Invariant Metrics Research Articles

Validation and noise robustness assessment of microscopic anisotropy estimation with clinically feasible double diffusion encoding MRI.

Double diffusion encoding (DDE) MRI enables the estimation of microscopic diffusion anisotropy, yielding valuable information on tissue microstructure. A recent study proposed that the acquisition of rotationally invariant DDE metrics, typically obtained using a spherical "5-design," could be greatly simplified by assuming Gaussian diffusion, facilitating reduced acquisition times that are more compatible with clinical settings. Here, we aim to validate the new minimal acquisition scheme against the standard DDE 5-design, and to quantify the proposed method's noise robustness to facilitate future clinical use. DDE MRI experiments were performed on both ex vivo and in vivo rat brains at 9.4 T using the 5-design and the proposed minimal design and taking into account the difference in the number of acquisitions. The ensuing microscopic fractional anisotropy (μFA) maps were compared over a range of b-values up to 5000 s/mm2 . Noise robustness was studied using analytical calculations and numerical simulations. The minimal protocol quantified μFA at an accuracy comparable to the estimates obtained by means of the more theoretically robust DDE 5-design. μFA's sensitivity to noise was found to strongly depend on compartment anisotropy and tensor magnitude in a nonlinear manner. When μFA < 0.75 or when mean diffusivity is particularly low, very high signal-to-noise ratio is required for precise quantification of µFA. Our work supports using DDE for quantifying microscopic diffusion anisotropy in clinical settings but raises hitherto overlooked precision issues when measuring μFA with DDE and typical clinical signal-to-noise ratio.

On the Geometry of Some (α, β)-Metrics on the Nilpotent Groups H(p, r)

In this paper we study the Riemann-Finsler geometry of the Lie groups H(p, r) which are a generalization of the Heisenberg Lie groups. For a certain Riemannian metric h·, ·i, the Levi-Civita connection and the sectional curvature are given. We classify all left invariant Randers metrics of Douglas type induced by h·, ·i, compute their flag curvatures and show that all of them are nonBerwaldian.

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 .

The ciconia metric on the tangent bundle of an almost Hermitian manifold

We find a new class of invariant metrics existing on the tangent bundle of any given almost-Hermitian manifold. We focus here on the case of Riemannian surfaces, which yield new examples of K\"ahlerian Ricci-flat manifolds in four real dimensions.

Homogeneous Einstein metrics on Stiefel manifolds associated to flag manifolds with two isotropy summands

We study invariant Einstein metrics on the Stiefel manifold VkRn≅SO(n)/SO(n−k) of all orthonormal k-frames in Rn. The isotropy representation of this homogeneous space contains equivalent summands, so a complete description of G-invariant metrics is not easy. In this paper we view the manifold V2pRn as total space over a classical generalized flag manifold with two isotropy summands and prove for 2≤p≤25n−1 it admits at least four invariant Einstein metrics determined by Ad(U(p)×SO(n−2p))-invariant scalar products. Two of the metrics are Jensen's metrics and the other two are new Einstein metrics. The Einstein equation reduces to a parametric system of polynomial equations, which we study by combining Gröbner bases and geometrical arguments.

The Poincaré half-space of a C$^*$-algebra

Let \mathcal{A} be a unital C ^* -algebra. Given a faithful representation \mathcal{A}\subset\mathcal B(\mathcal{L}) in a Hilbert space \mathcal{L} , the set G^+\subset\mathcal{A} of positive invertible elements can be thought of as the set of inner products in \mathcal{L} , related to \mathcal{A} , which are equivalent to the original inner product. The set G^+ has a rich geometry, it is a homogeneous space of the invertible group G of \mathcal{A} , with an invariant Finsler metric. In the present paper we study the tangent bundle TG^+ of G^+ , as a homogeneous Finsler space of a natural group of invertible matrices in M_2(\mathcal{A}) , identifying TG^+ with the Poincaré half-space \mathcal H of \mathcal{A} , \mathcal H=\{h\in\mathcal{A}: {\rm Im}(h)\ge 0, {\rm Im}(h) \hbox{ invertible}\}. We show that \mathcal H\simeq TG^+ has properties similar to those of a space of non-positive constant curvature.

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.

Diverging sequences of unit volume invariant metrics with bounded curvature

We study 1-parameter families in the space of \({\mathsf {G}}\)-invariant, unit volume metrics on a given compact, connected, almost-effective homogeneous space \(M={\mathsf {G}}/{\mathsf {H}}\). In particular, we focus on diverging sequences, i.e. that are not contained in any compact subset of , and we prove some structure results for those which have bounded curvature. We also relate our results to an algebraic version of collapse.

Peak functions and boundary behaviour of holomorphically invariant distances and metrics on strictly pseudoconvex domains

Peak functions and boundary behaviour of holomorphically invariant distances and metrics on strictly pseudoconvex domains

On the Existence of Homogeneous Geodesics in Homogeneous Kropina Spaces

Recently, it is shown that each regular homogeneous Finsler space $M$ admits at least one homogeneous geodesic through any point $o\in M$. The purpose of this article is to study the existence of homogeneous geodesics on singular homogeneous $(\alpha,\beta)$-spaces, specially, homogeneous Kropina spaces. We show that any homogeneous Kropina space admits at least one homogeneous geodesic through any point. It is shown that, under some conditions, the same result is true for any $(\alpha,\beta)$-homogeneous space. Also, in the case of homogeneous Kropina space of Douglas type, a necessary and sufficient condition for a vector to be a geodesic vector is given. Finally, as an example, homogeneous geodesics of $3$-dimensional non-unimodular real Lie groups equipped with a left invariant Randers metric of Douglas type are investigated.

Hermitian Curvature Flow on Compact Homogeneous Spaces

We study a version of the Hermitian curvature flow on compact homogeneous complex manifolds. We prove that the solution has a finite extinction time $$T>0$$ and we analyze its behavior when $$t\rightarrow T$$ . We also determine the invariant static metrics and we study the convergence of the normalized flow to one of them.

Invariant Ricci-flat metrics of cohomogeneity one with Wallach spaces as principal orbits

We construct a continuous 1-parameter family of smooth complete Ricci-flat metrics of cohomogeneity one on vector bundles over $\mathbb{CP}^2$, $\mathbb{HP}^2$ and $\mathbb{OP}^2$ with respective principal orbits $G/K$ the Wallach spaces $SU(3)/T^2$, $Sp(3)/(Sp(1)Sp(1)Sp(1))$ and $F_4/\mathrm{spin}(8)$. Almost all the Ricci-flat metrics constructed have generic holonomy. The only exception is the complete $G_2$ metric discovered in [BS89][GPP90]. It lies in the interior of the 1-parameter family on $\bigwedge_-^2\mathbb{CP}^2$. All the Ricci-flat metrics constructed have asymptotically conical limits given by the metric cone over a suitable multiple of the normal Einstein metric on $G/K$.

Nonnegatively curved quotient spaces with boundary

Let M be a compact nonnegatively curved Riemannian manifold admitting an isometric action by a compact Lie group $$\mathsf {G}$$ in a way that the quotient space $$M/\mathsf {G}$$ has nonempty boundary. Let $$\pi : M \rightarrow M/\mathsf {G}$$ denote the quotient map and B be any boundary stratum of $$M/\mathsf {G}$$. Via a specific soul construction for $$M/\mathsf {G}$$, we construct a smooth closed submanifold N of M such that $$M {\setminus } \pi ^{-1}(B)$$ is diffeomorphic to the normal bundle of N. As an application we show that a simply connected torus manifold admitting an invariant metric of nonnegative curvature is rationally elliptic.

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 $\nabla^gf$.

Curvature properties of anisotropic scale invariant metrics

The objective of this paper is to study the curvature restricted geometric properties of anisotropic nonrelativistic scale invariant metrics, namely, Lifshitz and Schrödinger spacetime metrics. It is found that the Lifshitz spacetime metric admits two important pseudosymmetric type curvature conditions [Formula: see text] and [Formula: see text]. Also, it is [Formula: see text]-quasi Einstein and generalized Roter type manifold. Finally, Lifshitz spacetime is compared with Schrödinger spacetime.

Sparse coding-based representation of LBP difference for 3D/4D facial expression recognition

This paper presents an effective method for automated 3D/4D facial expression recognition based on Mesh-Local Binary Pattern Difference (mesh-LBPD). In contrast to most of existing methods, the proposed mesh-LBPD is based on a unified set of geometric and appearance features of different facial regions. Indeed, multiple features are combined into a compact form using covariance matrices, namely Cov − 3D − LBP. Then, the Cov − 3D − LBP atoms are represented as sparse data combinations. To that end, a Riemannian optimization objective for dictionary learning and sparse coding is used, in order to reduce the complexity of the problem, and the representation loss is characterized via an affine invariant Riemannian metric. In order to prove the effectiveness of the proposed compact combination of geometric and appearance features, we conducted extensive experimental validations on real-world datasets. In fact, obtained results show the capability of the proposed method to significantly outperform, or achieve comparable performances with, the state-of-the-art methods.

Statistical origin of Legendre invariant metrics

Legendre invariant metrics have been introduced in Geometrothermodynamics to take into account the important fact that the thermodynamic properties of physical systems do not depend on the choice of thermodynamic potential from a geometric perspective. In this work, we show that these metrics also have a statistical origin which can be expressed in terms of the average and variance of the differential of the microscopic entropy. To show this, we use a particular reparametrization of the coordinates of the corresponding thermodynamic phase space.

Generic norms and metrics on countable abelian groups

For a countable abelian group G we investigate generic properties of the space of all invariant metrics on G. We prove that for every such an unbounded group G, i.e. group which has elements of arbitrarily high order, there is a dense set of invariant metrics on G which make G isometric to the rational Urysohn space, and a comeager set of invariant metrics such that the completion is isometric to the Urysohn space. This generalizes results of Cameron and Vershik, Niemiec, and the author. Then we prove that for every countable abelian G such that $$G\cong \bigoplus _\mathbb {N}G$$ there is a comeager set of invariant metrics on G such that all of them give rise to the same metric group after completion. If moreover G is unbounded, then using a result of Melleray and Tsankov we get that the completion is extremely amenable.

Fuzzy de Sitter Space from kappa‐Minkowski Space in Matrix Basis

AbstractWe consider the Lie group generated by the Lie algebra of κ‐Minkowski space. Imposing the invariance of the metric under the pull‐back of diffeomorphisms induced by right translations in the group, we show that a unique right invariant metric is associated with . This metric coincides with the metric of de Sitter space‐time. We analyze the structure of unitary representations of the group relevant for the realization of the non‐commutative κ‐Minkowski space by embedding into ‐dimensional Heisenberg algebra. Using a suitable set of generalized coherent states, we select the particular Hilbert space and realize the non‐commutative κ‐Minkowski space as an algebra of the Hilbert‐Schmidt operators. We define dequantization map and fuzzy variant of the Laplace‐Beltrami operator such that dequantization map relates fuzzy eigenvectors with the eigenfunctions of the Laplace‐Beltrami operator on the half of de Sitter space‐time.

Asymptotic Schur orthogonality in hyperbolic groups with application to monotony

We prove a generalization of Schur orthogonality relations for certain classes of representations of Gromov hyperbolic groups. We apply the obtained results to show that representations of non-abelian free groups associated to the Patterson-Sullivan measures corresponding to a wide class of invariant metrics on the group are monotonous in the sense introduced by Kuhn and Steger. This in particular includes representations associated to harmonic measures of a wide class of random walks.