Dimensional Manifold Research Articles

Statistical manifolds equipped with semi-symmetric connection and Ricci-soliton equations

Since the semi-symmetric connections have been categorized to metric and non metric cases we study both types of these connections on Ricci-solitons where the ambient manifold involves the statistical structure. We apply the statistical Kenmotsu structure on manifold and determine the constant value [Formula: see text] on Ricci-soliton equation via dimension of manifold. A non-trivial example is provided for these results.

The sectional curvature of the infinite dimensional manifold of Hölder equilibrium probabilities

Abstract Here we consider the discrete time dynamics described by a transformation $T:M \to M$ , where T is either the action of shift $T=\sigma$ on the symbolic space $M=\{1,2, \ldots,d\}^{\mathbb{N}}$ , or, T describes the action of a d to 1 expanding transformation $T:S^1 \to S^1$ of class $C^{1+\alpha}$ (for example $x \to T(x) =\mathrm{d} x $ (mod 1)), where $M=S^1$ is the unit circle. It is known that the infinite-dimensional manifold $\mathcal{N}$ of Hölder equilibrium probabilities is an analytical manifold and carries a natural Riemannian metric. Given a certain normalized Hölder potential A denote by $\mu_A \in \mathcal{N}$ the associated equilibrium probability. The set of tangent vectors X (functions $X: M \to \mathbb{R}$ ) to the manifold $\mathcal{N}$ at the point µA (a subspace of the Hilbert space $L^2(\mu_A)$ ) coincides with the kernel of the Ruelle operator for the normalized potential A. The Riemannian norm $|X|=|X|_A$ of the vector X, which is tangent to $\mathcal{N}$ at the point µA, is described via the asymptotic variance, that is, satisfies $ |X|^2 = \langle X, X \rangle = \lim_{n \to \infty} \frac{1}{n} \int (\sum_{i=0}^{n-1} X\circ T^i )^2 \,\mathrm{d} \mu_A$ . Consider an orthonormal basis Xi, $i \in \mathbb{N}$ , for the tangent space at µA. For any two orthonormal vectors X and Y on the basis, the curvature $K(X,Y)$ is \begin{equation*}K(X,Y) = \frac{1}{4}[ \sum_{i=1}^\infty (\int X Y X_i \,\mathrm{d} \mu_A)^2 - \sum_{i=1}^\infty \int X^2 X_i \,\mathrm{d} \mu_A \int Y^2 X_i \,\mathrm{d} \mu_A ].\end{equation*} When the equilibrium probabilities µA is the set of invariant Markov probabilities on $\{0,1\}^{\mathbb{N}}\subset \mathcal{N}$ , introducing an orthonormal basis $\hat{a}_y$ , indexed by finite words y, we show explicit expressions for $K(\hat{a}_x,\hat{a}_z)$ , which is a finite sum. These values can be positive or negative depending on A and the words x and z. Words $x,z$ with large length can eventually produce large negative curvature $K(\hat{a}_x,\hat{a}_z)$ . If $x, z$ do not begin with the same letter, then $K(\hat{a}_x,\hat{a}_z)=0$ .

Coding of hypersurfaces in Euclidean spaces by a constant vector

Abstract An n n -dimensional Riemannian manifold ( N n , g ) ({N}^{n},g) isometrically immersed in Euclidean space R n + 1 {R}^{n+1} with unit normal ζ \zeta and shape operator S S , for a fixed constant unit vector a → \overrightarrow{{\bf{a}}} in R n + 1 {R}^{n+1} induces a vector field v {\bf{v}} and a smooth function ρ \rho on N n {N}^{n} called its code vector and coding function, respectively. The sextuple ( N n , g , ζ , S , v , ρ ) ({N}^{n},g,\zeta ,S,{\bf{v}},\rho ) is called a coded hypersurface of Euclidean space R n + 1 {R}^{n+1} . In the first result of this article, we show that a compact and simply connected coded hypersurface ( N n , g , ζ , S , v , ρ ) ({N}^{n},g,\zeta ,S,{\bf{v}},\rho ) of R n + 1 {R}^{n+1} has integral of Ricci curvature Ric ( v , v ) {\rm{Ric}}({\bf{v}},{\bf{v}}) with a suitable lower bound if and only if it is isometric to the sphere S b n {S}_{b}^{n} of constant curvature b b . In the second result, we show that a complete and simply connected coded hypersurface ( N n , g , ζ , S , v , ρ ) ({N}^{n},g,\zeta ,S,{\bf{v}},\rho ) of R n + 1 {R}^{n+1} of positive Ricci curvature, with shape operator S S invariant under its code vector field v {\bf{v}} and coding function ρ \rho satisfying the static perfect fluid equation is isometric to S b n {S}_{b}^{n} and the converse also holds. Finally, we show that a complete and simply connected coded hypersurface ( N n , g , ζ , S , v , ρ ) ({N}^{n},g,\zeta ,S,{\bf{v}},\rho ) of R n + 1 {R}^{n+1} has a point p ∈ N n p\in {N}^{n} with Ric ( v , v ) ( p ) > 0 {\rm{Ric}}({\bf{v}},{\bf{v}})(p)\gt 0 and has mean curvature H H constant along integral curves of code vector v {\bf{v}} satisfies H 2 ≥ c {H}^{2}\ge c for a positive constant c c , and the coding function ρ \rho satisfies Δ ρ = − n c ρ \Delta \rho =-nc\rho if and only if it is isometric to S c n {S}_{c}^{n} .

A note on Stein fillability of circle bundles over symplectic manifolds

AbstractWe show that, given a closed integral symplectic manifold of dimension , for every integer , the Boothby–Wang bundle over carries no Stein fillable contact structure. This negatively answers a question raised by Eliashberg. A similar result holds for Boothby–Wang orbibundles. As an application, we prove the non‐smoothability of some isolated singularities.

Silver Structures on the Riemann Extensions

In the present paper we deal with an $n-$dimensional differentiable manifold $M$ with a torsion-free linear connection $\nabla $. Here we study some properties of a silver structure on the cotangent bundle ${{T}^{*}}M$ equipped with the Riemannian extension ${{}^{R}}\nabla$ and obtain a necessary condition for which the silver semi-Riemannian manifold $\left( {{T}^{*}}M{{,}^{R}}\nabla ,S \right)$ to be a locally decomposable.

An upper bound for the first nonzero Steklov eigenvalue

Let $(M^n,g)$ be a complete simply connected $n$-dimensional Riemannian manifold with curvature bounds $\Sect_g\leq \kappa$ for $\kappa\leq 0$ and $\Ric_g\geq(n-1)Kg$ for $K\leq 0$. We prove that for any bounded domain $\Omega \subset M^n$ with diameter $d$ and Lipschitz boundary, if $\Omega^*$ is a geodesic ball in the simply connected space form with constant sectional curvature $\kappa$ enclosing the same volume as $\Omega$, then $\sigma_1(\Omega) \leq C \sigma_1(\Omega^*)$, where $\sigma_1(\Omega)$ and $ \sigma_1(\Omega^*)$ denote the first nonzero Steklov eigenvalues of $\Omega$ and $\Omega^*$ respectively, and $C=C(n,\kappa, K, d)$ is an explicit constant. When $\kappa=K$, we have $C=1$ and recover the Brock-Weinstock inequality, asserting that geodesic balls uniquely maximize the first nonzero Steklov eigenvalue among domains of the same volume, in Euclidean space and the hyperbolic space.

On the Euler number of quasiregularly elliptic manifolds

A compact [Formula: see text]-dimensional Riemannian manifold [Formula: see text] is a quasiregularly elliptic manifold if it admits a nonconstant quasiregular map [Formula: see text]. In this paper, we show that the Euler number of a compact quasiregularly elliptic manifold with infinite fundamental group is zero. We also discuss Morse–Novikov cohomology groups of quasiregularly elliptic manifolds.

Metrics of positive Ricci curvature on simply‐connected manifolds of dimension 6k$6k$

AbstractA consequence of the surgery theorem of Gromov and Lawson is that every closed, simply‐connected 6‐manifold admits a Riemannian metric of positive scalar curvature. For metrics of positive Ricci curvature, it is widely open whether a similar result holds; there are no obstructions known for those manifolds to admit a metric of positive Ricci curvature, while the number of examples known is limited. In this article, we introduce a new description of certain ‐dimensional manifolds via labeled bipartite graphs and use an earlier result of the author to construct metrics of positive Ricci curvature on these manifolds. In this way, we obtain many new examples, both spin and nonspin, of ‐dimensional manifolds with a metric of positive Ricci curvature.

Recurrent Connectivity Shapes Spatial Coding in Hippocampal CA3 Subregions.

Stable and flexible neural representations of space in the hippocampus are crucial for navigating complex environments. However, how these distinct representations emerge from the underlying local circuit architecture remains unknown. Using two-photon imaging of CA3 subareas during active behavior, we reveal opposing coding strategies within specific CA3 subregions, with proximal neurons demonstrating stable and generalized representations and distal neurons showing dynamic and context-specific activity. We show in artificial neural network models that varying the recurrence level causes these differences in coding properties to emerge. We confirmed the contribution of recurrent connectivity to functional heterogeneity by characterizing the representational geometry of neural recordings and comparing it with theoretical predictions of neural manifold dimensionality. Our results indicate that local circuit organization, particularly recurrent connectivity among excitatory neurons, plays a key role in shaping complementary spatial representations within the hippocampus.

Existence and absence of Killing horizons in static solutions with symmetries

Abstract Without specifying a matter field nor imposing energy conditions, we study Killing horizons in $n(\ge 3)$-dimensional static solutions in general relativity with an $(n-2)$-dimensional Einstein base manifold. Assuming linear relations $p_{\rm r}\simeq\chi_{\rm r} \rho$ and $p_2\simeq\chi_{\rm t} \rho$ near a Killing horizon between the energy density $\rho$, radial pressure $p_{\rm r}$, and tangential pressure $p_2$ of the matter field, we prove that any non-vacuum solution satisfying $\chi_{\rm r}<-1/3$ ($\chi_{\rm r}\ne -1$) or $\chi_{\rm r}>0$ does not admit a horizon as it becomes a curvature singularity. For $\chi_{\rm r}=-1$ and $\chi_{\rm r}\in[-1/3,0)$, non-vacuum solutions admit Killing horizons, on which there exists a matter field only for $\chi_{\rm r}=-1$ and $-1/3$, which are of the Hawking-Ellis type~I and type~II, respectively. Differentiability of the metric on the horizon depends on the value of $\chi_{\rm r}$, and non-analytic extensions beyond the horizon are allowed for $\chi_{\rm r}\in[-1/3,0)$. In particular, solutions can be attached to the Schwarzschild-Tangherlini-type vacuum solution at the Killing horizon in at least a $C^{1,1}$ regular manner without a lightlike thin shell. We generalize some of those results in Lovelock gravity with a maximally symmetric base manifold.

On a stratification of positive scalar curvature compact manifolds

For a compact PSC Riemannian [Formula: see text]-manifold [Formula: see text], the metric constant [Formula: see text] is defined to be the infinimum over [Formula: see text] of the spectral scalar curvature [Formula: see text] of [Formula: see text], where [Formula: see text] are the eigenvalues of the curvature operator of [Formula: see text] and [Formula: see text] is the maximal eigenvalue. The functional [Formula: see text] is continuous, re-scale invariant and defines a stratification of the space of PSC metrics over [Formula: see text]. We introduce as well the smooth constant [Formula: see text], which is the supremum of [Formula: see text] over the set of all PSC Riemannian metrics [Formula: see text] on [Formula: see text]. In this paper, we show that in the top layer, compact manifolds with [Formula: see text] are positive space forms. No manifolds have their [Formula: see text] in the interval [Formula: see text]. The manifold [Formula: see text] and arbitrary connected sums of copies of it with connected sums of positive space forms all have [Formula: see text]. For [Formula: see text], we prove that the manifolds [Formula: see text] take the intermediate values [Formula: see text]. From the bottom, we prove that simply connected (resp. [Formula: see text]-connected, [Formula: see text]-connected and non-string) compact manifolds of dimension [Formula: see text] (resp. [Formula: see text], [Formula: see text]) have [Formula: see text] (resp. [Formula: see text], [Formula: see text]). The proof of these last three results is based on surgery. In fact, we prove that the smooth [Formula: see text] constant doesn’t decrease after a surgery on the manifold with adequate codimension.

Riemannian manifold-based disentangled representation learning for multi-site functional connectivity analysis

Functional connectivity (FC), derived from resting-state functional magnetic resonance imaging (rs-fMRI), has been widely used to characterize brain abnormalities in disorders. FC is usually defined as a correlation matrix that is a symmetric positive definite (SPD) matrix lying on the Riemannian manifold. Recently, a number of learning-based methods have been proposed for FC analysis, while the geometric properties of Riemannian manifold have not yet been fully explored in previous studies. Also, most existing methods are designed to target one imaging site of fMRI data, which may result in limited training data for learning reliable and robust models. In this paper, we propose a novel Riemannian Manifold-based Disentangled Representation Learning (RM-DRL) framework which is capable of learning invariant representations from fMRI data across multiple sites for brain disorder diagnosis. In RM-DRL, we first employ an SPD-based encoder module to learn a latent unified representation of FC from different sites, which can preserve the Riemannian geometry of the SPD matrices. In latent space, a disentangled representation module is then designed to split the learned features into domain-specific and domain-invariant parts, respectively. Finally, a decoder module is introduced to ensure that sufficient information can be preserved during disentanglement learning. These designs allow us to introduce four types of training objectives to improve the disentanglement learning. Our RM-DRL method is evaluated on the public multi-site ABIDE dataset, showing superior performance compared with several state-of-the-art methods.

On regularity of \overline∂-solutions on 𝑎_{𝑞} domains with 𝐶² boundary in complex manifolds

We study regularity of solutions u u to ∂ ¯ u = f \overline \partial u=f on a relatively compact C 2 C^2 domain D D in a complex manifold of dimension n n , where f f is a ( 0 , q ) (0,q) form. Assume that there are either ( q + 1 ) (q+1) negative or ( n − q ) (n-q) positive Levi eigenvalues at each point of boundary ∂ D \partial D . Under the necessary condition that a locally L 2 L^2 solution exists on the domain, we show the existence of the solutions on the closure of the domain that gain 1 / 2 1/2 derivative when q = 1 q=1 and f f is in the Hölder–Zygmund space Λ r ( D ) \Lambda ^r( D) with r > 1 r>1 . For q > 1 q>1 , the same regularity for the solutions is achieved when ∂ D \partial D is either sufficiently smooth or of ( n − q ) (n-q) positive Levi eigenvalues everywhere on ∂ D \partial D .

An extended two-parameter mixed-dimensional model of fractured porous media incorporating entrance flow and boundary-layer transition effects

We develop an enhanced reduced model for single-phase flow in fractured porous media capable of incorporating more realistic interface conditions at the fracture terminations. In addition to the traditional dimensional model reduction, where the elements of the discrete fracture network are treated as lower dimensional manifolds embedded in the porous matrix, we explore the microscale behavior of the boundary layer flow at the entrances of a fracture bounded by two parallel plates to construct a new set of interface conditions of Robin-type, giving rise to localized pressure jumps at the fracture edges. Within this enriched description, sharper reduced flow and tracer transport mixed-dimensional models are constructed in the asymptotic limit ruled by two small parameters related to the ratio between fracture aperture and entrance developing length and a macroscopic length scale. The discrete flow/transport mixed-dimensional model is discretized by a new discontinuous Galerkin(dG)-based formulation. An adequate version of the Galerkin-Newton method is developed for the numerical treatment of the non-linear Robin interface condition. Considering several fracture arrangements, numerical results illustrate the sharper description of the model proposed herein in predicting flow and tracer transport patterns in fractured media.

Spinor–Vector Duality and Mirror Symmetry

Mirror symmetry was first observed in worldsheet string constructions, and was shown to have profound implications in the Effective Field Theory (EFT) limit of string compactifications, and for the properties of Calabi–Yau manifolds. It opened up a new field in pure mathematics, and was utilised in the area of enumerative geometry. Spinor–Vector Duality (SVD) is an extension of mirror symmetry. This can be readily understood in terms of the moduli of toroidal compactification of the Heterotic String, which includes the metric the antisymmetric tensor field and the Wilson line moduli. In terms of the toroidal moduli, mirror symmetry corresponds to mappings of the internal space moduli, whereas Spinor–Vector Duality corresponds to maps of the Wilson line moduli. In the past few of years, we demonstrated the existence of Spinor–Vector Duality in the effective field theory compactifications of string theories. This was achieved by starting with a worldsheet orbifold construction that exhibited Spinor–Vector Duality and resolving the orbifold singularities, hence generating a smooth, effective field theory limit with an imprint of the Spinor–Vector Duality. Just like mirror symmetry, the Spinor–Vector Duality can be used to study the properties of complex manifolds with vector bundles. Spinor–Vector Duality offers a top-down approach to the “Swampland” program, by exploring the imprint of the symmetries of the ultra-violet complete worldsheet string constructions in the effective field theory limit. The SVD suggests a demarcation line between (2,0) EFTs that possess an ultra-violet complete embedding versus those that do not.

A space level light bulb theorem in all dimensions

Given a d -dimensional manifold M and a knotted sphere s\colon \mathbb{S}^{k-1}\hookrightarrow \partial M with 1\leq k\leq d , for which there exists a framed dual sphere G\colon \mathbb{S}^{d-k}\hookrightarrow \partial M , we show that the space of neat embeddings \mathbb{D}^{k} \hookrightarrow M with boundary s can be delooped by the space of neatly embedded (k-1) -disks, with a normal vector field, in the d -manifold obtained from M by attaching a handle to G . This increase in codimension significantly simplifies the homotopy type of such embedding spaces, and is of interest also in low-dimensional topology. In particular, we apply the work of Dax to describe the first interesting homotopy group of these embedding spaces, in degree d-2k . In a separate paper we use this to give a complete isotopy classification of 2-disks in a 4-manifold with such a boundary dual.

Generative learning for forecasting the dynamics of high-dimensional complex systems

We introduce generative models for accelerating simulations of high-dimensional systems through learning and evolving their effective dynamics. In the proposed Generative Learning of Effective Dynamics (G-LED), instances of high dimensional data are down sampled to a lower dimensional manifold that is evolved through an auto-regressive attention mechanism. In turn, Bayesian diffusion models, that map this low-dimensional manifold onto its corresponding high-dimensional space, operate on batches of physics correlated, time sequences of data and capture the statistics of the system dynamics. We demonstrate the capabilities and drawbacks of G-LED in simulations of several benchmark systems, including the Kuramoto-Sivashinsky (KS) equation, two-dimensional high Reynolds number flow over a backward-facing step, and simulations of three-dimensional turbulent channel flow. The results demonstrate that generative learning offers new frontiers for the accurate forecasting of the statistical properties of high-dimensional systems at a reduced computational cost.

On the role of the surface geometry in convex billiards

Abstract This work presents a framework for billiards in convex domains on two dimensional Riemannian manifolds. These domains are contained in connected, simply connected open subsets which are totally normal. In this context, some properties that have long been known for billiards on the plane are established. We prove the twist property of the billiard maps and establish some conditions for the existence and non-existence of rotational invariant curves. Although we prove that Lazutkin’s and Hubacher’s theorems are valid for general surfaces, we also find that Mather’s theorem does not apply to surfaces of non-negative curvature.

Heavenly metrics, hyper‐Lagrangians and Joyce structures

AbstractIn [Proc. Sympos. Pure Math., American Mathematical Society, Providence, RI, 2021, pp. 1–66], Bridgeland defined a geometric structure, named a Joyce structure, conjectured to exist on the space of stability conditions of a triangulated category. Given a non‐degeneracy assumption, a feature of this structure is a complex hyper‐Kähler metric with homothetic symmetry on the total space of the holomorphic tangent bundle. Generalising the isomonodromy calculation which leads to the Joyce structure in [Math. Ann. 385 (2023), 193–258], we obtain an explicit expression for a hyper‐Kähler metric with homothetic symmetry via construction of the isomonodromic flows of a Schrödinger equation with deformed polynomial oscillator potential of odd‐degree . The metric is defined on a total space of complex dimension and fibres over a ‐dimensional manifold which can be identified with the unfolding of the ‐singularity. The hyper‐Kähler structure is shown to be compatible with the natural symplectic structure on in the sense of admitting an affine symplectic fibration as defined in [Lett. Math. Phys. 111 (2021), 54]. Separately, using the additional conditions imposed by a Joyce structure, we consider reductions of Plebański's heavenly equations that govern the hyper‐Kähler condition. We introduce the notion of a projectable hyper‐Lagrangian foliation and show that in dimension four such a foliation of leads to a linearisation of the heavenly equation. The hyper‐Kähler metrics constructed here are shown to admit such a foliation.

Volume and Euler classes in bounded cohomology of transformation groups

Abstract Let $M$ be an oriented smooth manifold and $\operatorname{Homeo}\!(M,\omega )$ the group of measure preserving homeomorphisms of $M$ , where $\omega$ is a finite measure induced by a volume form. In this paper, we define volume and Euler classes in bounded cohomology of an infinite dimensional transformation group $\operatorname{Homeo}_0\!(M,\omega )$ and $\operatorname{Homeo}_+\!(M,\omega )$ , respectively, and in several cases prove their non-triviality. More precisely, we define: • Volume classes in $\operatorname{H}_b^n(\operatorname{Homeo}_0\!(M,\omega ))$ , where $M$ is a hyperbolic manifold of dimension $n$ . • Euler classes in $\operatorname{H}_b^2(\operatorname{Homeo}_+(S,\omega ))$ , where $S$ is an oriented closed hyperbolic surface. We show that Euler classes have positive norms for any closed hyperbolic surface and volume classes have positive norms for all hyperbolic surfaces and certain hyperbolic $3$ -manifolds; hence, they are non-trivial.