Riemannian Metric Research Articles

SSTD: A Novel Spatio-Temporal Demographic Network for EEG-Based Emotion Recognition

Emotion recognition is the key to making machines more intelligent. This study proposes a novel sing-link end-to-end spatio-temporal demographic network (SSTD) that fuses spatial, temporal, and demographic information for electroencephalography (EEG)-based emotion recognition. In the SSTD model, an adaptive time window using single-link hierarchical clustering based on Riemannian metrics was realized for data preprocessing to solve the problem of individual differences. Then, the preprocessed EEG data acted as a gate recurrent unit (GRU) network input to calculate high-level time-domain features. At the same time, the EEG covariance matrices were fed into the symmetric positive definite matrix network (SPDNet) to calculate high-level spatial features. Given the correlation between EEG signals and individual demographic information, gender and age factors were integrated into the spatio-temporal model, resulting in more effective high-level features for EEG-based emotion recognition. Finally, extensive comparative experiments were conducted on two public datasets: DEAP and DREAMER. The average accuracy of valence and arousal on the DEAP dataset are 68.28% and 71.48%, respectively. The average accuracy of valence and arousal on the DREAMER dataset are 76.81% and 81.64%, respectively. Experimental results show that the SSTD model has an excellent recognition performance.

4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data.

We propose a novel framework to learn the spatiotemporal variability in longitudinal 3D shape data sets, which contain observations of objects that evolve and deform over time. This problem is challenging since surfaces come with arbitrary parameterizations and thus, they need to be spatially registered. Also, different deforming objects, hereinafter referred to as 4D surfaces, evolve at different speeds and thus they need to be temporally aligned. We solve this spatiotemporal registration problem using a Riemannian approach. We treat a 3D surface as a point in a shape space equipped with an elastic Riemannian metric that measures the amount of bending and stretching that the surfaces undergo. A 4D surface can then be seen as a trajectory in this space. With this formulation, the statistical analysis of 4D surfaces can be cast as the problem of analyzing trajectories embedded in a nonlinear Riemannian manifold. However, performing the spatiotemporal registration, and subsequently computing statistics, on such nonlinear spaces is not straightforward as they rely on complex nonlinear optimizations. Our core contribution is the mapping of the surfaces to the space of Square-Root Normal Fields (SRNF) where the [Formula: see text] metric is equivalent to the partial elastic metric in the space of surfaces. Thus, by solving the spatial registration in the SRNF space, the problem of analyzing 4D surfaces becomes the problem of analyzing trajectories embedded in the SRNF space, which has a euclidean structure. In this paper, we develop the building blocks that enable such analysis. These include: (1) the spatiotemporal registration of arbitrarily parameterized 4D surfaces even in the presence of large elastic deformations and large variations in their execution rates; (2) the computation of geodesics between 4D surfaces; (3) the computation of statistical summaries, such as means and modes of variation, of collections of 4D surfaces; and (4) the synthesis of random 4D surfaces. We demonstrate the performance of the proposed framework using 4D facial surfaces and 4D human body shapes.

Gravitational lensing in Brill spacetimes

We consider the Brill metric which is an electrovacuum solution to Einstein's field equation. It depends on three parameters, a mass parameter $m$, a NUT parameter $l$ and a charge parameter $e$. If the charge parameter is small, the metric describes a black hole; if it is sufficiently big, it describes a wormhole. We determine the relevant lensing features both in the black-hole and in the wormhole case. In particular, we give formulas for the photon spheres, for the angular radius of the shadow and for the deflection angle. We illustrate the lensing features with the help of an effective potential and in terms of embedding diagrams. To that end we make use of the fact that each lightlike geodesic is contained in a (coordinate) cone and that it is a geodesic of a Riemannian optical metric on this cone. By the Gauss-Bonnet theorem, the sign of the Gaussian curvature of the optical metric determines the sign of the deflection angle. In the wormhole case the deflection angle may be negative which means that light rays are repelled from the center.

On Einstein-reversible mth root Finsler metrics

The theory of mth root Finsler metrics has been applied to Biology, Ecology, Gravitation, Seismic ray theory, etc. It is regarded as a direct generalization of Riemannian metric in a sense, namely, the second root metric is a Riemannian metric. On the other hand, the Riemannian curvature faithfully reveals the local geometric properties of a Riemann–Finsler metric. The reversibility of Riemannian and Ricci curvatures of Finsler metrics is an essential concept in Finsler geometry. Here, we study the Riemannian curvature of the class of third and fourth root [Formula: see text]-metrics. Then, we find the necessary and sufficient condition under which a cubic and fourth root [Formula: see text]-metric be Einstein-reversible.

On the space of riemannian metrics satisfying surgery stable curvature conditions

We utilize a condition for algebraic curvature operators called surgery stability as suggested by the work of Hoelzel to investigate the space of riemannian metrics over closed manifolds satisfying these conditions. Our main result is a parametrized Gromov–Lawson construction with not necessarily trivial normal bundles and shows that the homotopy type of this space of metrics is invariant under surgeries of a suitable codimension. This is a generalization of a well-known theorem by Chernysh and Walsh for metrics of positive scalar curvature. As an application of our method, we show that the space of metrics of positive scalar curvature on quaternionic projective spaces are homotopy equivalent to that on spheres.

On Asymptotically Locally Hyperbolic Metrics with Negative Mass

We construct families of asymptotically locally hyperbolic Riemannian metrics with constant scalar curvature (i.e., time symmetric vacuum general relativistic initial data sets with negative cosmological constant), with prescribed topology of apparent horizons and of the conformal boundary at infinity, and with controlled mass. In particular we obtain new classes of solutions with negative mass.

Sharp systolic inequalities for rotationally symmetric 2-orbifolds

We show that suitably defined systolic ratios are globally bounded from above on the space of rotationally symmetric spindle orbifolds and that the upper bound is attained precisely at so-called Besse metrics, i.e. Riemannian orbifold metrics all of whose geodesics are closed. The first type of systolic ratios that we consider are defined in terms of closed geodesics that lift to contractible loops on certain covers of the unit sphere bundle. The second type of systolic ratios are defined in terms of the kth shortest closed geodesic, where the number k depends on the underlying orbifold. Our results generalize a corresponding result of Abbondandolo, Bramham, Hryniewicz and Salomão for spheres of revolution, even in the manifold case. Moreover, they complement recent results by Abbondandolo, Mazzucchelli and the first named author on local systolic inequalities for Besse Reeb flows on closed 3-manifolds.

A Riemannian metric on hyperbolic components

A Riemannian metric on hyperbolic components

Moufang patterns and geometry of information

Technology of data collection and transmission is based on various mathematical models of encoding. The words Geometry of information refer to such models, whereas the words patterns refer to various sophisticated symmetries appearing naturally in such models. In this paper we show that the symmetries of spaces of probability distributions, endowed with their canonical Riemannian metric of geometry, have the structure of a commutative Moufang loop. We also show that the F-manifold structure on the space of probability distribution can be described in terms of differential 3-webs and Malcev algebras. We then present a new construction of (noncommutative) Moufang loops associated to almost-symplectic structures over finite fields, and use then to construct a new class of code loops with associated quantum error-correcting codes and networks of perfect tensors.

Obstructions to distinguished Riemannian metrics via Lorentzian geometry

Obstructions to distinguished Riemannian metrics via Lorentzian geometry

On isometric immersions of sub-Riemannian manifolds

We study curvature invariants of a sub-Riemannian manifold (i.e., a manifold with a Riemannian metric on a non-holonomic distribution) related to mutual curvature of several pairwise orthogonal subspaces of the distribution, and prove geometrical inequalities for a sub-Riemannian submanifold. As applications, inequalities are proved for submanifolds with mutually orthogonal distributions that include scalar and mutual curvature. For compact submanifolds, inequalities are obtained that are supported by known integral formulas for almost-product manifolds.

Semiclassical analysis and the Agmon-Finsler metric for discrete Schrödinger operators

We study the Agmon estimate and the exponential decay of eigenfunctions for multi-dimensional discrete Schrödinger operators with emphasis on the microlocal analysis on the torus. We first consider the semiclassical setting where semiclassical continuous Schrödinger operators are discretized with the mesh width proportional to the semiclassical parameter. For a general class of potentials, we prove the Agmon estimate for eigenfunctions using the Agmon metric which is a Finsler metric rather than a Riemannian metric. Klein and Rosenberger (2008) proved this by a different argument in the case of a potential minimum. Our argument seems to be simpler. We also prove the Agmon estimate and the optimal anisotropic exponential decay of eigenfunctions for discrete Schrödinger operators in the non-semiclassical standard setting.

Szabos algorithm and applications

In this paper, Szabos algorithm is used as the main tool to find locally symmetric left invariant Riemannian metrics on some $4$-dimensional Lie groups. Locally symmetric left invariant Riemannian Lie groups constitute an important subclass of Riemannian Lie groups with zero-divergence Weyl-tensor the so-called C-manifolds. Some properties of the curvature operator of these 4-dimensional C-manifolds are studied.

Almost Kähler structures on complex hyperbolic space

Complex hyperbolic space has the structure of a solvable Lie group. Based on the recent classification of Riemannian left-invariant metrics we describe all left-invariant almost K?hler structures on that group. The only K?hler structure is the standard structure of the complex hyperbolic space and all others are strictly almost K?hler. We calculated the Chern connection and obtained the characterization by its torsion tensor. It is proved that the only SCF-soliton is the K?hler one.

On the space of initial values strictly satisfying the dominant energy condition

The dominant energy condition imposes a restriction on initial value pairs found on a spacelike hypersurface of a Lorentzian manifold. In this article, we study the space of initial values that satisfy this condition strictly. To this aim, we introduce an index difference for initial value pairs and compare it to its classical counterpart for Riemannian metrics. Recent non-triviality results for the latter will then imply that this space has non-trivial homotopy groups.

Some calculations on Kaluza-Klein metric with respect to lifts in tangent bundle

In the present paper, a Riemannian metric on the tangent bundle, which is another generalization of Cheeger-Gromoll metric and Sasaki metric, is considered. This metric is known as Kaluza-Klein metric in literature which is completely determined by its action on vector fields of type X^{H} and Y^{V}. We obtain the covarient and Lie derivatives applied to the Kaluza-Klein metric with respect to the horizontal and vertical lifts of vector fields, respectively on tangent bundle TM.

Gradient Flows, Adjoint Orbits, and the Topology of Totally Nonnegative Flag Varieties

One can view a partial flag variety in \({\mathbb {C}}^n\) as an adjoint orbit \({\mathcal {O}}_{\lambda }\) inside the Lie algebra of \(n\times n\) skew-Hermitian matrices. We use the orbit context to study the totally nonnegative part of a partial flag variety from an algebraic, geometric, and dynamical perspective. The paper has three main parts: (1) We introduce the totally nonnegative part of \({\mathcal {O}}_{\lambda }\), and describe it explicitly in several cases. We define a twist map on it, which generalizes (in type A) a map of Bloch, Flaschka, and Ratiu (Duke Math. J. 61(1): 41–65, 1990) on an isospectral manifold of Jacobi matrices. (2) We study gradient flows on \({\mathcal {O}}_{\lambda }\) which preserve positivity, working in three natural Riemannian metrics. In the Kähler metric, positivity is preserved in many cases of interest, extending results of Galashin, Karp, and Lam (Adv. Math. 397: Paper No. 108123, 1–23, 2022; Adv. Math. 351: 614–620, 2019). In the normal metric, positivity is essentially never preserved on a generic orbit. In the induced metric, whether positivity is preserved appears to depends on the spacing of the eigenvalues defining the orbit. (3) We present two applications. First, we discuss the topology of totally nonnegative flag varieties and amplituhedra. Galashin, Karp, and Lam (2022, 2019) showed that the former are homeomorphic to closed balls, and we interpret their argument in the orbit framework. We also show that a new family of amplituhedra, which we call twisted Vandermonde amplituhedra, are homeomorphic to closed balls. Second, we discuss the symmetric Toda flow on \({\mathcal {O}}_{\lambda }\). We show that it preserves positivity, and that on the totally nonnegative part, it is a gradient flow in the Kähler metric up to applying the twist map. This extends a result of Bloch, Flaschka, and Ratiu (1990).

O(n)-invariant Riemannian metrics on SPD matrices

Symmetric Positive Definite (SPD) matrices are ubiquitous in data analysis under the form of covariance matrices or correlation matrices. Several O(n)-invariant Riemannian metrics were defined on the SPD cone, in particular the kernel metrics introduced by Hiai and Petz. The class of kernel metrics interpolates between many classical O(n)-invariant metrics and it satisfies key results of stability and completeness. However, it does not contain all the classical O(n)-invariant metrics. Therefore in this work, we investigate super-classes of kernel metrics and we study which key results remain true. We also introduce an additional key result called cometric-stability, a crucial property to implement geodesics with a Hamiltonian formulation. Our method to build intermediate embedded classes between O(n)-invariant metrics and kernel metrics is to give a characterization of the whole class of O(n)-invariant metrics on SPD matrices and to specify requirements on metrics one by one until we reach kernel metrics. As a secondary contribution, we synthesize the literature on the main O(n)-invariant metrics, we provide the complete formula of the sectional curvature of the affine-invariant metric and the formula of the geodesic parallel transport between commuting matrices for the Bures-Wasserstein metric.

Dixmier trace formulas and negative eigenvalues of Schrödinger operators on curved noncommutative tori

In a previous paper we established Cwikel-type estimates on noncommutative tori and used them to get analogues in this setting of the Cwikel-Lieb-Rozenblum (CLR) and Lieb-Thirring inequalities for negative eigenvalues of fractional Schrödinger operators. In this paper, we focus on “curved” NC tori, where the role of the usual Laplacian is played by Laplace-Beltrami operators associated with arbitrary Riemannian metrics. The Cwikel-type estimates of our previous paper are extended to pseudodifferential operators and powers of Laplace-Beltrami operators. There are several applications of these estimates. First, we get Lp-versions of the usual formula for the trace of ΨDOs on NC tori, i.e., for combinations of ΨDOs with Lp-position operators. Next, we get Lp-versions of the analogues for NC tori Connes' trace theorem and Connes' integration formula. They give formulas for the NC integrals (a.k.a. Dixmier traces) of products of Lp-position operators with ΨDOs or powers of the Laplace-Beltrami operators. Moreover, by combining our Cwikel-type estimates with suitable versions of the Birman-Schwinger principle we get versions of the CLR and Lieb-Thirring inequalities for negative eigenvalues of fractional Schrödinger operators associated with powers of Laplace-Beltrami operators and Lp-potentials. As in the original Euclidean case the Lieb-Thirring inequalities imply a dual Sobolev inequality for orthonormal families. Finally, we discuss spectral asymptotics and semiclassical Weyl's laws for our classes of operators on curved NC tori. This supersedes a previous conjecture of [63].

The spheres of Sol

Let Sol be the three-dimensional solvable Lie group equipped with its standard left-invariant Riemannian metric. We give a precise description of the cut locus of the identity, and a maximal domain in the Lie algebra on which the Riemannian exponential map is a diffeomorphism. As a consequence, we prove that the metric spheres in Sol are topological spheres, and we characterize their singular points almost exactly.