Riemannian Metric Research Articles

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.

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.

Some canonical metrics via Aubin's local deformations

English: In this paper, using special metric deformations introduced by Aubin, we construct Riemannian metrics satisfying non-vanishing conditions concerning the Weyl tensor, on every compact manifold. In particular, in dimension four, we show that there are no topological obstructions for the existence of metrics with non-vanishing Bach tensor.

Metric flows with neural networks

Abstract We develop a general theory of flows in the space of Riemannian metrics induced by neural network (NN) gradient descent. This is motivated in part by recent advances in approximating Calabi–Yau metrics with NNs and is enabled by recent advances in understanding flows in the space of NNs. We derive the corresponding metric flow equations, which are governed by a metric neural tangent kernel (NTK), a complicated, non-local object that evolves in time. However, many architectures admit an infinite-width limit in which the kernel becomes fixed and the dynamics simplify. Additional assumptions can induce locality in the flow, which allows for the realization of Perelman’s formulation of Ricci flow that was used to resolve the 3d Poincaré conjecture. We demonstrate that such fixed kernel regimes lead to poor learning of numerical Calabi–Yau metrics, as is expected since the associated NNs do not learn features. Conversely, we demonstrate that well-learned numerical metrics at finite-width exhibit an evolving metric-NTK, associated with feature learning. Our theory of NN metric flows therefore explains why NNs are better at learning Calabi–Yau metrics than fixed kernel methods, such as the Ricci flow.

Analysis of a Dry Friction Force Law for the Covariant Optimal Control of Mechanical Systems with Revolute Joints

This contribution shows a geometric optimal control procedure to solve the trajectory generation problem for the navigation (generic motion) of mechanical systems with revolute joints. The mechanical system is analyzed as a nonlinear Lagrangian system affected by dry friction at the joint level. Rayleigh’s dissipation function is used to model this dissipative effect of joint-level friction, and regarded as a potential. Rayleigh’s potential is an invariant scalar quantity from which friction forces derive and are represented by a smooth model that approaches the traditional Coulomb’s law in our proposal. For the optimal control procedure, an invariant cost function is formed with the motion equations and a Riemannian metric. The goal is to minimize the consumed energy per unit time of the system. Covariant control equations are obtained by applying Pontryagin’s principle, and time-integrated using a Finite Elements Method-based solver. The obtained solution is an optimal trajectory that is then applied to a mechanical system using a proportional–derivative plus feedforward controller to guarantee the trajectory tracking control problem. Simulations and experiments confirm that including joint-level friction forces at the modeling stage of the optimal control procedure increases performance, compared with scenarios where the friction is not taken into account, or when friction compensation is performed at the feedback level during motion control.

Is the Cyclic Model of the Universe Possible in the Relativistic Theory of Gravitation?

For the flat FLRW model of Universe evolution in RTG a new model of Dark Energy is proposed. It is a global scalar field Φ with the quadratic potential. It ensures cosmological acceleration at the present time and a bounce at the latе times. At the contraction stage Kazner-like growing anisotropy of Riemannian metrics will break a mass-of- the-graviton bounce mechanism near the Big Bang in FLRW case. There is also noncyclic option, when small enough graviton-mass-terms are significant only at the end of expansion. After bounce, during next contraction epoch, an anisotropy grows and the matter density finally reaches the Planck one.

Automatic Classification of Sleep Stages from EEG Signals Using Riemannian Metrics and Transformer Networks

Automatic Classification of Sleep Stages from EEG Signals Using Riemannian Metrics and Transformer Networks

Log-Cholesky filtering of diffusion tensor fields: Impact on noise reduction

Diffusion tensor imaging (DTI) is a powerful neuroimaging technique that provides valuable insights into the microstructure and connectivity of the brain. By measuring the diffusion of water molecules along neuronal fibers, DTI allows the visualization and study of intricate networks of neural pathways.DTI is a noise-sensitive method, where a low signal-to-noise ratio (SNR) results in significant errors in the estimated tensor field. Tensor field regularization is an effective solution for noise reduction.Diffusion tensors are represented by symmetric positive-definite (SPD) matrices. The space of SPD matrices may be viewed as a Riemannian manifold after defining a suitable metric on its tangent bundle. The Log-Cholesky metric is a recently developed concept with advantages over previously defined Riemannian metrics, such as the affine-invariant and Log-Euclidean metrics. The utility of the Log-Cholesky metric for tensor field regularization and noise reduction has not been investigated in detail.This manuscript provides a quantitative investigation of the impact of Log-Cholesky filtering on noise reduction in DTI. It also provides sufficient details of the linear algebra and abstract differential geometry concepts necessary to implement this technique as a simple and effective solution to filtering diffusion tensor fields.

A Riemannian geometric approach for timelike and null spacetime geodesics

The geodesic motion in a Lorentzian spacetime can be described by trajectories in a 3-dimensional Riemannian metric. In this article we present a generalized Jacobi metric obtained from projecting a Lorentzian metric over the directions of its Killing vectors. The resulting Riemannian metric inherits the geodesics for asymptotically flat spacetimes including the stationary and axisymmetric ones. The method allows us to find Riemannian metrics in three and two dimensions plus the radial geodesic equation when we project over three different directions. The 3-dimensional Riemannian metric reduces to the Jacobi metric when static, spherically symmetric and asymptotically flat spacetimes are considered. However, it can be calculated for a larger variety of metrics in any number of dimensions. We show that the geodesics of the original spacetime metrics are inherited by the projected Riemannian metric. We calculate the Gaussian and geodesic curvatures of the resulting 2-dimensional metric, we study its near horizon and asymptotic limit. We also show that this technique can be used for studying the violation of the strong cosmic censorship conjecture in the context of general relativity.

Analysis, Geometry and Topology of Positive Scalar Curvature Metrics

Riemannian metrics with positive scalar curvature play an important role in differential geometry and general relativity. To investigate these metrics, it is necessary to employ concepts and techniques from global analysis, geometric topology, metric geometry, index theory, and general relativity. This workshop brought together researchers from a variety of backgrounds to combine their expertise and promote cross-disciplinary exchange.

Information geometry and parameter sensitivity of non-Hermitian Hamiltonians

Information geometry is the application of differential geometry in statistics, where the Fisher-Rao metric serves as the Riemannian metric on the statistical manifold, providing an intrinsic property for parameter sensitivity. In this paper, we explore the application of information geometry in the realm of non-Hermitian quantum systems, focusing on the Fisher-Rao metric as a measure of parameter sensitivity. We approximate the Lindblad master equation for non-Hermitian Hamiltonians to analyze the temporal evolution of the quantum geometric metric. Utilizing the quantum spin Ising model with an imaginary magnetic field as an exemplar, we investigate the energy spectrum and geometric metric evolution within PT-symmetry Hamiltonians. We demonstrate that the detrimental effects of dissipation can be counteracted by introducing a control Hamiltonian, leading to improved accuracy in parameter estimation. Our work provides insights into the role of quantum control in mitigating dissipative impacts and enhancing the precision of quantum metrological tasks.

Deformable surface reconstruction via Riemannian metric preservation

Estimating the pose of an object from a monocular image is a fundamental inverse problem in computer vision. Due to its ill-posed nature, solving this problem requires incorporating deformation priors. In practice, many materials do not perceptibly shrink or extend when manipulated, constituting a reliable and well-known prior. Mathematically, this translates to the preservation of the Riemannian metric. Neural networks offer the perfect playground to solve the surface reconstruction problem as they can approximate surfaces with arbitrary precision and allow the computation of differential geometry quantities. This paper presents an approach for inferring continuous deformable surfaces from a sequence of images, which is benchmarked against several techniques and achieves state-of-the-art performance without the need for offline training. Being a method that performs per-frame optimization, our method can refine its estimates, contrary to those based on performing a single inference step. Despite enforcing differential geometry constraints at each update, our approach is the fastest of all the tested optimization-based methods.

Edge morphology attention mechanism and optimal geometric matching connection model for vascular segmentation

Over the past decades, medical image segmentation methods have been intensively developed, but there are still a number of unsolved challenging problems in vascular image segmentation, including broken vessels, insufficient vessel branches, and missing small vessels. In order to optimize the topology and accuracy of segmented vessels, we propose a novel Edge Morphology Attention Network (EMA-Net) for the segmentation of vessel-like structures, and an Optimal Geometric Matching Connection (OGMC) model to connect the broken vessel fragments. The EMA-Net has an edge attention module that is able to improve segmented performances of edges and small branches by morphology operation extracting boundary voxels on multi-scales. The OGMC model uses the geometry concept of curve touching to filter out fragmented vessel endpoints, and then employs minimal surfaces to determine the optimal connection order between blood vessels. Finally, we adopt geodesics to repair missing vessels under a Riemannian metric. Our method achieves superior or competitive results compared to state-of-the-art methods on four datasets of 3D vascular segmentation tasks, both effectively reducing vessel broken and increasing vessel branch richness, yielding blood vessels with more precise topological structures.

The phase space distance between collider events

How can one fully harness the power of physics encoded in relativistic N-body phase space? Topologically, phase space is isomorphic to the product space of a simplex and a hypersphere and can be equipped with explicit coordinates and a Riemannian metric. This natural structure that scaffolds the space on which all collider physics events live opens up new directions for machine learning applications and implementation. Here we present a detailed construction of the phase space manifold and its differential line element, identifying particle ordering prescriptions that ensure that the metric satisfies necessary properties. We apply the phase space metric to several binary classification tasks, including discrimination of high-multiplicity resonance decays or boosted hadronic decays of electroweak bosons from QCD processes, and demonstrate powerful performance on simulated data. Our work demonstrates the many benefits of promoting phase space from merely a background on which calculations take place to being geometrically entwined with a theory’s dynamics.

Dimension two holomorphic distributions on four-dimensional projective space

We study two-dimensional holomorphic distributions on P4. We classify dimension two distributions, of degree at most 2, with either locally free tangent sheaf or locally free conormal sheaf and whose singular scheme has pure dimension one. We show that the corresponding sheaves are split. Next, we investigate the geometry of such distributions, studying from maximally non–integrable to integrable distributions. In the maximally non–integrable case, we show that the distribution is either of Lorentzian type or a push-forward by a rational map of the Cartan prolongation of a singular contact structure on a weighted projective 3-fold. We study distributions of dimension two in P4 whose conormal sheaves are the Horrocks–Mumford sheaves, describing the numerical invariants of their singular schemes which are smooth and connected. Such distributions are maximally non–integrable, uniquely determined by their singular schemes and invariant by a group H5⋊SL(2,Z5)⊂Sp(4,Q), where H5 is the Heisenberg group of level 5. We prove that the moduli spaces of Horrocks–Mumford distributions are irreducible quasi-projective varieties and we determine their dimensions. Finally, we observe that the space of codimension one distributions, of degree d≥6, on P4 has a family of degenerate flat holomorphic Riemannian metrics. Moreover, the degeneracy divisors of such metrics consist of codimension one distributions invariant by H5⋊SL(2,Z5) and singular along a degenerate abelian surface with (1,5)-polarization and level-5-structure.

Feature-Based Interpolation and Geodesics in the Latent Spaces of Generative Models.

Interpolating between points is a problem connected simultaneously with finding geodesics and study of generative models. In the case of geodesics, we search for the curves with the shortest length, while in the case of generative models, we typically apply linear interpolation in the latent space. However, this interpolation uses implicitly the fact that Gaussian is unimodal. Thus, the problem of interpolating in the case when the latent density is non-Gaussian is an open problem. In this article, we present a general and unified approach to interpolation, which simultaneously allows us to search for geodesics and interpolating curves in latent space in the case of arbitrary density. Our results have a strong theoretical background based on the introduced quality measure of an interpolating curve. In particular, we show that maximizing the quality measure of the curve can be equivalently understood as a search of geodesic for a certain redefinition of the Riemannian metric on the space. We provide examples in three important cases. First, we show that our approach can be easily applied to finding geodesics on manifolds. Next, we focus our attention in finding interpolations in pretrained generative models. We show that our model effectively works in the case of arbitrary density. Moreover, we can interpolate in the subset of the space consisting of data possessing a given feature. The last case is focused on finding interpolation in the space of chemical compounds.

Twistor theory of the Chen–Teo gravitational instanton**Dedicated to Nick Woodhouse on the occasion of his 75th birthday.

Abstract Toric Ricci--flat metrics in dimension four correspond to certain holomorphic vector bundles over a
twistor space. We construct these bundles explicitly, by exhibiting and characterising their patching matrices, 
for the five--parameter family of Riemannian ALF metrics constructed by Chen and Teo. The Chen--Teo family contains
a two--parameter family of asymptotically flat gravitational instantons. The patching matrices for these 
instantons take a simple rational form.

Geodesic Anosov flows, hyperbolic closed geodesics and stable ergodicity

In this paper we show that the geodesic flow of a Finsler metric is Anosov if and only if there exists a C 2 C^2 open neighborhood of Finsler metrics all of whose closed geodesics are hyperbolic. For surfaces this result holds also for Riemannian metrics. This follows from a recent result of Contreras and Mazzucchelli [Duke Math. J. 173 (2024), pp. 347–390]. Furthermore, geodesic flows of Riemannian or Finsler metrics on surfaces are C 2 C^2 stably ergodic if and only if they are Anosov.

Quarter-symmetric connection on an almost Hermitian manifold and on a Kähler manifold

The paper observes an almost Hermitian manifold as an example of a generalized Riemannian manifold and examines the application of a quarter-symmetric connection on the almost Hermitian manifold. The almost Hermitian manifold with quarter-symmetric connection preserving the generalized Riemannian metric is actually the Kähler manifold. Observing the six linearly independent curvature tensors with respect to the quarter-symmetric connection, we construct tensors that do not depend on the quarter-symmetric connection generator. One of them coincides with the Weyl projective curvature tensor of symmetric metric $g$. Also, we obtain the relations between the Weyl projective curvature tensor and the holomorphically projective curvature tensor. Moreover, we examine the properties of curvature tensors when some tensors are hybrid.

On spectral simplicity of the Hodge Laplacian and curl operator along paths of metrics

We prove that the curl operator on closed oriented 3 3 -manifolds, i.e., the square root of the Hodge Laplacian on its coexact spectrum, generically has 1 1 -dimensional eigenspaces, even along 1 1 -parameter families of C k \mathcal {C}^k Riemannian metrics, where k ≥ 2 k\geq 2 . We show further that the Hodge Laplacian in dimension 3 3 has two possible sources for nonsimple eigenspaces along generic 1 1 -parameter families of Riemannian metrics: either eigenvalues coming from positive and from negative eigenvalues of the curl operator cross, or an exact and a coexact eigenvalue cross. We provide examples for both of these phenomena. In order to prove our results, we generalize a method of Teytel [Comm. Pure Appl. Math. 52 (1999), pp. 917–934], allowing us to compute the meagre codimension of the set of Riemannian metrics for which the curl operator and the Hodge Laplacian have certain eigenvalue multiplicities. A consequence of our results is that while the simplicity of the spectrum of the Hodge Laplacian in dimension 3 3 is a meagre codimension 1 1 property with respect to the C k \mathcal {C}^k topology as proven by Enciso and Peralta-Salas in [Trans. Amer. Math. Soc. 364 (2012), pp. 4207–4224], it is not a meagre codimension 2 2 property.