Canonical Metric Research Articles

Stress and Geometry for Isotropic Singularities.

We develop the mathematics needed to treat the interaction of geometry and stress at any isotropic spacetime singularity. This enables us to handle the Einstein equations at the initial singularity and characterize allowed general relativistic stress-energy tensors. Their leading behaviors are dictated by an initial hypersurface conformal embedding. We also show that an isotropic big bang determines a canonical nonsingular metric on and about the initial hypersurface as well as a cosmological time. This assigns a volume and energy to the initial point singularity.

Self-Supervised Lie Algebra Representation Learning via Optimal Canonical Metric

Self-Supervised Lie Algebra Representation Learning via Optimal Canonical Metric

SO(3)-Irreducible Geometry in Complex Dimension Five and Ternary Generalization of Pauli Exclusion Principle

Motivated by a ternary generalization of the Pauli exclusion principle proposed by R. Kerner, we propose a notion of a Z3-skew-symmetric covariant SO(3)-tensor of the third order, consider it as a 3-dimensional matrix, and study the geometry of the 10-dimensional complex space of these tensors. We split this 10-dimensional space into a direct sum of two 5-dimensional subspaces by means of a primitive third-order root of unity q, and in each subspace, there is an irreducible representation of the rotation group SO(3)↪SU(5). We find two SO(3)-invariants of Z3-skew-symmetric tensors: one is the canonical Hermitian metric in five-dimensional complex vector space and the other is a quadratic form denoted by K(z,z). We study the invariant properties of K(z,z) and find its stabilizer. Making use of these invariant properties, we define an SO(3)-irreducible geometric structure on a five-dimensional complex Hermitian manifold. We study a connection on a five-dimensional complex Hermitian manifold with an SO(3)-irreducible geometric structure and find its curvature and torsion.

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.

The Dudeney–Stockmeyer Conjecture

Consider a variant of the famous Tower of Hanoi problem where moves of the discs are allowed along the edges of a connected simple graph D whose vertices represent the vertical pegs of the puzzle. The task is to transfer a tower of n ∈ N discs from one peg to another according to the well-known rules of the classical game corresponding to D ≅ K 3 . Apart from its recreational appeal, the problem brings about fascinating integer sequences engendered by the respective minimum number of moves, i.e. the distance in the canonical metric of the state graph H D n . We construct an algorithm which solves the task for all cases but the exceptional instance of the Linear Tower of Hanoi (i.e. where D is a path graph) in which the transfer is between the two end vertices of D . Optimality of this algorithm is claimed in our central conjecture which is supported by large-scale computational explorations.

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.

Diameter estimate for closed manifolds with positive scalar curvature

For a simply connected closed Riemannian manifold with positive scalar curvature, we prove an upper diameter bound in terms of its scalar curvature integral, the Yamabe constant and the dimension of the manifold. When a manifold has a conformal immersion into a sphere, the dependency on the Yamabe constant is not necessary. The power of scalar curvature integral in these diameter estimates is sharp and it occurs at round spheres with canonical metric.

Almost paracomplex structures on 4-manifolds

Reflection in a line in Euclidean 3-space defines an almost paracomplex structure on the space of all oriented lines, isometric with respect to the canonical neutral Kaehler metric. Beyond Euclidean 3-space, the space of oriented geodesics of any real 3-dimensional space form admits both isometric and anti-isometric paracomplex structures.This paper considers the existence or otherwise of isometric and anti-isometric almost paracomplex structures j on a pseudo-Riemannian 4-manifold (M,g), such that j is parallel with respect to the Levi-Civita connection of g. It is shown that if an isometric or anti-isometric almost paracomplex structure on a conformally flat manifold is parallel, then the scalar curvature of the metric must be zero. In addition, it is found that j is parallel iff the eigenplanes are tangent to a pair of mutually orthogonal foliations by totally geodesic surfaces.The composition of a Riemannian metric with an isometric almost paracomplex structure j yields a neutral metric g′. It is proven that if j is parallel, then g is Einstein iff g′ is conformally flat and scalar flat.The vanishing of the Hirzebruch signature is found to be a necessary topological condition for a closed 4-manifold to admit an Einstein metric with a parallel isometric paracomplex structure. Thus, while the K3 manifold admits an Einstein metric with an isometric paracomplex structure, it cannot be parallel. The same holds true for certain connected sums of complex projective 2-space and its conjugate.

Geodesic Learning With Uniform Interpolation on Data Manifold

Recently with the development of deep learning on data representation and generation, how to sampling on a data manifold becomes a crucial problem for research. In this paper, we propose a method to learn a minimizing geodesic within a data manifold. Along the learned geodesic, our method is able to generate high-quality uniform interpolations with the shortest path between two given data samples. Specifically, we use an autoencoder network to map data samples into the latent space and perform interpolation in the latent space via an interpolation network. We add prior geometric information to regularize our autoencoder for a flat latent embedding. The Riemannian metric on the data manifold is induced by the canonical metric in the Euclidean space in which the data manifold is isometrically immersed. Based on this defined Riemannian metric, we introduce a constant-speed loss and a minimizing geodesic loss to regularize the interpolation network to generate uniform interpolations along the learned geodesic on the manifold. We provide a theoretical analysis of our model and use image interpolation as an example to demonstrate the effectiveness of our method.

Optimization flows landing on the Stiefel manifold⋆

We study a continuous-time system that solves optimization problems over the set of orthonormal matrices, which is also known as the Stiefel manifold. The resulting optimization flow follows a path that is not always on the manifold but asymptotically lands on the manifold. We introduce a generalized Stiefel manifold to which we extend the canonical metric of the Stiefel manifold. We show that the vector field of the proposed flow can be interpreted as the sum of a Riemannian gradient on a generalized Stiefel manifold and a normal vector. Moreover, we prove that the proposed flow globally converges to the set of critical points, and any local minimum and isolated critical point is asymptotically stable.

The Geometry of Off-the-Grid Compressed Sensing

Compressed sensing (CS) ensures the recovery of sparse vectors from a number of randomized measurements proportional to their sparsity. The initial theory considers discretized domains, and the randomness makes the physical positions of the grid nodes irrelevant. Most imaging devices, however, operate over some continuous physical domain, and it makes sense to consider Dirac masses with arbitrary positions. In this article, we consider such a continuous setup and analyze the performance of the BLASSO algorithm, which is the continuous extension of the celebrated LASSO \(\ell ^1\) regularization method. This approach is appealing from a numerical perspective because it avoids to discretize the domain of interest. Previous works considered translation-invariant measurements, such as randomized Fourier coefficients, in which it makes clear that the discrete theory should be extended by imposing a minimum distance separation constraint (often called “Rayleigh limit”) between the Diracs. These prior works, however, rule out many domains and sensing operators of interest, which are not translation invariant. This includes, for instance, Laplace measurements over the positive reals and Gaussian mixture models over the mean-covariance space. Our theoretical advances crucially rely on the introduction of a canonical metric associated with the measurement operator, which is the so-called Fisher geodesic distance. In the case of Fourier measurements, one recovers the Euclidean metric, but this metric can cope with arbitrary (possibly non-translation invariant) domains. Furthermore, it is naturally invariant under joint reparameterization of both the sensing operator and the Dirac locations. Our second and main contribution shows that if the Fisher distance between spikes is larger than a Rayleigh separation constant, then the BLASSO recovers in a stable way a stream of Diracs, provided that the number of measurements is proportional (up to log factors) to the number of Diracs. We measure the stability using an optimal transport distance constructed on top of the Fisher geodesic distance. Our result is (up to log factor) sharp and does not require any randomness assumption on the amplitudes of the underlying measure. Our proof technique relies on an infinite-dimensional extension of the so-called golfing scheme which operates over the space of measures and is of general interest.

Geodesic networks in Liouville quantum gravity surfaces

Recent work has shown that for $\gamma \in (0,2)$, a Liouville quantum gravity (LQG) surface can be endowed with a canonical metric. We prove several results concerning geodesics for this metric. In particular, we completely classify the possible networks of geodesics from a typical point on the surface to an arbitrary point on the surface, as well as the types of networks of geodesics joining two points which occur for a dense set of pairs of points on the surface. This latter result is the $\gamma$-LQG analog of the classification of geodesic networks in the Brownian map due to Angel, Kolesnik, and Miermont (2017). We also show that there is a deterministic $m\in\mathbb N$ such that almost surely any two points are joined by at most $m$ distinct LQG geodesics.

Riemannian Optimization on the Symplectic Stiefel Manifold

The symplectic Stiefel manifold, denoted by $\mathrm{Sp}(2p,2n)$, is the set of linear symplectic maps between the standard symplectic spaces $\mathbb{R}^{2p}$ and $\mathbb{R}^{2n}$. When $p=n$, it reduces to the well-known set of $2n\times 2n$ symplectic matrices. Optimization problems on $\mathrm{Sp}(2p,2n)$ find applications in various areas, such as optics, quantum physics, numerical linear algebra and model order reduction of dynamical systems. The purpose of this paper is to propose and analyze gradient-descent methods on $\mathrm{Sp}(2p,2n)$, where the notion of gradient stems from a Riemannian metric. We consider a novel Riemannian metric on $\mathrm{Sp}(2p,2n)$ akin to the canonical metric of the (standard) Stiefel manifold. In order to perform a feasible step along the antigradient, we develop two types of search strategies: one is based on quasi-geodesic curves, and the other one on the symplectic Cayley transform. The resulting optimization algorithms are proved to converge globally to critical points of the objective function. Numerical experiments illustrate the efficiency of the proposed methods.

ON THE RICCI ITERATION FOR HOMOGENEOUS METRICS ON SPHERES AND PROJECTIVE SPACES

We study the Ricci iteration for homogeneous metrics on spheres and complex projective spaces. Such metrics can be described in terms of modifying the canonical metric on the fibers of a Hopf fibration. When the fibers of the Hopf fibration are circles or spheres of dimension 2 or 7, we observe that the Ricci iteration as well as all ancient Ricci iterations can be completely described using known results. The remaining and most challenging case is when the fibers are spheres of dimension 3. On the 3-sphere itself, using a result of Hamilton on the prescribed Ricci curvature equation, we establish existence and convergence of the Ricci iteration and confirm in this setting a conjecture on the relationship between ancient Ricci iterations and ancient solutions to the Ricci ow. In higher dimensions we obtain sufficient conditions for the solvability of the prescribed Ricci curvature equation as well as partial results on the behavior of the Ricci iteration.

On the Koszul formula in noncommutative geometry

We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul formula, we show that our Levi-Civita connection is a bimodule connection. We construct a spectral triple on a fuzzy sphere and compute the scalar curvature for the Levi-Civita connection associated to a canonical metric.

A metric on the space of neutron star models in general relativity and modified gravity

Some pairs of neutron star models can intuitively be thought of as being ‘closer’ together than others, in the sense that more precise observations might be required to distinguish between them than would be necessary for other pairs. In this paper, borrowing ideas from the study of geometrodynamics, we introduce a mathematical formalism to define a geometric distance between stellar models, to provide a quantitative meaning to this notion of closeness. In particular, it is known that the set of all metrics on a Riemannian manifold itself admits the structure of a Riemannian manifold (‘configuration manifold’), which comes equipped with a canonical metric. By thinking of a stationary star as being a particular 3 + 1 metric, the structure of which is determined through the Tolman–Oppenheimer–Volkoff relations and their generalisations, points on a suitably restricted configuration manifold can be thought of as representing different stars, and distances between these points can be computed. We develop the necessary mathematical machinery to build the configuration manifold of neutron star models, and provide some worked examples to illustrate how this space might be used in future studies of stellar structure.

Linear invariants of complex manifolds and their plurisubharmonic variations

For a bounded domain D and a real number p>0, denote by Ap(D) the linear space of Lp integrable holomorphic functions on D, equipped with the Lp-pseudonorm. We prove that two bounded hyperconvex domains D1⊂Cn and D2⊂Cm are biholomorphic (in particular n=m) if there is a linear isometry between Ap(D1) and Ap(D2) for some 0<p<2. The same result holds for p>2,p≠2,4,⋯, provided that the p-Bergman kernels on D1 and D2 are exhaustive. With this as a motivation, we show that, for all p>0, the p-Bergman kernel on a strongly pseudoconvex domain with C2 boundary or a simply connected homogeneous regular domain is exhaustive. These results show that spaces of pluricanonical sections of complex manifolds equipped with canonical pseudonorms are important linear invariants of complex manifolds. The second part of the present work devotes to studying variations of these invariants. We show that the direct image sheaf of the twisted relative pluricanonical bundle associated to a holomorphic family of Stein manifolds or compact Kähler manifolds is positively curved, with respect to the canonical singular Finsler metric.

The Weyl problem in warped product spaces

In this paper, we discuss the Weyl problem in warped product spaces. We apply the method of continuity and prove the openness of the Weyl problem. A counterexample is constructed to show that the isometric embedding of the sphere with canonical metric is not unique up to an isometry if the ambient warped product space is not a space form. Then, we study the rigidity of the standard sphere if we fixed its geometric center in the ambient space. Finally, we discuss a Shi–Tam type of inequality for the Schwarzschild manifold as an application of our findings.

Scalar curvature of a Levi-Civita connection on the Cuntz algebra with three generators

A differential calculus on the Cuntz algebra with three generators coming from the action of rotation group in three dimensions is introduced. The differential calculus is shown to satisfy Assumptions I–IV of Bhowmick et al. (A new look at Levi-Civita Connection in NCG. arXiv:1606.08142) so that Levi-Civita connection exists uniquely for any pseudo-Riemannian metric in the sense of Bhowmick et al. [2] Scalar curvature is computed for the Levi-Civita connection corresponding to the canonical bilinear metric.

Homogeneous Einstein Finsler Metrics on -dimensional Spheres

AbstractIn this paper, we study a class of homogeneous Finsler metrics of vanishing $S$-curvature on a $(4n+3)$-dimensional sphere. We find a second order ordinary differential equation that characterizes Einstein metrics with constant Ricci curvature $1$ in this class. Using this equation we show that there are infinitely many homogeneous Einstein metrics on $S^{4n+3}$ of constant Ricci curvature $1$ and vanishing $S$-curvature. They contain the canonical metric on $S^{4n+3}$ of constant sectional curvature $1$ and the Einstein metric of non-constant sectional curvature given by Jensen in 1973.