Spherical Space Forms Research Articles

New curvature characterizations for spherical space forms and complex projective spaces

In this paper, we introduce a new positivity notion for curvature of Riemannian manifolds and obtain characterizations for spherical space forms and the complex projective space C P n \mathbb {C}\mathbb {P}^n .

Real-time anomaly detection of the stochastically excited systems on spherical ([formula omitted]) manifold

Advanced analytical tools have become crucial in today’s constantly changing and complex systems. Real-time Principal Geodesic Analysis (RPGA) is a novel technique that provides a unique perspective for analyzing nonlinear data on differentiable manifolds. Traditional linear methods are often inadequate when exploring the complexities of such data. Orthogonal transformation techniques such as Principal Component Analysis (PCA) and Principal Geodesic Analysis (PGA) are highly desirable for condition monitoring stochastically excited systems in domains like mechanical, aerospace, and civil engineering. However, uncertainties and dynamic fluctuations necessitate robust analytical methods for early change detection to ensure safety, performance, and cost-effectiveness. Limitations posed by linear orthogonal transformation techniques such as PCA and its recursive counterparts make it imperative to adapt these techniques to nonlinear situations where data does not evolve in a flat Euclidean space. Significant advancements have been made in this field over recent decades, with data-driven real-time algorithms such as RPCA, RCCA, and RSSA providing reliable solutions for complex multidimensional problems. However, for curved space, the nonlinear RPGA technique takes center stage. It is known for its effectiveness in extracting meaningful information from the complex data stream. This paper explores the foundational concepts and methodologies underlying the transition from linear to nonlinear data analysis. By examining examples such as stochastic geometric oscillator on S2, and the inverted spherical pendulum cart system navigating a rough surface, we illustrate the significance of reliable, real-time damage detection techniques provided by tools such as RPGA.

A New Differentiable Sphere Theorem and Its Applications

In this paper, we use the Lichnerowicz Laplacian to prove new results: the sphere theorem and the integral inequality for Einstein's infinitesimal deformations, which allow us to characterize spherical space forms. Our version of the sphere theorem states that a closed connected Riemannian manifold $(M, g)$ of even dimension $n>3$ is diffeomorphic to a Euclidean sphere or a real projective space if the inequality $Ric_{\rm max}(x) < n K_{\rm min}(x) g$ is true at each point $x\in M$, where $Ric_{\rm max}(x)$ is the maximum of the Ricci curvature, and $K_{\rm min}(x)$ is the minimum of the sectional curvature of $(M, g)$ at $x$. Since this inequality implies positive sectional curvature; therefore, our result partially answers Hopf's old open question.

Metrics for Domain Shift Characterization: Comparisons and New Directions

Domain adaptation is an important area of research, as it aims to remedy the effects of the domain shift due to differences in the distribution between the source domain used for training and the target domain where prediction takes place. However, methods for characterizing the domain shift across datasets are lacking. In this work, we propose a domain shift metric called SpOT, which stands for spherical optimal transport, by operating on the spherical manifold. We realize our approach with a spherical network, used to obtain features, and an orthogonal projection loss, used to impose orthogonality in the feature space. The resulting spherical features have better inter-class separation and lower intra-class variation compared to features in Euclidean space. This type of feature clustering makes each domain representation more compact and more suitable for further analysis. The domain shift between the datasets is calculated using the optimal transport on the spherical features, which has a sound theoretical basis. Our results are further supported by experiments that show the correlation of SpOT with a new gain of transfer measure across domain adaptation datasets.

Differentiable and accelerated spherical harmonic and Wigner transforms

Many areas of science and engineering encounter data defined on spherical manifolds. Modelling and analysis of spherical data often necessitates spherical harmonic transforms, at high degrees, and increasingly requires efficient computation of gradients for machine learning or other differentiable programming tasks. We develop novel algorithmic structures for accelerated and differentiable computation of generalised Fourier transforms on the sphere S2 and rotation group SO(3), i.e. spherical harmonic and Wigner transforms, respectively. We present a recursive algorithm for the calculation of Wigner d-functions that is both stable to high harmonic degrees and extremely parallelisable. By tightly coupling this with separable spherical transforms, we obtain algorithms that exhibit an extremely parallelisable structure that is well-suited for the high throughput computing of modern hardware accelerators (e.g. GPUs). We also develop a hybrid automatic and manual differentiation approach so that gradients can be computed efficiently. Our algorithms are implemented within the JAX differentiable programming framework in the S2FFT▪ software code. Numerous samplings of the sphere are supported, including equiangular and HEALPix sampling. Computational errors are at the order of machine precision for spherical samplings that admit a sampling theorem. When benchmarked against alternative C codes we observe up to a 400-fold acceleration. Furthermore, when distributing over multiple GPUs we achieve very close to optimal linear scaling with increasing number of GPUs due to the highly parallelised and balanced nature of our algorithms. Provided access to sufficiently many GPUs our transforms thus exhibit an unprecedented effective linear time complexity.

Теоремы о дифференцируемых сферах для многообразий с ограниченными сверху кривизнами Риччи

In the present paper, we prove that if is an -dimensional compact Riemannian manifold and if where , and are the sectional and Ricci curvatures of respectively, then is diffeomorphic to a spherical space form where is a finite group of isometries acting freely. In particular, if is simply connected, then it is diffeo­mor­phic to the Euclidian sphere

Topological complexity of $S^3/Q_8$ as fibrewise L-S category

In 2010, M. Sakai and the first author showed that the topological complexity of a space $X$ coincides with the fibrewise unpointed L-S category of a pointed fibrewise space $\proj_{1} \colon X \times X \to X$ with the diagonal map $\Delta \colon X \to X \times X$ as its section. In this paper, we describe our algorithm how to determine the fibrewise L-S category or the Topological Complexity of a topological spherical space form. Especially, for $S^3/Q_8$ where $Q_8$ is the quaternion group, we write a python code to realise the algorithm to determine its Topological Complexity.

Linear Classification of Neural Manifolds with Correlated Variability.

Understanding how the statistical and geometric properties of neural activity relate to performance is a key problem in theoretical neuroscience and deep learning. Here, we calculate how correlations between object representations affect the capacity, a measure of linear separability. We show that for spherical object manifolds, introducing correlations between centroids effectively pushes the spheres closer together, while introducing correlations between the axes effectively shrinks their radii, revealing a duality between correlations and geometry with respect to the problem of classification. We then apply our results to accurately estimate the capacity of deep network data.

Estimating the correlation dimension of a fractal on a sphere

The correlation dimension is a fundamental parameter in order to characterize fractal geometries. The estimation of the correlation dimension on a spherical manifold is of major interest in many applications, especially in climate science and geophysics. In this paper we investigate the use of the Takens estimator of the correlation dimension on a sphere. By relying on two analytically tractable cases, we show how using geodetic and Euclidean metrics ensues different kinds of bias. As a major application, our analysis provides a cue for investigations of fractal geometries in seismology.

Geometrical, topological, and dynamical description of N interacting spin- s particles in a long-range Ising model and their interplay with quantum entanglement

Comprehending the connections between the geometric, topological, and dynamical structures of integrable quantum systems with quantum phenomena exploitable in quantum information tasks, such as quantum entanglement, is a major problem in geometric information science. In this work we investigate these issues in a physical system of $\mathcal{N}$ interacting spin-$\mathtt{s}$ under long-range Ising model. We discover the relevant dynamics, identify the corresponding quantum phase space and we derive the associated Fubini-Study metric. Through the application of the Gauss-Bonnet theorem and the derivation of the Gaussian curvature, we have proved that the dynamics occurs on a spherical topology manifold. Afterwards, we analyze the gained geometrical phase under the arbitrary and cyclic evolution processes and solve the quantum brachistochrone problem by establishing the time-optimal evolution. Moreover, by narrowing the system to a two spin-$\mathtt{s}$ system, we explore the relevant entanglement from two different perspectives; The first is geometrical in nature and involves the investigation of the interplay between the entanglement degree and the geometrical structures, such as the Fubini-Study metric, the Gaussian curvature and the geometrical phase. The second is dynamical in nature and tackles the entanglement effect on the evolution speed and geodesic distance. Additionally, we resolve the quantum brachistochrone problem based on the entanglement degree.

Manifolds with nonnegative curvature operator of the second kind

We investigate the curvature operator of the second kind on Riemannian manifolds and prove several classification results. The first one asserts that a closed Riemannian manifold with three-positive curvature operator of the second kind is diffeomorphic to a spherical space form, improving a recent result of Cao–Gursky–Tran assuming two-positivity. The second one states that a closed Riemannian manifold with three-nonnegative curvature operator of the second kind is either diffeomorphic to a spherical space form, or flat, or isometric to a quotient of a compact irreducible symmetric space. This settles the nonnegativity part of Nishikawa’s conjecture under a weaker assumption.

Half-harmonic gradient flow: aspects of a non-local geometric PDE

<abstract><p>The goal of this paper is to discuss some of the results in the author's previous papers and expand upon the work there by proving two new results: a global weak existence result as well as a first bubbling analysis for the half-harmonic gradient flow in finite time. In addition, an alternative local existence proof to the one provided in <sup>[<xref ref-type="bibr" rid="b47">47</xref>]</sup> is presented based on a fixed-point argument. This preliminary bubbling analysis leads to two potential outcomes for the possibility of finite-time bubbling until a conjecture by Sire, Wei and Zheng, see <sup>[<xref ref-type="bibr" rid="b40">40</xref>]</sup>, is settled: Either there always exists a global smooth solution to the half-harmonic gradient flow without concentration of energy in finite-time, which still allows for the formation of half-harmonic bubbles as $ t \to +\infty $, or finite-time bubbling may occur in a similar way as for the harmonic gradient flow due to energy concentration in finitely many points. In the first part of the introduction to this paper, we provide a survey of the theory of harmonic and fractional harmonic maps and the associated gradient flows. For clarity's sake, we restrict our attention to the case of spherical target manifolds $ S^{n-1} $, but our discussion extends to the general case after taking care of technicalities associated with arbitrary closed target manifolds $ N $ (cf. <sup>[<xref ref-type="bibr" rid="b48">48</xref>]</sup>).</p></abstract>

Data-Efficient Learning via Minimizing Hyperspherical Energy.

Deep learning on large-scale data is currently dominant nowadays. The unprecedented scale of data has been arguably one of the most important driving forces behind its success. However, there still exist scenarios where collecting data or labels could be extremely expensive, e.g., medical imaging and robotics. To fill up this gap, this paper considers the problem of data-efficient learning from scratch using a small amount of representative data. First, we characterize this problem by active learning on homeomorphic tubes of spherical manifolds. This naturally generates feasible hypothesis class. With homologous topological properties, we identify an important connection - finding tube manifolds is equivalent to minimizing hyperspherical energy (MHE) in physical geometry. Inspired by this connection, we propose a MHE-based active learning (MHEAL) algorithm, and provide comprehensive theoretical guarantees for MHEAL, covering convergence and generalization analysis. Finally, we demonstrate the empirical performance of MHEAL in a wide range of applications for data-efficient learning, including deep clustering, distribution matching, version space sampling, and deep active learning.

Spherical contact toric manifolds

Let ( M , α ) (M, \alpha ) be a 2 n + 1 2n+1 -dimensional connected compact contact toric manifold of Reeb type. Suppose the contact form α \alpha is regular, we find conditions under which M M is homeomorphic to S 2 n + 1 S^{2n+1} .

Manifolds with $$4\frac{1}{2}$$-Positive Curvature Operator of the Second Kind

We show that a closed four-manifold with \(4\frac{1}{2}\)-positive curvature operator of the second kind is diffeomorphic to a spherical space form. The curvature assumption is sharp as both \({\mathbb{CP}\mathbb{}}^2\) and \({\mathbb {S}}^3 \times {\mathbb {S}}^1\) have \(4\frac{1}{2}\)-nonnegative curvature operator of the second kind. In higher dimensions \(n\ge 5\), we show that closed Riemannian manifolds with \(4\frac{1}{2}\)-positive curvature operator of the second kind are homeomorphic to spherical space forms. These results are proved by showing that \(4\frac{1}{2}\)-positive curvature operator of the second kind implies both positive isotropic curvature and positive Ricci curvature. Rigidity results for \(4\frac{1}{2}\)-nonnegative curvature operator of the second kind are also obtained.

The prism manifold realization problem III

Every prism manifold can be parametrized by a pair of relatively prime integers p > 1 $p>1$ and q $q$ . In our earlier papers, we determined a complete list of prism manifolds P ( p , q ) $P(p, q)$ that can be realized by positive integral surgeries on knots in S 3 $S^3$ when q < 0 $q<0$ or q > p $q>p$ ; in the present work, we solve the case when 0 < q < p $0<q<p$ . This completes the solution of the realization problem for prism manifolds.

Ricci flow and diffeomorphism groups of 3-manifolds

We complete the proof of the Generalized Smale Conjecture, apart from the case of R P 3 RP^3 , and give a new proof of Gabai’s theorem for hyperbolic 3 3 -manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms, except S 3 S^3 and R P 3 RP^3 , as well as hyperbolic manifolds, to prove that the space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3 3 -manifold X X , the inclusion Isom ⁡ ( X , g ) → Diff ⁡ ( X ) \operatorname {Isom}(X,g)\rightarrow \operatorname {Diff}(X) is a homotopy equivalence for any Riemannian metric g g of constant sectional curvature.

On four-dimensional steady gradient Ricci solitons that dimension reduce

In this paper, we will study the asymptotic geometry of 4-dimensional steady gradient Ricci solitons under the condition that they dimension reduce to 3-manifolds. We will show that such solitons either strongly dimension reduce to a spherical space form S3/Γ or weakly dimension reduce to the 3-dimensional Bryant soliton. We also show that 4-dimensional steady gradient Ricci soliton singularity models with nonnegative Ricci curvature outside a compact set either are Ricci-flat ALE 4-manifolds or dimension reduce to 3-dimensional manifolds. As a further application, we prove that any steady gradient Kähler-Ricci soliton singularity models on complex surfaces with nonnegative Ricci curvature outside a compact set must be hyperkähler ALE 4-manifolds.

Chern classes of spherical CR manifolds

We first construct closed spherical CR manifolds of dimension at least five having non-trivial first Chern class with real coefficients. We next prove a constraint on Chern classes with real coefficients of (not necessarily closed) spherical CR manifolds. Finally, we obtain a topological obstruction to the existence of spherical CR structures on co-oriented contact manifolds.

SPHERICAL TRANSFORMER FOR QUALITY ASSESSMENT OF PEDIATRIC CORTICAL SURFACES.

Brain cortical surfaces, which have an intrinsic spherical topology, are typically represented by triangular meshes and mapped onto a spherical manifold in neuroimaging analysis. Inspired by the strong capability of feature learning in Convolutional Neural Networks (CNNs), spherical CNNs have been developed accordingly and achieved many successes in cortical surface analysis. Motivated by the recent success of the transformer, in this paper, for the first of time, we extend the transformer into the spherical space and propose the spherical transformer, which can better learn contextual and structural features than spherical CNNs. We applied the spherical transformer in the important task of automatic quality assessment of infant cortical surfaces, which is a necessary procedure to identify problematic cases due to extremely low tissue contrast and strong motion effects in pediatric brain MRI studies. Experiments on 1,860 infant cortical surfaces validated its superior effectiveness and efficiency in comparison with spherical CNNs.