Riemannian Metric Research Articles

Three-Dimensional Nonunimodular Lie Groups with a Riemannian Metric of an Invariant Ricci Soliton and a Semisymmetric Metric Connection

Three-Dimensional Nonunimodular Lie Groups with a Riemannian Metric of an Invariant Ricci Soliton and a Semisymmetric Metric Connection

Variational formulas for curves of fixed degree

We consider a length functional for $C^1$ curves of fixed degree in graded manifolds equipped with a Riemannian metric. The first variation of this length functional can be computed only if the curve can be deformed in a suitable sense, and this condition is expressed via a differential equation along the curve. In the classical differential geometry setting, the analogous condition was considered by Bryant and Hsu in [7, 21] who proved that it is equivalent to the surjectivity of a holonomy map. The purpose of this paper is to extend this deformation theory to curves of fixed degree providing several examples and applications. In particular, we give a useful sufficient condition to guarantee the possibility of deforming a curve.

Quantum geometric confinement and dynamical transmission in Grushin cylinder

We classify the self-adjoint realizations of the Laplace–Beltrami operator minimally defined on an infinite cylinder equipped with an incomplete Riemannian metric of Grushin type, in the class of metrics yielding an infinite deficiency index. Such realizations are naturally interpreted as Hamiltonians governing the geometric confinement of a Schrödinger quantum particle away from the singularity, or the dynamical transmission across the singularity. In particular, we characterize all physically meaningful extensions qualified by explicit local boundary conditions at the singularity. Within our general classification we retrieve those distinguished extensions previously identified in the recent literature, namely the most confining and the most transmitting one.

Finslerian Viewpoint to the Rectifying, Normal, and Osculating Curves

The theory of Finsler metric was introduced by Paul Finsler, in 1918. The author defines this metric using the Minkowski norm instead of the inner product. Therefore, this geometry is a more general metric and includes the Riemannian metric. In the present work, using the Finsler metric, we investigate the position vector of the rectifying, normal and osculating curves in Finslerian 3-space $\mathbb{F}^{3}$. We obtain the general characterizations of these curves in $\mathbb{F}^{3}$. Furthermore, we show that rectifying curves are extremal curves derived from the Finslerian spherical curve. We also plotted various examples by using the Randers metrics.

$$C^0$$-robustness of topological entropy for geodesic flows

In this paper, we study the regularity of topological entropy, as a function on the space of Riemannian metrics endowed with the \(C^0\) topology. We establish several instances of entropy robustness (persistence of positive entropy after small \(C^0\) perturbations). A large part of this paper is dedicated to metrics on the two-dimensional torus, for which our main results are that metrics with a contractible closed geodesic have robust entropy (thus, generalizing and quantifying a result of Denvir–Mackay) and that metrics with robust positive entropy on the torus are \(C^{\infty }\) generic. Moreover, we quantify the asymptotic behavior of volume entropy in the Teichmüller space of hyperbolic metrics on a punctured torus, which bounds from below the topological entropy for these metrics. For general closed manifolds of dimension at least 2, we prove that the set of metrics with robust and large positive entropy is \(C^0\)-large in the sense that it is dense and contains cones and arbitrarily large balls.

On the structure of mathrm {RCD} spaces with upper curvature bounds

We develop a structure theory for mathrm {RCD} spaces with curvature bounded above in Alexandrov sense. In particular, we show that any such space is a topological manifold with boundary whose interior is equal to the set of regular points. Further the set of regular points is a smooth manifold and is geodesically convex. Around regular points there are mathrm {DC} coordinates and the distance is induced by a continuous mathrm {BV} Riemannian metric.

Area-minimizing properties of Pansu spheres in the sub-Riemannian 3-sphere

Abstract We consider the sub-Riemannian 3-sphere ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} obtained by restriction of the Riemannian metric of constant curvature 1 to the planar distribution orthogonal to the vertical Hopf vector field. It was shown in [A. Hurtado and C. Rosales, Area-stationary surfaces inside the sub-Riemannian three-sphere, Math. Ann. 340 2008, 3, 675–708] that ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} contains a family of spherical surfaces { 𝒮 λ } λ ⩾ 0 {\{\mathcal{S}_{\lambda}\}_{\lambda\geqslant 0}} with constant mean curvature λ. In this work, we first prove that the two closed half-spheres of 𝒮 0 {\mathcal{S}_{0}} with boundary C 0 = { 0 } × 𝕊 1 {C_{0}=\{0\}\times\mathbb{S}^{1}} minimize the sub-Riemannian area among compact C 1 {C^{1}} surfaces with the same boundary. We also see that the only C 2 {C^{2}} solutions to this Plateau problem are vertical translations of such half-spheres. Second, we establish that the closed 3-ball enclosed by a sphere 𝒮 λ {\mathcal{S}_{\lambda}} with λ > 0 {\lambda>0} uniquely solves the isoperimetric problem in ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} for C 1 {C^{1}} sets inside a vertical solid tube and containing a horizontal section of the tube. The proofs mainly rely on calibration arguments.

Dark energy and accelerating cosmological evolution from osculating Barthel–Kropina geometry

Finsler geometry is an important extension of Riemann geometry, in which each point of the spacetime manifold is associated with an arbitrary internal variable. Two interesting Finsler geometries with many physical applications are the Randers and Kropina type geometries. A subclass of Finsler geometries is represented by the osculating Finsler spaces, in which the internal variable is a function of the base manifold coordinates only. In an osculating Finsler geometry, we introduce the Barthel connection, with the remarkable property that it is the Levi–Civita connection of a Riemannian metric. In the present work we consider the gravitational and cosmological implications of a Barthel–Kropina type geometry. We assume that in this geometry the Ricci type curvatures are related to the matter energy–momentum tensor by the standard Einstein equations. The generalized Friedmann equations in the Barthel–Kropina geometry are obtained by considering that the background Riemannian metric is of Friedmann–Lemaitre–Robertson–Walker type. The matter energy balance equation is also derived. 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 dark energy component can also be generated. Several cosmological solutions are also obtained by numerically integrating the generalized Friedmann equations. A comparison of two specific classes of models with the observational data and with the standard Lambda CDM model is also performed, and it is found that the Barthel–Kropina type models give a satisfactory description of the observations.

Left invariant Riemannian metrics with harmonic curvature are Ricci-parallel in solvable Lie groups and Lie groups of dimension ≤6

We show that any left invariant metric with harmonic curvature on a solvable Lie group is Ricci-parallel. We show the same result for any Lie group of dimension ≤ 6.

Harmonic Hermitian Structures on Riemannian Manifolds with Skew Torsion

We find geometric conditions on a four-dimensional Hermitian manifold endowed with a metric connection with totally skew-symmetric torsion under which the complex structure is a harmonic map from the manifold into its twistor space considered with a natural family of Riemannian metrics defined by means of the metric and the given connection on the base manifold.

Berezin–Toeplitz quantization associated with higher Landau levels of the Bochner Laplacian

In this paper, we construct a family of Berezin–Toeplitz type quantizations of a compact symplectic manifold. For this, we choose a Riemannian metric on the manifold such that the associated Bochner Laplacian has the same local model at each point (this is slightly more general than in almost-Kähler quantization). Then the spectrum of the Bochner Laplacian on high tensor powers L^p of the prequantum line bundle L asymptotically splits into clusters of size {\mathcal O}(p^{3/4}) around the points p\Lambda , where \Lambda is an eigenvalue of the model operator (which can be naturally called a Landau level). We develop the Toeplitz operator calculus with the quantum space, which is the eigenspace of the Bochner Laplacian corresponding to the eigenvalues from the cluster.We show that it provides a Berezin–Toeplitz quantization. If the cluster corresponds to a Landau level of multiplicity one, we obtain an algebra of Toeplitz operators and a formal star-product. For the lowest Landau level, it recovers the almost Kähler quantization.

On the Cerbo Conjecture on Lie Groups with the Left-Invariant Lorentzian Metric

Manifolds with constraints on tensor fields include Einstein manifolds, Einstein-like manifolds, conformally flat manifolds, and a number of other important classes of manifolds. The work of many mathematicians is devoted to the study of such manifolds, which is reflected in the monographs of A. Besse, M. Berger, M.-D. Cao, M. Wang.
 Ricci solitons are one of the natural generalizations of Einstein's metrics. If a Riemannian manifold is a Lie group, one speaks of invariant Ricci solitons.
 Invariant Ricci solitons were studied in most detail in the case of unimodular Lie groups with left-invariant Riemannian metrics and the case of low dimension. Thus, L. Cerbo proved that all invariant Ricci solitons are trivial on unimodular Lie groups with left-invariant Riemannian metric and Levi-Civita connection.A similar result up to dimension four was obtained by P.N. Klepikov and D.N. Oskorbin for the non-unimodular case.
 We study invariant Ricci solitons on three-dimensional unimodular Lie groups with the Lorentzian metric.The study results show that unimodular Lie groups with left-invariant Lorentzian metric admit invariant Ricci solitons other than trivial ones. In this paper, a complete classification of invariant Ricci solitons on three-dimensional unimodular Lie groups with leftinvariant Lorentzian metric is obtained.

Invariant Einstein metrics on real flag manifolds with two or three isotropy summands

We study the existence of invariant Einstein metrics on real flag manifolds associated to simple and non-compact split real forms of complex classical Lie algebras whose isotropy representation decomposes into two or three irreducible subrepresentations. In this situation, one can have equivalent submodules, leading to the existence of non-diagonal homogeneous Riemannian metrics. In particular, we prove the existence of non-diagonal Einstein metrics on real flag manifolds.

The geometry of mixed-Euclidean metrics on symmetric positive definite matrices

Several Riemannian metrics and families of Riemannian metrics were defined on the manifold of Symmetric Positive Definite (SPD) matrices. Firstly, we formalize a common general process to define families of metrics: the principle of deformed metrics. We relate the recently introduced family of alpha-Procrustes metrics to the general class of mean kernel metrics by providing a sufficient condition under which elements of the former belong to the latter. Secondly, we focus on the principle of balanced bilinear forms that we recently introduced. We give a new sufficient condition under which the balanced bilinear form is a metric. It allows us to introduce the Mixed-Euclidean (ME) metrics which generalize the Mixed-Power-Euclidean (MPE) metrics. We unveal their link with the (u,v)-divergences and the (α,β)-divergences of information geometry and we provide an explicit formula of the Riemann curvature tensor. We show that the sectional curvature of all ME metrics can take negative values and we show experimentally that the sectional curvature of all MPE metrics but the log-Euclidean, power-Euclidean and power-affine metrics can take positive values.

Closed geodesics on connected sums and $3$-manifolds

We study the asymptotics of the number $N(t)$ of geometrically distinct closed geodesics of length $\leq t$ of a Riemannian or Finsler metric on a connected sum of two compact manifolds of dimension at least three with non-trivial fundamental groups, and apply the results to the prime decomposition of a three-manifold. In particular we show that the function $N(t)$ grows at least like the prime numbers on a compact $3$-manifold with infinite fundamental group. It follows that a generic Riemannian metric on a compact $3$-manifold has infinitely many geometrically distinct closed geodesics. We also consider the case of a connected sum of a compact manifold with positive first Betti number and a simplyconnected manifold which is not homeomorphic to a sphere.

Modified LPP based on Riemannian metric for feature extraction and fault detection

Dimensionality reduction methods based on manifold learning are widely adopted for industrial process monitoring. However, in many situations, these methods fail to preserve manifold intrinsic features in low-dimensional space, resulting in reduced process monitoring efficacy. To overcome this problem, a modified locality preserving projection (MLPP) based on the Riemannian metric is put forward. First, the Riemannian metric, which embodies a manifold’s geometric information, is estimated from process data. Then, the low dimensional embedding coordinates obtained from LPP are supplemented with an estimate of the Riemannian metric. Finally, a process monitoring model is developed, and kernel density estimation is utilized to approximate confidence bounds for T2 and SPE statistics. The proposed MLPP method is applied to the feature extraction of Twin-Peaks dataset, fault detection of hot strip mill, steam turbine system and Tennessee Eastman processes. The effectiveness of MLPP method is compared with both the manifold learning and deep learning approaches. In addition, the proposed method is evaluated under various noisy conditions. The average fault detection rate of 98.9%, 99.6% and 84.4% in hot strip mill, steam turbine system and Tennessee Eastman processes, respectively, are higher than the existing methods. Quantitative results indicate the superiority of the proposed MLPP method.

Scattering Theory for the Hodge Laplacian

We prove using an integral criterion the existence and completeness of the wave operators \(W_{\pm }(\Delta _h^{(k)}, \Delta _g^{(k)}, I_{g,h}^{(k)})\) corresponding to the Hodge Laplacians \(\Delta _\nu ^{(k)}\) acting on differential \(k\)-forms, for \(\nu \in \{g,h\}\), induced by two quasi-isometric Riemannian metrics g and h on a complete open smooth manifold M. In particular, this result provides a criterion for the absolutely continuous spectra \(\sigma _{\mathrm {ac}}(\Delta _g^{(k)}) = \sigma _{\mathrm {ac}}(\Delta _h^{(k)})\) of \(\Delta _\nu ^{(k)}\) to coincide. The proof is based on gradient estimates obtained by probabilistic Bismut-type formulae for the heat semigroup defined by spectral calculus. By these localised formulae, the integral criterion requires local curvature bounds and some upper local control on the heat kernel acting on functions provided the Weitzenböck curvature endomorphism is in the Kato class, but no control on the injectivity radii. A consequence is a stability result of the absolutely continuous spectrum under a Ricci flow. As an application we concentrate on the important case of conformal perturbations.

Thermodynamic metrics on outer space

AbstractIn this paper we consider two piecewise Riemannian metrics defined on the Culler–Vogtmann outer space which we call the entropy metric and the pressure metric. As a result of work of McMullen, these metrics can be seen as analogs of the Weil–Petersson metric on the Teichmüller space of a closed surface. We show that while the geometric analysis of these metrics is similar to that of the Weil–Petersson metric, from the point of view of geometric group theory, these metrics behave very differently than the Weil–Petersson metric. Specifically, we show that when the rank r is at least 4, the action of $\operatorname {\mathrm {Out}}(\mathbb {F}_r)$ on the completion of the Culler–Vogtmann outer space using the entropy metric has a fixed point. A similar statement also holds for the pressure metric.

Stationary solutions to the Keller–Segel equation on curved planes

We study stationary solutions to the Keller–Segel equation on curved planes. We prove the necessity of the mass being $8 \pi$ and a sharp decay bound. Notably, our results do not require the solutions to have a finite second moment, and thus are novel already in the flat case. Furthermore, we provide a correspondence between stationary solutions to the static Keller–Segel equation on curved planes and positively curved Riemannian metrics on the sphere. We use this duality to show the nonexistence of solutions in certain situations. In particular, we show the existence of metrics, arbitrarily close to the flat one on the plane, that do not support stationary solutions to the static Keller–Segel equation (with any mass). Finally, as a complementary result, we prove a curved version of the logarithmic Hardy–Littlewood–Sobolev inequality and use it to show that the Keller–Segel free energy is bounded from below exactly when the mass is $8 \pi$, even in the curved case.

On the Signature of the Ricci Curvature on Nilmanifolds

We completely describe the signatures of the Ricci curvature of left-invariant Riemannian metrics on arbitrary real nilpotent Lie groups. The main idea in the proof is to exploit a link between the kernel of the Ricci endomorphism and closed orbits in a certain representation of the general linear group, which we prove using the ‘real GIT’ framework for the Ricci curvature of nilmanifolds.