Related papers: Log-Euclidean Lie Groups

In this work we find the Killing vector fields of the riemannian submanifolds of the thermodynamic phase space of an ideal gas and show that the isometry group corresponding to them is homomorphic to the euclidean group $E(2)$. We also give…

In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient…

We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension $\leq6$, every nice nilpotent Lie group of dimension $\leq7$ and every…

Accurate quantification of complex human movements, such as gait, is essential for clinical diagnosis and rehabilitation but is often limited by traditional linear models rooted in Euclidean geometry. These frameworks frequently fail to…

We address the problem of computing Riemannian normal coordinates on the real, compact Stiefel manifold of orthogonal frames. The Riemannian normal coordinates are based on the so-called Riemannian exponential and the Riemannian logarithm…

We study in this paper previously defined by V.N. Berestovskii and C.P. Plaut $\delta$-homogeneous spaces in the case of Riemannian manifolds. Every such manifold has non-negative sectional curvature. The universal covering of any…

We introduce an approach based on moving frames for polygon recognition and symmetry detection. We present detailed algorithms for recognition of polygons modulo the special Euclidean, Euclidean, equi-affine, skewed-affine and similarity…

We develop a new approach to the classical problem on isotopy classification of embeddings of manifolds into Euclidean spaces. This approach involves studying of a new embedding invariant, of almost-embeddings and of smoothing, as well as…

Let N be a nilpotent Lie group and let S be an invariant geometric structure on N (cf. symplectic, complex or hypercomplex). We define a left invariant Riemannian metric on N compatible with S to be "minimal", if it minimizes the norm of…

Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form. Assuming certain conditions on the fundamental group $\pi_1(M)$ we construct quasi-isometric embeddings of either free Abelian or…

We investigate connections between pairs of (pseudo-)Riemannian metrics whose sum is a (tensor) product of a covector field with itself. A bijective mapping between the classes of Euclidean and Lorentzian metrics is constructed as a special…

The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related:…

We consider left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, \infty)\rightarrow\mathtt{Aut}(G)$, $\lambda\mapsto\delta_\lambda$, so that $…

We study Cartan-Schouten metrics, explore invariant dual connections, and propose them as models for Information Geometry. Based on the underlying Riemannian barycenter and the biinvariant mean of Lie groups, we subsequently propose a new…

This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive…

Following our approach to metric Lie algebras developed in math.DG/0312243 we propose a way of understanding pseudo-Riemannian symmetric spaces which are not semi-simple. We introduce cohomology sets (called quadratic cohomology) associated…

Latent space geometry provides a rigorous and empirically valuable framework for interacting with the latent variables of deep generative models. This approach reinterprets Euclidean latent spaces as Riemannian through a pull-back metric,…

We consider left-invariant (purely) coclosed G$_2$-structures on 7-dimensional 2-step nilpotent Lie groups. According to the dimension of the commutator subgroup, we obtain various criteria characterizing the Riemannian metrics induced by…

The real symplectic Stiefel manifold is the manifold of symplectic bases of symplectic subspaces of a fixed dimension. It features in a large variety of applications in physics and engineering. In this work, we study this manifold with the…

The Symmetric Positive Definite (SPD) matrices have received wide attention for data representation in many scientific areas. Although there are many different attempts to develop effective deep architectures for data processing on the…