Related papers: Geodesic Orbit Metrics on Compact Simple Lie Group…

In this article we study the Hofer geometry of a compact Lie group $K$ which acts by Hamiltonian diffeomorphisms on a symplectic manifold $M$. Generalized Hofer norms on the Lie algebra of $K$ are introduced and analyzed with tools from…

The geodesic distance vanishes on the group of compactly supported diffeomorphisms of a Riemannian manifold $M$ of bounded geometry, for the right invariant weak Riemannian metric which is induced by the Sobolev metric $H^s$ of order $0\le…

We investigate the geodesic orbit property of pseudo-Riemannian nilmanifolds, specifically those known in the literature as pseudo $H$-type Lie groups -- i.e., 2-step nilpotent Lie groups of Heisenberg type equipped with a left invariant…

We compute the full isometry group of any left invariant metric on a simply connected, non-unimodular Lie group of dimension three. As an application, we determine the index of symmetry of such metrics and prove that the singularities of…

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…

If $G$ is a compact Lie group endowed with a left invariant metric $g$, then $G$ acts via pullback by isometries on each eigenspace of the associated Laplace operator $\Delta_g$. We establish algebraic criteria for the existence of left…

This is a survey on left invariant semi-Riemannian metrics on compact Lie groups.

We study cohomogeneity one Riemannian manifolds and we establish some simple criterium to test when a singular orbit is totally geodesic. As an application, we classify compact, positively curved Riemannian manifolds which are acted on…

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and…

The authors found geodesics, shortest arcs, cut loci, and conjugate sets for left-invariant sub-Riemannian matric on the Lie group $SL(2)$, which is right-invariant relative to the Lie subgroup $SO(2)\subset SL(2)$ (in other words, for…

Gromov asked if the bi-invariant metrics on a compact Lie group are extremal compared to any other metrics. In this note, we prove that the bi-invariant metrics on a compact connected semi-simple Lie group $G$ are extremal (in fact rigid)…

Let (N,g) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their `almost' versions). We define a left invariant Riemannian metric on N compatible with g to be minimal,…

We obtain new invariant Einstein metrics on the compact Lie group $\SU(N)$ which are not naturally reductive. This is achieved by using the generalized flag manifold $G/K=\SU(k_1+\cdots +k_p)/\s(\U(k_1)\times\cdots\times\U(k_p))$ and by…

We study isometric actions of compact Lie groups on complete orientable positively curved $n$-manifolds whose orbit spaces have non-empty boundary in the sense of Alexandrov geometry. In particular, we classify quotients of the unit sphere…

We prove sufficient conditions for the existence of conjugate points along geodesics of a left-invariant metric on a Lie group, using a reformulation of the index form in terms of the adjoint action. In the compact semisimple case, with an…

The paper presents the complete classification of Automorphic Lie Algebras based on $\mathfrak{sl}_n (\mathbb{C})$, where the symmetry group $G$ is finite and the orbit is any of the exceptional $G$-orbits in $\overline{\mathbb{C}}$. A key…

Let $M=G/K$ be a generalized flag manifold, that is the adjoint orbit of a compact semisimple Lie group $G$. We use the variational approach to find invariant Einstein metrics for all flag manifolds with two isotropy summands. We also…

We study Wick-rotations of left-invariant metrics on Lie groups, using results from real GIT (\cite{1}, \cite{2}, \cite{3}). An invariant for Wick-rotation of Lie groups is given, and we describe when a pseudo-Riemannian Lie group can be…

In this paper, we establish a sufficient condition for a geodesic in a Riemannian manifold to be homogeneous, i.e. an orbit of an $1$-parameter isometry group. As an application of this result, we provide a new proof of the fact that every…

Let $G$ be a connected, simply-connected, compact simple Lie group. In this paper, we show that the isometry group of $G$ with a left-invariant pseudo-Riemannan metric is compact. Furthermore, the identity component of the isometry group is…