Related papers: SO(n)\SO_0(n,1) has Positive Curvatures

In this paper, we establish a complete structural description of flat Lorentzian Lie groups, i.e., Lie groups endowed with a flat left invariant Lorentzian metric, thereby resolving a long-standing open problem in the theory of…

We present a new class of stochastic, geometrically-driven optimization algorithms on the orthogonal group $O(d)$ and naturally reductive homogeneous manifolds obtained from the action of the rotation group $SO(d)$. We theoretically and…

SO(3), SO(5), and SO(6)-models are singular elliptic fibrations with Mordell--Weil torsion Z/2Z and singular fibers whose dual fibers correspond to affine Dynkin diagrams of type A1, C2, and A3 respectively, where we emphasize the…

Let $ S $ be a hyperbolic surface. We investigate the topology of the space of all curves on $ S $ which start and end at given points in given directions, and whose curvatures are constrained to lie in a given interval $…

M/string theory on noncompact, negatively curved, cosets which generalize $AdS_{D+1}=SO(D,2)/SO(D,1)$ is considered. Holographic descriptions in terms of a conformal field theory on the boundary of the spacetime are proposed. Examples…

The nine two-dimensional Cayley-Klein geometries are firstly reviewed by following a graded contraction approach. Each geometry is considered as a set of three symmetrical homogeneous spaces (of points and two kinds of lines), in such a…

We prove that the group of isometries of a metric measure space that satisfies the Riemannian curvature condition, $RCD^*(K,N),$ is in fact a Lie group. We obtain an optimal upper bound on its dimension and classify the spaces where this…

We study Poisson symmetric spaces of group type with Cartan subalgebra "adapted" to the Lie cobracket.

We study the positive Hermitian curvature flow for left-invariant metrics on $2$-step nilpotent Lie groups with a left-invariant complex structure $J$. We describe the long-time behavior of the flow under the assumption that…

We show the following result: Let $(M,g_0)$ be a compact manifold of dimension $n\geq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or a quotient manifold of $\mathbb{S}^{n-1}\times \mathbb{R}$…

We study isometric immersions of surfaces into simply connected 3-dimensional unimodular Lie groups endowed with either Riemannian or Lorentzian left-invariant metrics, assuming that Milnor's operator is diagonalizable in the Lorentzian…

We develope a new scheme for the construction of explicit complex-valued proper biharmonic functions on Riemannian Lie groups. We exploit this and manufacture many infinite series of uncountable families of new solutions on the special…

We study the decomposition as an $\textrm{SO}(3)$-module of the multiplicity space corresponding to the branching from $\textrm{SO}(n+3)$ to $\textrm{SO}(n)$. Here, $\textrm{SO}(n)$ (resp.\ $\textrm{SO}(3)$) is considered embedded in…

The Cartan development takes a Lie algebra valued 1-form satisfying the Maurer-Cartan equation on a simply connected manifold $M$ to a smooth mapping from $M$ into the Lie group. In this paper this is generalized to infinite dimensional $M$…

We study 4-dimensional simply connected Lie groups $G$ with left-invariant Riemannian metric $g$ admitting non-trivial conformal Killing 2-forms. We show that either the real line defined by such a form is invariant under the group action,…

For homogeneous reductive spaces G/H with reductive complements decomposable into an orthogonal sum \mathfrak{m}=\mathfrak{m}_1 \oplus \mathfrak{m}_2 \oplus \mathfrak{m}_3 of three Ad(H)-invariant irreducible mutually inequivalent…

Let group generators having finite-dimensional representation be realized as Hermitian linear differential operators without nhomogeneous terms as takes place, for example, for the SO(n) group. Then orresponding group Hamiltonians…

We study completeness properties of Sobolev metrics on the space of immersed curves and on the shape space of unparametrized curves. We show that Sobolev metrics of order $n\geq 2$ are metrically complete on the space $\mathcal…

Over an algebraically closed base field $k$ of characteristic 2, the ring $R^G$ of invariants is studied, $G$ being the orthogonal group O(n) or the special orthogonal group SO(n) and acting naturally on the coordinate ring $R$ of the…

An $SO(9,1)$ invariant formulation of an 11-dimensional supermembrane is presented by combining an $SO(10,1)$ invariant treatment of reparametrization symmetry with an $SO(9,1)$ invariant $\theta_{R} = 0$ gauge of $\kappa$-symmetry. The…