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

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...

We give a necessary and sufficient condition for set of left invariant metrics on compact Heisenberg manifold to be relatively in the corresponding moduli space.

The aim of this paper is to determine left-invariant strictly almost Kähler structures on 4-dimensional Lie groups (g, J,Ω) such that the Ricci tensor is J-invariant.

The three-dimensional Heisenberg group H3 has three left-invariant Lorentz metrics g1 , g2 and g3 as in [R92] . They are not isometric each other. In this paper, we characterize the left-invariant Lorentzian metric g1 as a Lorentz Ricci soliton. This Ricci soliton g1 is a shrinking non-gradient Ricci soliton. Likewise we prove that the isometry group of flat Euclid plane E(2) has Lorentz Ricci ...

Inverse sampling design is generally considered to be appropriate technique when the population is divided into two subpopulations, one of which contains only few units. In this paper, we derive the Horvitz-Thompson estimator for the population mean under inverse sampling designs, where subpopulation sizes are known. We then introduce an alternative unbiased estimator, corresponding to post-st...

When performing statistics on elements of sets that possess a particular geometric structure, it is desirable to respect this structure. For instance in a Lie group, it would be judicious to have a notion of a mean which is stable by the group operations (composition and inversion). Such a property is ensured for Riemannian center of mass in Lie groups endowed with a bi-invariant Riemannian met...

We construct the space of vector fields on a generic quantum group. Its elements are products of elements of the quantum group itself with left invariant vector fields. We study the duality between vector fields and 1-forms and generalize the construction to tensor fields. A Lie derivative along any (also non left invariant) vector field is proposed and a puzzling ambiguity in its definition di...

We provide the solutions of Linear Left-invariant 2nd-order Evolution Equations on the 2D-Euclidean motion group. These solutions are given by group convolution with the corresponding Green’s functions which we derive in explicit form. A particular case coincides with the Forward Kolmogorov equation of the direction process, the exact solution of which was strongly required in the field of imag...

We study hermitian structures, with respect to the standard neutral metric on the cotangent bundle T ∗G of a 2n-dimensional Lie group G, which are left invariant with respect to the Lie group structure on T ∗G induced by the coadjoint action. These are in one-to-one correspondence with left invariant generalized complex structures on G. Using this correspondence and results of [8] and [10], it ...