Invariant Riemannian Metrics Research Articles

A harmonic property of right invariant priors

AbstractIt is shown that, on any Lie group, the density ratio of the right invariant measure to the left invariant measure is harmonic with respect to the left invariant Riemannian metric. This result is applied to the Bayesian prediction theory on group invariant statistical models. A method of constructing Bayesian prior distributions that asymptotically dominate the right invariant priors is provided.

Invariant contact metric structures on tangent sphere bundles of compact symmetric spaces

A new characterization is provided for the class of compact rank-one symmetric spaces. Such spaces are the only symmetric spaces of compact type for which the standard vector field xi ^{S} on their sphere bundles is Killing with respect to some invariant Riemannian metric. The set of all these metrics is determined, as well as the set of all those invariant contact metric structures with characteristic vector field xi ^{S}. Moreover, on tangent sphere bundles of compact symmetric spaces with rank greater than or equal to two, a family of invariant contact metric structures, which contains the standard structure, is obtained.

Geometry of cotangent bundle of Heisenberg group

In this paper, the classification of left invariant Riemannian metrics on the cotangent bundle of the (2n+1)-dimensional Heisenberg group up to the action of the automorphism group is presented. Moreover, it is proved that the complex structure on this group is unique, and the corresponding pseudo-Kähler metrics are described and shown to be Ricci flat. It is known that this algebra admits an ad-invariant metric of neutral signature. Here, the uniqueness of such metric is proved.

Szabos algorithm and applications

In this paper, Szabos algorithm is used as the main tool to find locally symmetric left invariant Riemannian metrics on some $4$-dimensional Lie groups. Locally symmetric left invariant Riemannian Lie groups constitute an important subclass of Riemannian Lie groups with zero-divergence Weyl-tensor the so-called C-manifolds. Some properties of the curvature operator of these 4-dimensional C-manifolds are studied.

O(n)-invariant Riemannian metrics on SPD matrices

Symmetric Positive Definite (SPD) matrices are ubiquitous in data analysis under the form of covariance matrices or correlation matrices. Several O(n)-invariant Riemannian metrics were defined on the SPD cone, in particular the kernel metrics introduced by Hiai and Petz. The class of kernel metrics interpolates between many classical O(n)-invariant metrics and it satisfies key results of stability and completeness. However, it does not contain all the classical O(n)-invariant metrics. Therefore in this work, we investigate super-classes of kernel metrics and we study which key results remain true. We also introduce an additional key result called cometric-stability, a crucial property to implement geodesics with a Hamiltonian formulation. Our method to build intermediate embedded classes between O(n)-invariant metrics and kernel metrics is to give a characterization of the whole class of O(n)-invariant metrics on SPD matrices and to specify requirements on metrics one by one until we reach kernel metrics. As a secondary contribution, we synthesize the literature on the main O(n)-invariant metrics, we provide the complete formula of the sectional curvature of the affine-invariant metric and the formula of the geodesic parallel transport between commuting matrices for the Bures-Wasserstein metric.

Zimmer's conjecture: Subexponential growth, measure rigidity, and strong property (T)

We prove several cases of Zimmer's conjecture for actions of higher-rank, cocompact lattices on low-dimensional manifolds. For example, if $\Gamma$ is a cocompact lattice in $\mathrm{SL}(n, \mathbb{R})$, $M$ is a compact manifold, and $\omega$ a volume form on $M$, we show that any homomorphism $\alpha\colon \Gamma \rightarrow \mathrm{Diff}(M)$ has finite image if the dimension of $M$ is less than $n-1$ and that any homomorphism $\alpha\colon \Gamma \rightarrow\mathrm{Diff}(M,\omega)$ has finite image if the dimension of $M$ is less than $n$. The key step in the proof is to show that any such action has uniform subexponential growth of derivatives. This is established using ideas from the smooth ergodic theory of higher-rank abelian groups, structure theory of semisimple groups, and results from homogeneous dynamics. Having established uniform subexponential growth of derivatives, we apply Lafforgue's strong property (T) to establish the existence of an invariant Riemannian metric.

Classification of Left Invariant Riemannian Metrics on Complex Hyperbolic Space

It is well known that \({\mathbb {C}}H^n\) has the structure of a solvable Lie group with left invariant metric of constant holomorphic sectional curvature. In this paper we give the full classification of all possible left invariant Riemannian metrics on this Lie group. We prove that each of those metrics is of constant negative scalar curvature, only one of them being Einstein (up to isometry and scaling).

On the Michor–Mumford phenomenon in the infinite dimensional Heisenberg group

In the infinite dimensional Heisenberg group, we construct a left invariant weak Riemannian metric that gives a degenerate geodesic distance. The same construction yields a degenerate sub-Riemannian distance. We show how the standard notion of sectional curvature adapts to our framework, but it cannot be defined everywhere and it is unbounded on suitable sequences of planes. The vanishing of the distance precisely occurs along this sequence of planes, so that the degenerate Riemannian distance appears in connection with an unbounded sectional curvature. In the 2005 paper by Michor and Mumford, this phenomenon was first observed in some specific Fréchet manifolds.

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.

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.

New Non-Naturally Reductive Einstein Metrics on Sp(n)

In this paper, we consider a class of left invariant Riemannian metrics on Sp(n), which is invariant under the adjoint action of the subgroup Sp(n − 3) × Sp(1) × Sp(1) × Sp(1). Based on the related formulae in the literature, we show that the existence of Einstein metrics is equivalent to the existence of solutions of some homogeneous Einstein equations. Then we use a technique of the Grobner basis to get a sufficient condition for the existence, and show that this method will lead to new non-naturally reductive metrics.

Local and global homogeneity for three obstinate spheres

In this note we complete a study of globally homogeneous Riemannian quotients Γ\\(M,ds2) in positive curvature. Specifically, M is a homogeneous space G/H that admits a G–invariant Riemannian metric of strictly positive sectional curvature, and ds2 is a G–invariant Riemannian metric on M, not necessarily normal and not necessarily positively curved. The Homogeneity Conjecture is that Γ\\(M,ds2) is (globally) homogeneous if and only if (M,ds2) is homogeneous and every γ∈Γ is of constant displacement on (M,ds2). In an earlier paper we verified that conjecture for all homogeneous spaces that admit an invariant Riemannian metric of positive curvature — with three exceptions, all of them odd dimensional spheres, which surprisingly did not yield to the earlier approaches. In this note we deal with those exceptions: we develop some methods that let us verify the Homogeneity Conjecture for those three obstinate spheres. That completes verification of the Homogeneity Conjecture in positive curvature.

Prescribed Riemannian Symmetries

Given a smooth free action of a compact connected Lie group $G$ on a smooth compact manifold $M$, we show that the space of $G$-invariant Riemannian metrics on $M$ whose automorphism group is precisely $G$ is open dense in the space of all $G$-invariant metrics, provided the dimension of $M$ is "sufficiently large" compared to that of $G$. As a consequence, it follows that every compact connected Lie group can be realized as the automorphism group of some compact connected Riemannian manifold; this recovers prior work by Bedford-Dadok and Saerens-Zame under less stringent dimension conditions. Along the way we also show, under less restrictive conditions on both dimensions and actions, that the space of $G$-invariant metrics whose automorphism groups preserve the $G$-orbits is dense $G_{\delta}$ in the space of all $G$-invariant metrics.

Biharmonic and harmonic homomorphisms between Riemannian three dimensional unimodular Lie groups

We classify biharmonic and harmonic homomorphisms f:(G,g1)⟶(G,g2) where G is a connected and simply connected three-dimensional unimodular Lie group and g1 and g2 are left invariant Riemannian metrics.

Almost non-negatively curved 4-manifolds with torus symmetry

We prove that if a closed, smooth, simply-connected 4-manifold with a circle action admits an almost non-negatively curved sequence of invariant Riemannian metrics, then it also admits a non-negatively curved Riemannian metric invariant with respect to the same action. The same is shown for torus actions of higher rank, giving a classification of closed, smooth, simply-connected 4-manifolds of almost non-negative curvature under the assumption of torus symmetry.

Semi-invariant Riemannian metrics in hydrodynamics

Many models in mathematical physics are given as non-linear partial differential equation of hydrodynamic type; the incompressible Euler, KdV, and Camassa–Holm equations are well-studied examples. A beautiful approach to well-posedness is to go from the Eulerian to a Lagrangian description. Geometrically it corresponds to a geodesic initial value problem on the infinite-dimensional group of diffeomorphisms with a right invariant Riemannian metric. By establishing regularity properties of the Riemannian spray one can then obtain local, and sometimes global, existence and uniqueness results. There are, however, many hydrodynamic-type equations, notably shallow water models and compressible Euler equations, where the underlying infinite-dimensional Riemannian structure is not fully right invariant, but still semi-invariant with respect to the subgroup of volume preserving diffeomorphisms. Here we study such metrics. For semi-invariant metrics of Sobolev H^k-type we give local and some global well-posedness results for the geodesic initial value problem. We also give results in the presence of a potential functional (corresponding to the fluid’s internal energy). Our study reveals many pitfalls in going from fully right invariant to semi-invariant Sobolev metrics; the regularity requirements, for example, are higher. Nevertheless the key results, such as no loss or gain in regularity along geodesics, can be adopted.

Non-embedding theorems of nilpotent Lie groups and sub-Riemannian manifolds

We prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the RCD(0,N), with N > 1. In fact, we can prove that a subRiemannian manifold whose generic degree of nonholonomy is not smaller than 2 can not be biLipschitzly embedded in any Banach space with the Radon-Nikodym property. We also get that every regular sub-Riemannian manifold do not satisfy the CD(K,N) with N > 1. We also prove that the subRiemannian manifold is infinitesimally Hilbert space.

Positive scalar curvature and Poincaré duality for proper actions

For G an almost-connected Lie group, we study G -equivariant index theory for proper co-compact actions with various applications, including obstructions to and existence of G -invariant Riemannian metrics of positive scalar curvature. We prove a rigidity result for almost-complex manifolds, generalising Hattori’s results, and an analogue of Petrie’s conjecture. When G is an almost-connected Lie group or a discrete group, we establish Poincaré duality between G -equivariant K -homology and K -theory, observing that Poincaré duality does not necessarily hold for general G .

Classification of left invariant Hermitian structures on 4-dimensional non-compact rank one symmetric spaces

The only 4-dimensional non-compact rank one symmetric spaces are CH2 and RH4. By the classical results of Heintze, one can model these spaces by real solvable Lie groups with left invariant metrics. In this paper we classify all possible left invariant Hermitian structures on these Lie groups, i.e., left invariant Riemannian metrics and the corresponding Hermitian complex structures. We show that each metric from the classification on CH2 admits at least four Hermitian complex structures. One class of metrics on CH2 and all the metrics on RH4 admit 2-spheres of Hermitian complex structures. The standard metric of CH2 is the only Einstein metric from the classification, and also the only metric that admits K¨ahler structure, while on RH4 all the metrics are Einstein. Finally, we examine the geometry of these Lie groups: curvature properties, self-duality, and holonomy.

Automorphisms of generic gradient vector fields with prescribed finite symmetries

Let $M$ be a compact and connected smooth manifold endowed with a smooth action of a finite group $\Gamma$, and let $f$ be a $\Gamma$-invariant Morse function on $M$. We prove that the space of $\Gamma$-invariant Riemannian metrics on $M$ contains a residual subset ${\mathcal M\mathrm{et}}\_f$ with the following property. Let $g\in\mathcal{M}\mathrm{et}\_f$ and let $\nabla^gf$ be the gradient vector field of $f$ with respect to $g$. For any diffeomorphism $\phi\in \mathrm {Diff}(M)$ preserving $\nabla^gf$ there exists some $t\in\mathbb R$ and some $\gamma\in\Gamma$ such that for every $x\in M$ we have $\phi(x)=\gamma,\Phi\_t^g(x)$, where $\Phi\_t^g$ is the time-$t$ flow of the vector field $\nabla^gf$.