Riemannian Metric Research Articles

Besse conjecture with positive isotropic curvature

The critical point equation arises as a critical point of the total scalar curvature functional defined on the space of constant scalar curvature metrics of a unit volume on a compact manifold. In this equation, there exists a function f on the manifold that satisfies the following $$\begin{aligned} (1+f)\mathrm{Ric} = Ddf + \frac{nf +n-1}{n(n-1)}sg. \end{aligned}$$ It has been conjectured that if (g, f) is a solution of the critical point equation, then g is Einstein and so (M, g) is isometric to a standard sphere. In this paper, we show that this conjecture is true if the given Riemannian metric has positive isotropic curvature.

The second closed geodesic, the fundamental group, and generic Finsler metrics

For compact manifolds with infinite fundamental group we present sufficient topological or metric conditions ensuring the existence of two geometrically distinct closed geodesics. We also extend results about generic Riemannian metrics to Finsler metrics. We show a bumpy metrics theorem for Finsler metrics and prove that a C^4-generic Finsler metric on a compact and simply-connected manifold carries infinitely many closed geodesics.

Macroscopic scalar curvature and local collapsing

Consider a closed Riemannian $n$-manifold $M$ admitting a negatively curved Riemannian metric. We show that for every Riemannian metric on $M$ of sufficiently small volume, there is a point in the universal cover of $M$ such that the volume of every ball of radius $r \geq 1$ centered at this point is greater or equal to the volume of the ball of the same radius in the hyperbolic $n$-space. We also give an interpretation of this result in terms of macroscopic scalar curvature. This result, which holds more generally in the context of polyhedral length spaces, is related to a question of Guth. Its proof relies on a generalization of recent progress in metric geometry about the Alexandrov/Urysohn width involving the volume of balls of radius in a certain range with collapsing at different scales.

Equivariant Callias index theory via coarse geometry

The equivariant coarse index is well-understood and widely used for actions by discrete groups. We extend the definition of this index to general locally compact groups. We use a suitable notion of admissible modules over $C^*$-algebras of continuous functions to obtain a meaningful index. Inspired by work by Roe, we then develop a localised variant, with values in the $K$-theory of a group $C^*$-algebra. This generalises the Baum-Connes assembly map to non-cocompact actions. We show that an equivariant index for Callias-type operators is a special case of this localised index, obtain results on existence and non-existence of Riemannian metrics of positive scalar curvature invariant under proper group actions, and show that a localised version of the Baum-Connes conjecture is weaker than the original conjecture, while still giving a conceptual description of the $K$-theory of a group $C^*$-algebra.

Some Results On Silver Riemannian Structures

Our aim in this paper is to study of silver Riemannian structures on manifold and bundle. An integrability condition and curvature properties for silver Riemannian structure are investigated via the Tachibana operator. Twin silver Riemannian metric is defined and some properties of twin silver Riemannian metric are investigated. Examples of silver structure are given on tangent and cotangent bundles.

Weight metamorphosis of varifolds and the LDDMM-Fisher-Rao metric

This paper introduces and studies a metamorphosis framework for geometric measures known as varifolds, which extends the diffeomorphic registration model for objects such as curves, surfaces and measures by complementing diffeomorphic deformations with a transformation process on the varifold weights. We consider two classes of cost functionals to penalize those combined transformations, in particular the LDDMM-Fisher-Rao energy which, as we show, leads to a well-defined Riemannian metric on the space of varifolds with existence of corresponding geodesics. We further introduce relaxed formulations of the respective optimal control problems, study their well-posedness and derive optimality conditions for the solutions. From these, we propose a numerical approach to compute optimal metamorphoses between discrete varifolds and illustrate the interest of this model in the situation of partially missing data.

Sharp upper bounds on the length of the shortest closed geodesic on complete punctured spheres of finite area

We establish sharp universal upper bounds on the length of the shortest closed geodesic on a punctured sphere with three or four ends endowed with a complete Riemannian metric of finite area. These sharp curvature-free upper bounds are expressed in terms of the area of the punctured sphere. In both cases, we describe the extremal metrics, which are modeled on the Calabi–Croke sphere or the tetrahedral sphere. We also extend these optimal inequalities for reversible and non-necessarily reversible Finsler metrics. In this setting, we obtain optimal bounds for spheres with a larger number of punctures. Finally, we present a roughly asymptotically optimal upper bound on the length of the shortest closed geodesic for spheres/surfaces with a large number of punctures in terms of the area.

Integrated Construction of Multimodal Atlases with Structural Connectomes in the Space of Riemannian Metrics

The structural network of the brain, or structural connectome, can be represented by fiber bundles generated by a variety of tractography methods. While such methods give qualitative insights into brain structure, there is controversy over whether they can provide quantitative information, especially at the population level. In order to enable population-level statistical analysis of the structural connectome, we propose representing a connectome as a Riemannian metric, which is a point on an infinite-dimensional manifold. We equip this manifold with the Ebin metric, a natural metric structure for this space, to get a Riemannian manifold along with its associated geometric properties. We then use this Riemannian framework to apply object-oriented statistical analysis to define an atlas as the Fréchet mean of a population of Riemannian metrics. This formulation ties into the existing framework for diffeomorphic construction of image atlases, allowing us to construct a multimodal atlas by simultaneously integrating complementary white matter structure details from DWMRI and cortical details from T1-weighted MRI. We illustrate our framework with 2D data examples of connectome registration and atlas formation. Finally, we build an example 3D multimodal atlas using T1 images and connectomes derived from diffusion tensors estimated from a subset of subjects from the Human Connectome Project.

Feature extraction and fault detection scheme via improved locality preserving projection and SVDD

Manifold learning is widely adopted for the fault detection of industrial processes. However, the quality of low-dimensional embedding coordinates can be adversely affected by ill-constructed graph Laplacian. An improved locality preserving projection (ILPP) scheme is proposed. ILPP is built on a geometrically inspired Laplacian, and the Riemannian metric is used to find the suitable bandwidth parameter. The proposed approach combines the advantages of ILPP in preserving manifold data structures and those of support vector data description (SVDD) in handling complex process data distributions. Case studies on helix data, hot strip mill, and Pensim benchmark processes demonstrate the utility and feasibility of the proposed approach. The average fault detection rate for proposed ILPP is 99%, which is higher than locality preserving projection (LPP; 87.8%), local tangent space alignment (LTSA; 74.9%), and principal component analysis (PCA; 90.6%).

Computing the Riemannian Logarithm on the Stiefel Manifold: Metrics, Methods, and Performance

We address the problem of computing Riemannian normal coordinates on the real, compact Stiefel manifold of orthonormal frames. The Riemannian normal coordinates are based on the so-called Riemannian exponential and the associated Riemannian logarithm map and enable one to transfer almost any computational procedure to the realm of the Stiefel manifold. To compute the Riemannian logarithm is to solve the (local) geodesic endpoint problem. Instead of restricting the consideration to geodesics with respect to a single selected metric, we consider a family of Riemannian metrics introduced by Hüper, Markina, and Silva Leite that includes the Euclidean and the canonical metric as prominent examples. As main contributions, we provide (1) a unified, structured, reduced formula for the Stiefel geodesics. The formula is unified in the sense that it works for the full family of metrics under consideration. It is structured in the sense that it relies on matrix exponentials of skew-symmetric matrices exclusively. It is reduced in relation to the dimension of the matrices of which matrix exponentials have to be calculated. We provide (2) a unified method to tackle the geodesic endpoint problem numerically, and (3) we improve the existing Riemannian log algorithm under the canonical metric in terms of the computational efficiency. The findings are illustrated by means of numerical examples, where the novel algorithms prove to be the most efficient methods known to this date.

On the construction of Riemannian three-spaces with smooth inverse mean curvature foliation

Consider a one-parameter family of smooth Riemannian metrics on a two-sphere, {mathscr {S}}. By choosing a one-parameter family of smooth lapse and shift, these Riemannian two-spheres can always be assembled into smooth Riemannian three-space, with metric h_{ij} on a three-manifold Sigma foliated by a one-parameter family of two-spheres {mathscr {S}}_rho . It is shown first that we can always choose the shift such that the {mathscr {S}}_rho surfaces form a smooth inverse mean curvature foliation of Sigma . An integrodifferential expression, referring only to the area of the level sets and the lapse function, is also derived that can be used to quantify the Geroch mass. If the constructed Riemannian three-space happens to be asymptotically flat and the rho -integral of the integrodifferential expression is non-negative, then not only the positive mass theorem but, if one of the {mathscr {S}}_{rho } level sets is a minimal surface, the Penrose inequality also holds. Notably, neither of the above results requires the scalar curvature of the constructed three-metric to be non-negative.

Standard homogeneous (α1,α2)${\big(\alpha _1,\alpha _2\big)}$‐metrics and geodesic orbit property

AbstractIn this paper, we introduce the notion of standard homogeneous ‐metric, as a generalization for the normal homogeneity in Finsler geometry with relatively good computability. We explore the geodesic orbit (g.o. in short) property of this metric. Especially, when it is associated with a triple of compact Lie groups, we find new examples of g.o. Finsler spaces from H. Tamaru's classification list. Meanwhile, we prove that all standard g.o. ‐metrics on the Wallach spaces, , and , must be the normal homogeneous Riemannian metrics.

3-dimensional Levi-Civita metrics with projective vector fields

Projective vector fields are the infinitesimal transformations whose local flow preserves geodesics up to reparametrisation. In 1882, Sophus Lie posed the problem of describing 2-dimensional metrics admitting a non-trivial projective vector field, which was solved in recent years. In the present paper, we solve the analog of Lie's problem in dimension 3, for Riemannian metrics and, more generally, for Levi-Civita metrics of arbitrary signature.

Motor Imagery Classification via Kernel-Based Domain Adaptation on an SPD Manifold.

Background: Recording the calibration data of a brain–computer interface is a laborious process and is an unpleasant experience for the subjects. Domain adaptation is an effective technology to remedy the shortage of target data by leveraging rich labeled data from the sources. However, most prior methods have needed to extract the features of the EEG signal first, which triggers another challenge in BCI classification, due to small sample sets or a lack of labels for the target. Methods: In this paper, we propose a novel domain adaptation framework, referred to as kernel-based Riemannian manifold domain adaptation (KMDA). KMDA circumvents the tedious feature extraction process by analyzing the covariance matrices of electroencephalogram (EEG) signals. Covariance matrices define a symmetric positive definite space (SPD) that can be described by Riemannian metrics. In KMDA, the covariance matrices are aligned in the Riemannian manifold, and then are mapped to a high dimensional space by a log-Euclidean metric Gaussian kernel, where subspace learning is performed by minimizing the conditional distribution distance between the sources and the target while preserving the target discriminative information. We also present an approach to convert the EEG trials into 2D frames (E-frames) to further lower the dimension of covariance descriptors. Results: Experiments on three EEG datasets demonstrated that KMDA outperforms several state-of-the-art domain adaptation methods in classification accuracy, with an average Kappa of 0.56 for BCI competition IV dataset IIa, 0.75 for BCI competition IV dataset IIIa, and an average accuracy of 81.56% for BCI competition III dataset IVa. Additionally, the overall accuracy was further improved by 5.28% with the E-frames. KMDA showed potential in addressing subject dependence and shortening the calibration time of motor imagery-based brain–computer interfaces.

A random cover of a compact hyperbolic surface has relative spectral gap frac{3}{16}-varepsilon

Let X be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature -1. For each nin {mathbf {N}}, let X_{n} be a random degree-n cover of X sampled uniformly from all degree-n Riemannian covering spaces of X. An eigenvalue of X or X_{n} is an eigenvalue of the associated Laplacian operator Delta _{X} or Delta _{X_{n}}. We say that an eigenvalue of X_{n} is new if it occurs with greater multiplicity than in X. We prove that for any varepsilon >0, with probability tending to 1 as nrightarrow infty , there are no new eigenvalues of X_{n} below frac{3}{16}-varepsilon . We conjecture that the same result holds with frac{3}{16} replaced by frac{1}{4}.

Information geometry of operator scaling

Matrix scaling is a classical problem with a wide range of applications. It is known that the Sinkhorn algorithm for matrix scaling is interpreted as alternating e-projections from the viewpoint of classical information geometry. Recently, a generalization of matrix scaling to completely positive maps called operator scaling has been found to appear in various fields of mathematics and computer science, and the Sinkhorn algorithm has been extended to operator scaling. In this study, the operator Sinkhorn algorithm is studied from the viewpoint of quantum information geometry through the Choi representation of completely positive maps. The operator Sinkhorn algorithm is shown to coincide with alternating e-projections with respect to the symmetric logarithmic derivative metric, which is a Riemannian metric on the space of quantum states relevant to quantum estimation theory.

On the Michor–Mumford phenomenon in the infinite dimensional Heisenberg group

In the infinite dimensional Heisenberg group, we construct a left invariant weak Riemannian metric that gives a degenerate geodesic distance. The same construction yields a degenerate sub-Riemannian distance. We show how the standard notion of sectional curvature adapts to our framework, but it cannot be defined everywhere and it is unbounded on suitable sequences of planes. The vanishing of the distance precisely occurs along this sequence of planes, so that the degenerate Riemannian distance appears in connection with an unbounded sectional curvature. In the 2005 paper by Michor and Mumford, this phenomenon was first observed in some specific Fréchet manifolds.

Discrete curve model for non-elastic shape analysis on shape manifold

In this paper, we construct a novel finite dimensional shape manifold for shape analyses. Elements of the shape manifold are a set of discrete, planar, and closed curves, which stand for object boundaries and are represented by direction function. On this manifold, we use a set of N-dimensional Fourier basis to construct the tangent space of the shape manifold as a finite dimensional space. Furthermore, we construct the shape manifold as a Riemannian manifold, in which the Riemannian metric is interpreted as an l2 metric. Our method improves the performance of bending-only models in the issues of shape analysis including the shape synthesis, comparison, and statistic analysis. We evaluate the performance of the manifold via the following applications: 1) shape interpolation and extrapolation between curves, 2) shape retrieval on the Flavia leaf database, 3) shape synthesis using an estimated probability distribution on the manifold, and 4) a novel application named shape arithmetic. All the above experiments clearly demonstrate our approach achieves superior performance to state-of-the-art methods.

Gaussian free fields and Riemannian rigidity

In the present paper, we show that on a compact Riemannian manifold $(M,g)$ of dimension $d\leqslant 4$ whose metric has negative curvature, the renormalized partition function $Z_g(\lambda)$ of a massive Gaussian Free Field determines the length spectrum of $(M,g)$ and imposes some strong geometric constraints on the Riemannian structure of $(M,g)$. In any finite dimensional family of Riemannian metrics of negative sectional curvature bounded from below and above and whose isometry group is trivial, there is only a \textbf{finite number of isometry classes} of metrics with given partition function $Z_g(\lambda)$. When $d<4$, the same result holds true if the random variable $\int_M:\phi^2:dv$ has given probability distribution and without the lower bound on the sectional curvatures.

On Properties of Riemannian Metrics Associated with B-Elliptic Operators

In this paper, we consider a Riemannian metric in which the Laplace–Beltrami operator coincides with a B-elliptic operator up to a factor.