Abstract In this paper, we classify the compact locally homogeneous non-gradient m -quasi Einstein 3- manifolds. Along way, also prove that given a quotient of Lie group any dimension is Einstein, potential vector field X must be left invariant and Killing. We nontrivial metrics are product two metrics. show S 1 only manifold which admits metric nontrivially Einstein.

The 2-parameter family of certain homogeneous Lorentzian 3-manifolds, which includes Minkowski 3-space and anti-de Sitter 3-space, is considered. Each 3-manifold in the has a solvable Lie group structure with left invariant metric. A generalized integral representation formula for maximal spacelike surfaces 3-manifolds obtained. normal Gauß map its harmonicity are discussed.

In computational anatomy, one needs to perform statistics on shapes and transformations, and to transport these statistics from one geometry (e.g. a given subject) to another (e.g. the template). The geometric structure that appeared to be the best suited so far was the Riemannian setting. The statistical Riemannian framework was indeed pretty well developped for finite-dimensional manifolds an...

Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their ‘almost’ versions). We define a left invariant Riemannian metric on N compatible with γ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics...

In contrast to SPD matrices, few tools exist perform Riemannian statistics on the open elliptope of full-rank correlation matrices. The quotient-affine metric was recently built as quotient affine-invariant by congruence action positive diagonal space matrices had always been thought a homogeneous space. contrast, we view in this work Lie group and left-invariant metric. This unexpected new vie...

We show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N, g) satisfy the matrix equation Uv = [1], where [1] is a vector with every entry a one. We associate a generalized Cartan matrix to the matrix U and use the theory of Kac-Moody algebras to analyze the solution spaces for such linear systems. We ...