Related papers: Riemannian Geometry on Contact Lie Groups

We establish the existence of solvable Lie groups of dimension 4 and left-invariant Riemannian metrics with zero Bach tensor which are neither conformally Einstein nor half conformally flat.

In this paper, we use the powerful tool Milnor bases to classify all the $3-$dimensional connected and locally symmetric Riemannian Lie Groups by solving system of polynomial equations of structure constants of each Lie algebra . Moreover,…

In this work we deal with left invariant complex and symplectic structures on simply connected four dimensional solvable real Lie groups. We search the general form of such structures, when they exist and we make use of this information to…

A Lorentzian Lie group is a Lie group endowed with a left invariant Lorentzian metric. We study left-invariant Codazzi tensors on Lorentzian Lie groups. We obtain new results on left-invariant Lorentzian metrics with harmonic curvature and…

Almost contact manifolds with B-metric are considered. There are studied three natural connections (i.e. linear connections preserving the structure tensors) determined by conditions for their torsions. These connections are investigated on…

Geodesics on Riemannian manifolds are precisely the locally length-minimizing curves, but their explicit description via simple functions is rarely possible. Geodesics of the simplest form, such as lines on Euclidean space and great circles…

In this paper we generalize the main notions from the geometry of (almost) contact manifolds in the category of Lie algebroids. Also, using the framework of generalized geometry, we obtain an (almost) contact Riemannian Lie algebroid…

Properties of invariant, anti-invariant and slant isometrically immersed submanifolds of metallic Riemannian manifolds are given with a special view towards the induced $\Sigma$-structure. Examples of such metallic manifolds are also given.

A Riemannian geometry of noncommutative n-dimensional surfaces is developed as a first step towards the construction of a consistent noncommutative gravitational theory. Historically, as well, Riemannian geometry was recognized to be the…

Let $M$ be pseudo-Riemannian homogeneous Einstein manifold of finite volume, and suppose a connected Lie group $G$ acts transitively and isometrically on $M$. In this situation, the metric on $M$ induces a bilinear form…

In the context of a connected, simply connected, nilpotent Lie group, whose representations are square-integrable modulo the center, we find characterization results of extra-invariant spaces under the left translations associated with the…

A conformal Lie group is a conformal manifold $(M,c)$ such that $M$ has a Lie group structure and $c$ is the conformal structure defined by a left-invariant metric $g$ on $M$. We study Weyl-Einstein structures on conformal solvable Lie…

Left-invariant Hermitian and Gauduchon connections are studied on an arbitrary Lie group $G$ equipped with an arbitrary left-invariant almost Hermitian structure $(\langle\cdot,\cdot\rangle,J)$. The space of left-invariant Hermitian…

We derive a curvature-variation formula for a path of left-invariant metrics on a compact Lie group, beginning at a bi-invariant metric. We prove rigidity theorems for paths which remain nonnegatively curved, and we make progress towards a…

Let M be a real analytic Riemannian manifold. An adapted complex structure on TM is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. This structure is called…

Let $ G $ be a connected Lie group with real Lie algebra $ \mathfrak{g}$. Suppose $G$ is also a complex manifold. We obtain explicit holomorphic sectional and bisectional curvature formulas of left-invariant strongly pseudoconvex complex…

Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the…

In this article, we study Einstein Kropina metrics on Lie groups and homogeneous spaces. We give a method to construct Einstein Kropina metrics on Lie groups. As an example of this method, a family of non-Riemannian Einstein Kropina metrics…

This paper is an adaptation of a chapter from an upcoming monograph on noncommutative geometry and quantum groups. We present examples of non compact quantum groups which are deformations of low dimensional Lie groups. The paper is of…

We prove that on closed Riemannian manifolds with infinite abelian, but not cyclic, fundamental group, any isometry that is homotopic to the identity possesses infinitely many invariant geodesics. We conjecture that the result remains true…