ABSTRACT There are various useful metrics for finding the distance between two points in Euclidean space. Metrics for finding the distance between two rigid body locations in Euclidean space depend on both the coordinate frame and units used. A metric independent of these choices is desirable. This paper presents a metric for a finite set of rigid body displacements. The methodology uses the pr...

Examples of almost-positively and quasi-positively curved spaces of the form M = H\((G, h) × F ) were discovered recently [9],[8]. Here h is a left-invariant metric on a compact Lie group G, F is a compact Riemannian manifold on which the subgroup H ⊂ G acts isometrically on the left, and M is the orbit space of the diagonal left action of H on (G, h)×F with the induced Riemannian submersion me...

Abstract In the present paper, we calculate Yano connection, its curvature and Lie derivative of metric associated to it on three-dimensional Lorentzian groups with some product structure. We introduce affine generalized Ricci solitons connection classify left-invariant groups.

This note studies the relationship between the complete integrability of geodesic flow on SO(4) with a left-invariant metric and the geometry of the intersection of four quadrics in P(6).

A Hermitian metric on a complex manifold $(M, I)$ of dimension $n$ is called Calabi-Yau with torsion (CYT) or Bismut-Ricci flat, if the restricted holonomy associated Bismut connection contained in ${\rm SU}(n)$ and it strong K\"ahler (SKT) pluriclosed fundamental form $F$ $\partial \overline \partial$-closed. In paper we study existence left-invariant SKT CYT metrics compact semi-simple Lie gr...

We show that the existence of a left-invariant pluriclosed Hermitian metric on unimodular Lie group with abelian complex structure forces to be 2-step nilpotent. Moreover, we prove flow starting from nilpotent preserves Strominger Kähler–like condition.

In this paper, we investigate the concept of topological stationary for locally compact semigroups. In [4], T. Mitchell proved that a semigroup S is right stationary if and only if m(S) has a left Invariant mean. In this case, the set of values ?(f) where ? runs over all left invariant means on m(S) coincides with the set of constants in the weak* closed convex hull of right translates of f. Th...

It is shown that the Hermitian-symmetric space CP1 × CP1 × CP1 and the flag manifold F1,2 endowed with any left invariant metric admit no compatible integrable almost complex structures (even locally) different from the invariant ones. As an application it is proved that any stable harmonic immersion from F1,2 equipped with an invariant metric into an irreducible Hermitian symmetric space of co...

We define the concept of a partial translation structure T on a metric space X and we show that there is a natural C *-algebra C * (T) associated with it which is a subalgebra of the uniform Roe algebra C * u (X). We introduce a coarse invariant of the metric which provides an obstruction to embedding the space in a group. When the space is sufficiently group-like, as determined by our invarian...