terms.

Logarithmic negativity in quantum Lifshitz theories

We investigate quantum entanglement in a non-relativistic critical system by calculating the logarithmic negativity of a class of mixed states in the quantum Lifshitz model in one and two spatial dimensions. In 1+1 dimensions we employ a correlator approach to obtain analytic results for both open and periodic biharmonic chains. In 2+1 dimensions we use a replica method and consider spherical and toroidal spatial manifolds. In all cases, the universal finite part of the logarithmic negativity vanishes for mixed states defined on two disjoint components. For mixed states defined on adjacent components, we find a non-trivial logarithmic negativity reminiscent of two-dimensional conformal field theories. As a byproduct of our calculations, we obtain exact results for the odd entanglement entropy in 2+1 dimensions.