Abstract The variational theory of higher-power energy is developed for mappings between Riemannian manifolds, and more generally sections submersions applied to vector bundles their sphere subbundles. A complete classification then given left-invariant fields on three-dimensional unimodular Lie groups equipped with an arbitrary metric.

We develop the theory of left-invariant generalized pseudo-Riemannian metrics on Lie groups. Such a metric accompanied by choice divergence operator gives rise to Ricci curvature tensor and we study corresponding Einstein equation. compute in terms tensors (on sum algebra its dual) encoding Courant algebroid structure, operator. The resulting expression is polynomial homogeneous degree two coef...

We develop techniques for classifying the nonnegatively curved left-invariant metrics on a compact Lie group G. We prove rigidity theorems for general G and a partial classification for G = SO(4). Our approach is to reduce the general question to an infinitesimal version; namely, to classify the directions one can move away from a fixed bi-invariant metric such that curvature variation formulas...

We show that the groups of finite energy loops and paths (that is, those Sobolev class H 1 ) with values in a compact connected Lie group, as well their central extensions, satisfy an amenability-like property: they admit left-invariant mean on space bounded functions uniformly continuous regard to metric. Every strongly unitary representation π such group (which we call skew-amenable) has conj...

In this paper, we study inextensible ows in three dimensional Lie groups with a bi-invariant metric. The necessary and sucient conditions for inextensible curve ow are expressed as a partial dierential equation involving the curvatures. Also, we give some results for special cases of Lie groups.

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 ...

We study the speed of convergence to asymptotic cone for a finitely generated nilpotent group endowed with word metric. The first result on this theme is given by Burago who showed that an abelian metric converges normed space $$O\left( \frac{1}{n}\right) $$ in sense Gromov–Hausdorff distance. Later Krat same statement Heisenberg group, and Breuillard Le Donne constructed example, direct produc...

The subject of left-invariant Ricci soliton metrics on nilpotent Lie groups has enjoyed quite a bit of attention in the past several years. These metrics are intimately related to left-invariant Einstein metrics on non-unimodular solvable Lie groups. In fact, a classification of one is equivalent to a classification of the other. In this note, we focus our attention on nilpotent Lie groups and ...

We study left invariant conformal Killing–Yano (CKY) 2-forms on Lie groups endowed with a metric. classify all 5-dimensional metric algebras carrying CKY tensors that are obtained as one-dimensional central extension of 4-dimensional an invertible parallel skew-symmetric tensor. On the other hand, we also center dimension greater than one admitting strict tensors. In addition, determine possibl...