Homogeneous Metrics Research Articles

Four-dimensional homogeneous critical metrics for quadratic curvature functionals

We determine all homogeneous metrics which are critical for some quadratic curvature functional in dimension four.

Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds

We prove two rigidity results on holomorphic isometries into homogeneous Kähler manifolds. The first shows that a Kähler-Ricci soliton induced by the homogeneous metric of the Kähler product of a special generalized flag manifold (i.e. a flag of classical type or integral type) with a bounded homogeneous domain is trivial, i.e. Kähler-Einstein. In the second one we prove that: (i) a flat space is not relative to the Kähler product of a special generalized flag manifold with a homogeneous bounded domain, (ii) a special generalized flag manifold is not relative to the Kähler product of a flat space with a homogeneous bounded domain and (iii) a homogeneous bounded domain is not relative to the Kähler product of a flat space with a special generalized flag manifold. Our theorems strongly extend the results of Cheng and Hao [Ann. Global Anal. Geom. 60 (2021), pp. 167–180], Cheng, Di Scala, and Yuan [Internat. J. Math. 28 (2017), p. 1750027], Loi and Mossa [Proc. Amer. Math. Soc. 149 (2021), pp. 4931–4941], Loi and Mossa [Proc. Amer. Math. Soc. 151 (2023), pp. 3975–3984] and Umehara [Tokyo J. Math. 10 (1987), pp. 203–214].

Universality in long-distance geometry and quantum complexity

In physics, two systems that radically differ at short scales can exhibit strikingly similar macroscopic behaviour: they are part of the same long-distance universality class1. Here we apply this viewpoint to geometry and initiate a program of classifying homogeneous metrics on group manifolds2 by their long-distance properties. We show that many metrics on low-dimensional Lie groups have markedly different short-distance properties but nearly identical distance functions at long distances, and provide evidence that this phenomenon is even more robust in high dimensions. An application of these ideas of particular interest to physics and computer science is complexity geometry3–7—the study of quantum computational complexity using Riemannian geometry. We argue for the existence of a large universality class of definitions of quantum complexity, each linearly related to the other, a much finer-grained equivalence than typically considered. We conjecture that a new effective metric emerges at larger complexities that describes a broad class of complexity geometries, insensitive to various choices of microscopic penalty factors. We discuss the implications for recent conjectures in quantum gravity.

Experimental One-Sided Choppers Relating Neuromuscular Human Abilities to Heart Rates and Technological Evolution

The length of time it takes to experimentally make one-sided choppers, as found in the fossil record, bears a linear relationship to the knapping process of fabricating them. In addition, this temporal frame appears to be related to human heart rates measured as beats per minute, which act as a physiological metronome. We achieved these observations, assuming that any paleolithic one-sided chopper has the information needed to estimate, quantitatively, the number of strikes on it. The experimental data allow us to establish the total timing needed for the standard fabricating of any one-sided chopper. We discuss issues derived from these experimental results, showing the evolution of human neurological abilities from 2.4 million years ago to the Modern period via the duration of time needed for making one chopper to that needed to play a 19th-century music score on a piano. Given that the neuronal and physiological distance between both actions differs by a factor of 6, we propose the concept of “technome” to measure human evolution by using methodological homogeneous metrics applied to these two human technologic objects: the chopper and the piano.

Holomorphic isometries into homogeneous bounded domains

We prove two rigidity theorems on holomorphic isometries into homogeneous bounded domains. The first shows that a Kähler-Ricci soliton induced by the homogeneous metric of a homogeneous bounded domain is trivial, i.e. Kähler-Einstein. In the second one we prove that a homogeneous bounded domain and the flat (definite or indefinite) complex Euclidean space are not relatives, i.e. they do not share a common Kähler submanifold (of positive dimension). Our theorems extend the results proved by us earlier [Proc. Amer. Math. Soc. 149 (2021), pp. 4931–4941] and by Xiaoliang Cheng and Yihong Hao [Ann. Global Anal. Geom. 60 (2021), pp. 167–180].

Infinite families of manifolds of positive kmathrm{th}-intermediate Ricci curvature with k small

Positive kmathrm{th}-intermediate Ricci curvature on a Riemannian n-manifold, to be denoted by {{,mathrm{Ric},}}_k>0, is a condition that interpolates between positive sectional and positive Ricci curvature (when k =1 and k=n-1 respectively). In this work, we produce many examples of manifolds of {{,mathrm{Ric},}}_k>0 with k small by examining symmetric and normal homogeneous spaces, along with certain metric deformations of fat homogeneous bundles. As a consequence, we show that every dimension nge 7 congruent to 3,{{,mathrm{mod},}}4 supports infinitely many closed simply connected manifolds of pairwise distinct homotopy type, all of which admit homogeneous metrics of {{,mathrm{Ric},}}_k>0 for some k<n/2. We also prove that each dimension nge 4 congruent to 0 or 1,{{,mathrm{mod},}}4 supports closed manifolds which carry metrics of {{,mathrm{Ric},}}_k>0 with kle n/2, but do not admit metrics of positive sectional curvature.

Homogeneous and curvature homogeneous Lorentzian critical metrics

We determine all three-dimensional homogeneous and $1$-curvature homogeneous Lorentzian metrics which are critical for a quadratic curvature functional. As a result, we show that any quadratic curvature functional admits different non-Einstein homogeneous critical metrics and that there exist homogeneous metrics which are critical for all quadratic curvature functionals without being Einstein.

Einstein flow with matter sources: stability and convergence

Two recent articles (Moncrief V. 2015 In General relativity and gravitation-A centennial perspective (eds A Asthekar, B Berger, J Isenberg, M MacCallum), pp. 480-498. Cambridge, UK: Cambridge University Press; Moncrief V, Mondal P. 2019 Pure Appl. Math. Q. 15, 921-965. (doi:10.4310/PAMQ.2019.v15.n3.a7)) suggested an interesting dynamical mechanism within the framework of the vacuum Einstein flow (or [Formula: see text] flow if a positive cosmological constant [Formula: see text] is included) which suggests that many closed (compact without boundary) manifolds that do not support homogeneous and isotropic metrics at all will nevertheless evolve to be asymptotically compatible with the observed approximate homogeneity and isotropy of the physical universe. These studies however did not include matter sources. Therefore, the aim of the present study is to include suitable matter sources and investigate whether one is able to draw a similar conclusion. This article is part of the theme issue 'The future of mathematical cosmology, Volume 1'.

On the Ricci flow of homogeneous metrics on spheres

We study the Ricci flow of the four-parameter family of Sp $$(n+1)$$ -invariant metrics on spheres. We determine their forward behaviour and also classify ancient solutions. In doing so, we exhibit a new one-parameter family of ancient solutions on spheres. This completes the classification of homogeneous ancient solutions on spheres.

The First Eigenvalue of a Homogeneous CROSS

We provide explicit formulae for the first eigenvalue of the Laplace-Beltrami operator on a compact rank one symmetric space (CROSS) endowed with any homogeneous metric. As consequences, we prove that homogeneous metrics on CROSSes are isospectral if and only if they are isometric, and also discuss their stability (or lack thereof) as solutions to the Yamabe problem.

Homogeneous metrics with prescribed Ricci curvature on spaces with non-maximal isotropy

Homogeneous metrics with prescribed Ricci curvature on spaces with non-maximal isotropy

Geometric structures and the Laplace spectrum, Part II

We classify the locally homogeneous Riemannian metrics on elliptic three-manifolds, which contributes to our classification of the compact homogeneous Riemannian three-manifolds. Then, we apply the former result to our continued exploration of the extent to which the spectrum encodes the local geometry of a locally homogeneous three-manifold. Specifically, using the first four heat invariants, we find that any collection of isospectral locally homogeneous metrics on an elliptic three-manifold Γ ∖ S 3 \Gamma \backslash S^3 contains at most two isometry classes and these classes are necessarily locally isometric. In fact, if the elliptic three-manifold is S 3 S^3 , R P 3 \mathbb {R}P^3 or has non-cyclic fundamental group, then (up to isometry) its locally homogeneous Riemannian metrics can be mutually distinguished via their spectra. Currently, there is no example of an isospectral pair consisting of locally homogeneous elliptic three-manifolds with non-isometric universal cover; however, we show that if such a pair exists, then it satisfies certain restrictive geometric conditions. Finally, we note that our classification of locally homogeneous elliptic three-manifolds shows that, for q ≥ 3 q \geq 3 , the lens space L ( q ; 1 , 1 ) L(q;1,1) admits pairs of locally isometric locally homogeneous metrics where only one of the metrics is homogeneous. While this phenomenon cannot occur in dimension two, these lens spaces account for all such examples in dimension three.

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.

The great emptiness at the beginning of the Universe

The great emptiness is a possible beginning of the Universe in the infinite past of physical time. For the epoch of great emptiness particles are extremely rare and effectively massless. Only expectation values of fields and average fluctuations characterize the lightlike vacuum of this empty Universe. The physical content of the early stages of standard inflationary cosmological models is the lightlike vacuum. Towards the beginning, the Universe is almost scale invariant. This is best seen by an appropriate choice of the metric field – the primordial flat frame – for which the beginning of a homogeneous metric is flat Minkowski space. We suggest that our observed inhomogeneous Universe can evolve from the lightlike vacuum in the infinite past, and therefore can have lasted eternally. Then no physical big bang singularity is present.

Curvature Homogeneous Manifolds in Dimension 4

We classify complete curvature homogeneous metrics on simply connected four-dimensional manifolds which are invariant under a cohomogeneity one action.

On the δ-Pinching Function of the Sectional Curvature of a Compact Connected Lie Group G with a Bi-Invariant Riemannian Metric and a Vectorial Torsion Connection

One of the important problems of Riemannian geometry is the problem of establishing connections between curvature and the topology of a Riemannian manifold, and, in particular, the influence of the sign of sectional curvature on the topological structure of a Riemannian manifold. Of particular importance in these studies is the question of the influence of d-pinching of Riemannian metrics of positive sectional curvature on the geometric and topological structure of the Riemannian manifold. This question is most studied for the homogeneous Riemannian case. In this direction, the classification of homogeneous Riemannian manifolds of positive sectional curvature, obtained by M. Berger, N. Wallach, L. Bergeri, as well as a number of results on d- pinching of homogeneous Riemannian metrics of positive sectional curvature, is well known.&#x0D; In this paper, we investigate Riemannian manifolds with metric connection being a connection with vectorial torsion. The Levi-Civita connection falls into this class of connections. Although the curvature tensor of these connections does not possess the symmetries of the Levi-Civita connection curvature tensor, it seems possible to determine sectional curvature. This paper studies the d-pinch function of the sectional curvature of a compact connected Lie group G with a biinvariant Riemannian metric and a connection with vectorial torsion. It is proved that it takes the values d(||V ||)∈(0,1].

Zermelo Deformation of Hermitian Metrics by Holomorphic Vector Fields

In this paper we first present the real homogeneous complex Finsler metrics ( $$\mathbb {R}$$ -complex Finsler metrics) and make use of them to generalize the Zermelo navigation on Hermitian manifolds. $$\mathbb {R}$$ -complex Randers metrics are obtained by the Zermelo deformation of Hermitian metrics, with variable space-dependent ship’s relative speed under action of weak complex vector fields. Next, we indicate how some properties of a Hermitian metric, e.g. Kahler property, holomorphic sectional curvature behave by the Zermelo deformation in a special holomorphic wind. Lastly, the results are illustrated with some relevant examples.

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.

Power spectrum of stochastic wave and diffusion equations in the warm inflation models

We discuss dissipative stochastic wave and diffusion equations resulting from an interaction of the inflaton with an environment in an external expanding homogeneous metric. We show that a diffusion equation well approximates the wave equation in a strong friction limit. We calculate the long wave power spectrum of the wave equation under the assumption that the perturbations are slowly varying in time and the expansion is almost exponential. Under the assumption that the noise has a form invariant under the coordinate transformations we obtain the power spectrum close to the scale invariant one. In the diffusion approximation we go beyond the slow variation assumption. We calculate the power spectrum exactly in models with exponential inflation and polynomial potentials and with power-law inflation and exponential potentials.

Homogeneous Einstein-like metrics on symmetric spaces

In this paper, we study homogeneous Einstein-like metrics on the compact irreducible symmetric space M, which is not isometric to a compact Lie group and has rank greater than 1. Whenever there exists a closed proper subgroup G′ of G = Isom0(M) acting transitively on M, we find all G′-invariant $${\mathcal A}$$ -metrics and $${\mathcal B}$$ -metrics on M. More precisely, we prove that G′-invariant metrics on M must be $${\mathcal A}$$ -metrics, and G′-invariant $${\mathcal B}$$ -metrics on M are always Einstein.