Prescribing Ricci curvature on homogeneous spaces

The prescribed Ricci curvature problem in the context of G-invariant metrics on a homogeneous space M = G / K {M=G/K} is studied. We focus on the metrics at which the map g ↦ Rc ⁡ ( g ) {g\mapsto\operatorname{Rc}(g)} is, locally, as injective and surjective as it can be. Our main result is that such property is generic in the compact case. Our main tool is a formula for the Lichnerowicz Laplacian we prove in terms of the moment map for the variety of algebras.

Geometric Invariants Under the Möbius Action of the Group SL(2;R)

In this paper we have introduced new invariant geometric objects in the homogeneous spaces of complex, dual and double numbers for the principal group SL(2; ℝ), in the Klein’s Erlangen Program. We have considered the action as the Möbius action and have taken the spaces as the spaces of complex, dual and double numbers. Some new decompositions of SL(2; ℝ) have been used.

Translates of homogeneous measures associated with observable subgroups on some homogeneous spaces

In the present article, we study the following problem. Let $\boldsymbol {G}$ be a linear algebraic group over $\mathbb {Q}$ , let $\Gamma$ be an arithmetic lattice, and let $\boldsymbol {H}$ be an observable $\mathbb {Q}$ -subgroup. There is a $H$ -invariant measure $\mu _H$ supported on the closed submanifold $H\Gamma /\Gamma$ . Given a sequence $(g_n)$ in $G$ , we study the limiting behavior of $(g_n)_*\mu _H$ under the weak- $*$ topology. In the non-divergent case, we give a rather complete classification. We further supplement this by giving a criterion of non-divergence and prove non-divergence for arbitrary sequence $(g_n)$ for certain large $\boldsymbol {H}$ . We also discuss some examples and applications of our result. This work can be viewed as a natural extension of the work of Eskin–Mozes–Shah and Shapira–Zheng.

Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups

The complete classification of classical r-matrices generating quantum deformations of the (3+1)-dimensional (A)dS and Poincar ́e groups such that their Lorentz sector is a quantum sub-group is presented. It is found that there exists three classes of such r-matrices, one of them being a novel two-parametric one. The (A)dS and Minkowskian Poisson homogeneous spaces corresponding to these three deformations are explicitly constructed in both local and ambient coordinates. Their quantization is performed, thus giving rise to the associated noncommutative spacetimes, that in the Minkowski case are naturally expressed in terms of quantum null-plane coordinates, and they are always defined by homogeneous quadratic algebras. Finally, non-relativistic and ultra-relativistic limits giving rise to novel Newtonian and Carrollian noncommutative spacetimes are also presented.