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Register a new userOn isometry groups of pseudo-Riemannian compact Lie groups
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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 compact if $G$ is not simply-connected.
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The notion of $Gamma$-symmetric space is a natural generalization of the classical notion of symmetric space based on $Z_2$-grading of Lie algebras. In our case, we consider homogeneous spaces $G/H$ such that the Lie algebra $g$ of $G$ admits a $Gamma$-grading where $Gamma$ is a finite abelian group. In this work we study Riemannian metrics and Lorentzian metrics on the Heisenberg group $mathbb{H}_3$ adapted to the symmetries of a $Gamma$-symmetric structure on $mathbb{H}_3$. We prove that the classification of $z$-symmetric Riemannian and Lorentzian metrics on $mathbb{H}_3$ corresponds to the classification of left-invariant Riemannian and Lorentzian metrics, up to isometry. We study also the $Z_2^k$-symmetric structures on $G/H$ when $G$ is the $(2p+1)$-dimensional Heisenberg group for $k geq 1$. This gives examples of non riemannian symmetric spaces. When $k geq 1$, we show that there exists a family of flat and torsion free affine connections adapted to the $Z_2^k$-symmetric structures.
We show that the compact quotient $Gammabackslashmathrm{G}$ of a seven-dimensional simply connected Lie group $mathrm{G}$ by a co-compact discrete subgroup $Gammasubsetmathrm{G}$ does not admit any exact $mathrm{G}_2$-structure which is induced by a left-invariant one on $mathrm{G}$.
We prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the RCD(0,N), with N > 1. In fact, we can prove that a subRiemannian manifold whose generic degree of nonholonomy is not smaller than 2 can not be biLipschitzly embedded in any Banach space with the Radon-Nikodym property. We also get that every regular sub-Riemannian manifold do not satisfy the CD(K,N) with N > 1. We also prove that the subRiemannian manifold is infinitesimally Hilbert space.
In the paper Einstein metrics on compact simple Lie groups attached to standard triples, the authors introduced the definition of standard triples and proved that every compact simple Lie group $G$ attached to a standard triple $(G,K,H)$ admits a left-invariant Einstein metric which is not naturally reductive except the standard triple $(Sp(4),2Sp(2),4Sp(1))$. For the triple $(Sp(4),2Sp(2),4Sp(1))$, we find there exists an involution pair of $sp(4)$ such that $4sp(1)$ is the fixed point of the pair, and then give the decomposition of $sp(4)$ as a direct sum of irreducible $ad(4sp(1))$-modules. But $Sp(4)/4Sp(1)$ is not a generalized Wallach space. Furthermore we give left-invariant Einstein metrics on $Sp(4)$ which are non-naturally reductive and $Ad(4Sp(1))$-invariant. For the general case $(Sp(2n_1n_2),2Sp(n_1n_2),2n_2Sp(n_1))$, there exist $2n_2-1$ involutions of $sp(2n_1n_2)$ such that $2n_2sp(n_1))$ is the fixed point of these $2n_2-1$ involutions, and it follows the decomposition of $sp(2n_1n_2)$ as a direct sum of irreducible $ad(2n_2sp(n_1))$-modules. In order to give new non-naturally reductive and $Ad(2n_2Sp(n_1)))$-invariant Einstein metrics on $Sp(2n_1n_2)$, we prove a general result, i.e. $Sp(2k+l)$ admits at least two non-naturally reductive Einstein metrics which are $Ad(Sp(k)timesSp(k)timesSp(l))$-invariant if $k<l$. It implies that every compact simple Lie group $Sp(n)$ for $ngeq 4$ admits at least $2[frac{n-1}{3}]$ non-naturally reductive left-invariant Einstein metrics.
We show that Lorentzian manifolds whose isometry group is of dimension at least $frac{1}{2}n(n-1)+1$ are expanding, steady and shrinking Ricci solitons and steady gradient Ricci solitons. This provides examples of complete locally conformally flat and symmetric Lorentzian Ricci solitons which are not rigid.
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