The geodesic motion on a Lie group equipped with a left or right invariant Riemannian metric is governed by the Euler–Arnold equation. This paper investigates conditions on the metric in order for a given subgroup to be totally geodesic. Results on the construction and characterisation of such metrics are given, especially in the special case of easy totally geodesic submanifolds that we introd...

Abstract We study four-dimensional second Chern-Einstein almost-Hermitian manifolds. In the compact case, we observe that under a certain hypothesis, Riemannian dual of Lee form is Killing vector field. use observation to describe locally conformally symplectic manifolds, and give some examples such Finally, problem on unimodular almost-abelian Lie algebras, classifying those admit left-invaria...

Many brain morphometry studies have been performed in order to characterize the brain atrophy pattern of Alzheimer's disease (AD). The earliest studies focused on the volume of particular brain structures, such as hippocampus and entorhinal cortex. Even though volumetry is a powerful, robust and intuitive technique that has yielded a wealth of findings, more complex shape descriptors have been ...

The geodesic motion on a Lie group equipped with a left or right invariant Riemannian metric is governed by the Euler-Arnold equation. This paper investigates conditions on the metric in order for a given subgroup to be totally geodesic. Results on the construction and characterisation of such metrics are given. The setting works both in the classical finite dimensional case, and in the categor...

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Abstract Stratified groups are those simply connected Lie whose algebras admit a derivation for which the eigenspace with eigenvalue 1 is generating. When stratified group equipped left-invariant path distance that homogeneous respect to automorphisms induced by derivation, this metric space known as Carnot group. appear in several mathematical contexts. To understand their algebraic structure,...

Let G be a compact connected Lie group of dimension m. Once bi-invariant metric on is fixed, left-invariant metrics are in correspondence with m × positive definite symmetric matrices. We estimate the diameter and smallest eigenvalue Laplace-Beltrami operator associated to terms eigenvalues corresponding matrix. As consequence, we give partial answers conjecture by Eldredge, Gordina Saloff-Cost...

We study “warped Berger” solutions ( S1×S3, G(t) ) of Ricci flow: generalized warped products with the metric induced on each fiber {s}×SU(2) a left-invariant Berger metric. We prove that this structure is preserved by the flow, that these solutions develop finite-time neckpinch singularities, and that they asymptotically approach round product metrics in space-time neighborhoods of their singu...

In this paper, we study the set of homogeneous geodesics of a leftinvariant Finsler metric on Lie groups. We first give a simple criterion that characterizes geodesic vectors. As an application, we study some geometric properties of bi-invariant Finsler metrics on Lie groups. In particular a necessary and sufficient condition that left-invariant Randers metrics are of Berwald type is given. Fin...

We construct a one-parameter family of properly embedded minimal annuli in the Heisenberg group Nil3 endowed with a left-invariant Riemannian metric. These annuli are not rotationally invariant. This family gives a vertical half-space theorem and proves that each complete minimal graph in Nil3 is entire. Also, the sister surface of an entire minimal graph in Nil3 is an entire constant mean curv...