Riemannian Metric Research Articles

Scalar curvature on some open 3-manifolds

This article is a survey of recent results about the existence of Riemannian metrics of positive scalar curvature on some open [Formula: see text]-manifolds. This results culminate in a rigidity theorem obtained using the theory of stable minimal surfaces.

Decompositions of the space of Riemannian metrics on a compact manifold with boundary

In this paper, for a compact manifold M with non-empty boundary \(\partial M\), we give a Koiso-type decomposition theorem, as well as an Ebin-type slice theorem, for the space of all Riemannian metrics on M endowed with a fixed conformal class on \(\partial M\). As a corollary, we give a characterization of relative Einstein metrics.

The Information Geometry of Sensor Configuration.

In problems of parameter estimation from sensor data, the Fisher information provides a measure of the performance of the sensor; effectively, in an infinitesimal sense, how much information about the parameters can be obtained from the measurements. From the geometric viewpoint, it is a Riemannian metric on the manifold of parameters of the observed system. In this paper, we consider the case of parameterized sensors and answer the question, “How best to reconfigure a sensor (vary the parameters of the sensor) to optimize the information collected?” A change in the sensor parameters results in a corresponding change to the metric. We show that the change in information due to reconfiguration exactly corresponds to the natural metric on the infinite-dimensional space of Riemannian metrics on the parameter manifold, restricted to finite-dimensional sub-manifold determined by the sensor parameters. The distance measure on this configuration manifold is shown to provide optimal, dynamic sensor reconfiguration based on an information criterion. Geodesics on the configuration manifold are shown to optimize the information gain but only if the change is made at a certain rate. An example of configuring two bearings-only sensors to optimally locate a target is developed in detail to illustrate the mathematical machinery, with Fast Marching methods employed to efficiently calculate the geodesics and illustrate the practicality of using this approach.

Construction of Initial Data Sets for Lorentzian Manifolds with Lightlike Parallel Spinors

Lorentzian manifolds with parallel spinors are important objects of study in several branches of geometry, analysis and mathematical physics. Their Cauchy problem has recently been discussed by Baum, Leistner and Lischewski, who proved that the problem locally has a unique solution up to diffeomorphisms, provided that the intial data given on a space-like hypersurface satisfy some constraint equations. In this article we provide a method to solve these constraint equations. In particular, any curve (resp. closed curve) in the moduli space of Riemannian metrics on M with a parallel spinor gives rise to a solution of the constraint equations on Mtimes (a,b) (resp. Mtimes S^1).

Positive scalar curvature and an equivariant Callias-type index theorem for proper actions

For a proper action by a locally compact group $G$ on a manifold $M$ with a $G$-equivariant Spin-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature. We focus on the case where $M/G$ is noncompact. The obstructions follow from a Callias-type index theorem, and relate to positive scalar curvature near hypersurfaces in $M$. We also deduce some other applications of this index theorem. If $G$ is a connected Lie group, then the obstructions to positive scalar curvature vanish under a mild assumption on the action. In that case, we generalise a construction by Lawson and Yau to obtain complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature, under an equivariant bounded geometry assumption.

Cosmological evolution and dark energy in osculating Barthel\u2013Randers geometry

We consider the cosmological evolution in an osculating point Barthel–Randers type geometry, in which to each point of the space-time manifold an arbitrary point vector field is associated. This Finsler type geometry is assumed to describe the physical properties of the gravitational field, as well as the cosmological dynamics. For the Barthel–Randers geometry the connection is given by the Levi-Civita connection of the associated Riemann metric. The generalized Friedmann equations in the Barthel–Randers geometry are obtained by considering that the background Riemannian metric in the Randers line element is of Friedmann–Lemaitre–Robertson–Walker type. The matter energy balance equation is derived, and it is interpreted from the point of view of the thermodynamics of irreversible processes in the presence of particle creation. The cosmological properties of the model are investigated in detail, and it is shown that the model admits a de Sitter type solution, and that an effective cosmological constant can also be generated. Several exact cosmological solutions are also obtained. A comparison of three specific models with the observational data and with the standard Lambda CDM model is also performed by fitting the observed values of the Hubble parameter, with the models giving a satisfactory description of the observations.

Pinned Geometric Configurations in Euclidean Space and Riemannian Manifolds

Let M be a compact d-dimensional Riemannian manifold without a boundary. Given a compact set E⊂M, we study the set of distances from the set E to a fixed point x∈E. This set is Δρx(E)={ρ(x,y):y∈E}, where ρ is the Riemannian metric on M. We prove that if the Hausdorff dimension of E is greater than d+12, then there exist many x∈E such that the Lebesgue measure of Δρx(E) is positive. This result was previously established by Peres and Schlag in the Euclidean setting. We give a simple proof of the Peres–Schlag result and generalize it to a wide range of distance type functions. Moreover, we extend our result to the setting of chains studied in our previous work and obtain a pinned estimate in this context.

Periodic magnetic geodesics on Heisenberg manifolds

We study the dynamics of magnetic flows on Heisenberg groups, investigating the extent to which properties of the underlying Riemannian geometry are reflected in the magnetic flow. Much of the analysis, including a calculation of the Mañé critical value, is carried out for $$(2n+1)$$ -dimensional Heisenberg groups endowed with any left invariant metric and any exact, left-invariant magnetic field. In the three-dimensional Heisenberg case, we obtain a complete analysis of left-invariant, exact magnetic flows. This is interesting in and of itself, because of the difficulty of determining geodesic information on manifolds in general. We use this analysis to establish two primary results. We first show that the vectors tangent to periodic magnetic geodesics are dense for sufficiently large energy levels and that the lower bound for these energy levels coincides with the Mañé critical value. We then show that the marked magnetic length spectrum of left-invariant magnetic systems on compact quotients of the Heisenberg group determines the Riemannian metric. Both results confirm that this class of magnetic flows carries significant information about the underlying geometry. Finally, we provide an example to show that extending this analysis of magnetic flows to the Heisenberg-type setting is considerably more difficult.

Estimating the probability of coincidental similarity between atomic displacement parameters with machine learning

High-resolution diffraction studies of macromolecules incorporate the tensor form of the anisotropic displacement parameter (ADP) of atoms from their mean position. The comparison of these parameters requires a statistical framework that can handle the experimental and modeling errors linked to structure determination. Here, a Bayesian machine learning model is introduced that approximates ADPs with the random Wishart distribution. This model allows for the comparison of random samples from a distribution that is trained on experimental structures. The comparison revealed that the experimental similarity between atoms is larger than predicted by the random model for a substantial fraction of the comparisons. Different metrics between ADPs were evaluated and categorized based on how useful they are at detecting non-accidental similarity and whether they can be replaced by other metrics. The most complementary comparisons were provided by Euclidean, Riemann and Wasserstein metrics. The analysis of ADP similarity and the positional distance of atoms in bovine trypsin revealed a set of atoms with striking ADP similarity over a long physical distance, and generally the physical distance between atoms and their ADP similarity do not correlate strongly. A substantial fraction of long- and short-range ADP similarities does not form by coincidence and are reproducibly observed in different crystal structures of the same protein.

Scalar curvature and the multiconformal class of a direct product Riemannian manifold

For a closed, connected direct product Riemannian manifold (M, g)=(M_1, g_1) times cdots times (M_l, g_l), we define its multiconformal class [![ g ]!] as the totality {f_1^2g_1oplus cdots oplus f_l^2g_l} of all Riemannian metrics obtained from multiplying the metric g_i of each factor M_i by a positive function f_i on the total space M. A multiconformal class [![ g ]!] contains not only all warped product type deformations of g but also the whole conformal class [tilde{g}] of every tilde{g}in [![ g ]!]. In this article, we prove that [![ g ]!] contains a metric of positive scalar curvature if and only if the conformal class of some factor (M_i, g_i) does, under the technical assumption dim M_ige 2. We also show that, even in the case where every factor (M_i, g_i) has positive scalar curvature, [![ g ]!] contains a metric of scalar curvature constantly equal to -1 and with arbitrarily large volume, provided lge 2 and dim Mge 3.

Kähler geometry and Chern insulators: Relations between topology and the quantum metric

We study Chern insulators from the point of view of K\"ahler geometry, i.e. the geometry of smooth manifolds equipped with a compatible triple consisting of a symplectic form, an integrable almost complex structure and a Riemannian metric. The Fermi projector, i.e. the projector onto the occupied bands, provides a map to a K\"ahler manifold. The quantum metric and Berry curvature of the occupied bands are then related to the Riemannian metric and symplectic form, respectively, on the target space of quantum states. We find that the minimal volume of a parameter space with respect to the quantum metric is $\pi |\mathcal{C}|$, where $\mathcal{C}$ is the first Chern number. We determine the conditions under which the minimal volume is achieved both for the Brillouin zone and the twist-angle space. The minimal volume of the Brillouin zone, provided the quantum metric is everywhere non-degenerate, is achieved when the latter is endowed with the structure of a K\"ahler manifold inherited from the one of the space of quantum states. If the quantum volume of the twist-angle torus is minimal, then both parameter spaces have the structure of a K\"ahler manifold inherited from the space of quantum states. These conditions turn out to be related to the stability of fractional Chern insulators. For two-band systems, the volume of the Brillouin zone is naturally minimal provided the Berry curvature is everywhere non-negative or nonpositive, and we additionally show how the latter, which in this case is proportional to the quantum volume form, necessarily has zeros due to topological constraints.

Fixed angle inverse scattering in the presence of a Riemannian metric

Abstract We consider a fixed angle inverse scattering problem in the presence of a known Riemannian metric. First, assuming a no caustics condition, we study the direct problem by utilizing the progressing wave expansion. Under a symmetry assumption on the metric, we obtain uniqueness and stability results in the inverse scattering problem for a potential with data generated by two incident waves from opposite directions. Further, similar results are given using one measurement provided the potential also satisfies a symmetry assumption. This work extends the results of [Rakesh and M. Salo, Fixed angle inverse scattering for almost symmetric or controlled perturbations, SIAM J. Math. Anal. 52 2020, 6, 5467–5499] and [Rakesh and M. Salo, The fixed angle scattering problem and wave equation inverse problems with two measurements, Inverse Problems 36 2020, 3, Article ID 035005] from the Euclidean case to certain Riemannian metrics.

Hydrodynamic derivation of the Gross–Pitaevskii equation in general Riemannian metric

Here we show that the Gross–Pitaevskii equation (GPE) for Bose–Einstein condensates (BECs) admits hydrodynamic interpretation in a general Riemannian metric, and show that in this metric the momentum equation has a new term that is associated with local curvature and density distribution profile. In particular conditions of steady state a new Einstein’s field equation is determined in presence of negative curvature. Since GPE governs BECs defects that are useful, analogue models in cosmology, a relativistic form of GPE is also considered to show connection with models of analogue gravity, thus providing further grounds for future investigations of black hole dynamics in cosmology.

On the existence of the global conformal gauge in string theory

The global conformal gauge is playing the crucial role in string theory providing the basis for quantization. Its existence for two-dimensional Lorentzian metric is known locally for a long time. We prove that if a Lorentzian metric is given on a plain then the conformal gauge exists globally on the whole {{mathbb {R}}}^2. Moreover, we prove the existence of the conformal gauge globally on the whole worldsheets represented by infinite strips with straight boundaries for open and closed bosonic strings. The global existence of the conformal gauge on the whole plane is also proved for the positive definite Riemannian metric.

Geodesic Random Walks, Diffusion Processes and Brownian Motion on Finsler Manifolds

We show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric.

Spaces of positive intermediate curvature metrics

In this paper we study spaces of Riemannian metrics with lower bounds on intermediate curvatures. We show that the spaces of metrics of positive p-curvature and k-positive Ricci curvature on a given high-dimensional mathrm {Spin}-manifold have many non-trivial homotopy groups provided that the manifold admits such a metric.

Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition

Abstract We study the interaction between the g.o. property and certain flag curvature conditions. A Finsler manifold is called g.o. if each constant speed geodesic is the orbit of a one-parameter subgroup. Besides the non-negatively curved condition, we also consider the condition (FP) for the flag curvature, i.e. in any flag we find a flag pole such that the flag curvature is positive. By our main theorem, if a g.o. Finsler space (M, F) has non-negative flag curvature and satisfies (FP), then M is compact. If M = G/H where G has a compact Lie algebra, then the rank inequality rk 𝔤 ≤ rk 𝔥+1 holds. As an application,we prove that any even-dimensional g.o. Finsler space which has non-negative flag curvature and satisfies (FP) is a smooth coset space admitting a positively curved homogeneous Riemannian or Finsler metric.

Variations of Weyl\u2019s Tube Formula

In 1939 Weyl showed that the volume of spherical tubes around compact submanifolds M of Euclidean space depends solely on the induced Riemannian metric on M. Can this intrinsic nature of the tube volume be preserved for tubes with more general cross sections mathbb {D} than the round ball? Under sufficiently strong symmetry conditions on mathbb {D} the answer turns out to be yes.

Wick Rotation and the Positivity of Energy in Quantum Field Theory

Abstract We propose a new axiom system for unitary quantum field theories on curved space-time backgrounds, by postulating that the partition function and the correlators extend analytically to a certain domain of complex-valued metrics. Ordinary Riemannian metrics are contained in the allowable domain, while Lorentzian metrics lie on its boundary.

A probabilistic representation for heat flow of harmonic map on manifolds with time-dependent Riemannian metric

In this paper we will give a probabilistic representation for the heat flow of harmonic map with time-dependent Riemannian metric via a forward–backward stochastic differential equation on Riemannian manifolds. Moreover, we can provide an alternative stochastic method for the proof of existence of a unique local solution for heat flow of harmonic map with time-dependent Riemannian metric.