arXiv subjects

Shaoxiang Zhang

Publications and source records attributed to Shaoxiang Zhang.

4 recordsLinked to original sources

arXiv2025

Invariant Einstein metrics on basic classical Lie supergroups

This paper presents a systematic study of invariant Einstein metrics on basic classical Lie supergroups, whose Lie superalgebras belong to the Kac's classification of finite dimensional classical simple Lie superalgebras over $\mathbb{R}$. We consider a natural family of left invariant metrics parameterized by scaling factors on the simple and Abelian components of the reductive even part, using the canonical bi-invariant bilinear form. Explicit expressions for the Levi-Civita connection and Ricci tensor are derived, and the Einstein condition is reduced to a solvable algebraic system. Our main result shows that, except for the cases of $\mathbf{A}(m,n)$ with $m\neq n$, $\mathbf{F}(4)$, and their real forms, every real basic classical Lie superalgebra admits at least two distinct Einstein metrics. Notably, for $\mathbf{D}(n+1,n)$ and $\mathbf{D}(2,1;α)$, we obtain both Ricci flat and non Ricci flat Einstein metrics, a phenomenon not observed in the non-super setting.

math.DG↗

arXiv2023

Pseudo-Riemannian geodesic orbit nilmanifolds of signature $\boldsymbol{(n-2,2)}$

The geodesic orbit property is useful and interesting in itself, and it plays a key role in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here we extend Riemannian and Lorentz results to trans-Lorentz nilmanifolds. Those are the geodesic orbit pseudo Riemannian manifolds $M = G/H$ of signature $(n-2,2)$ such that a nilpotent analytic subgroup of $G$ is transitive on $M$. For that we suppose that there is a reductive decomposition $\g = \h \oplus \n \text{ (vector space direct sum) with } [\h,\n] \subset \n$ and $\n$ nilpotent. When the metric is nondegenerate on $[\n,\n]$ we show that $\n$ is abelian or 2-step nilpotent. That is the same result as for geodesic orbit Riemannian and Lorentz nilmanifolds. When the metric is degenerate on $[\n,\n]$ we show that $\n$ is a double extension of a geodesic orbit nilmanifold of either Riemannian or Lorentz signature.

math.DG↗

arXiv2021

On the Geometric Orbit Property for Lorentz Manifolds

The geodesic orbit property has been studied intensively for Riemannian manifolds. Geodesic orbit spaces are homogeneous and allow simplifications of many structural questions using the Lie algebra of the isometry group. Weakly symmetric Riemannian manifolds are geodesic orbit spaces. Here we define "naturally reductive" for pseudo-Riemannian manifolds and note that they are geodesic orbit spaces. A few years ago two of the authors proved that weakly symmetric pseudo-Riemannian manifolds are geodesic orbit spaces. In particular these results apply to pseudo-Riemannian Lorentz manifolds. There our main results are Theorems 4.2 and 5.1. In the Riemannian case the nilpotent isometry group for a geodesic orbit nilmanifold is abelian or $2$-step nilpotent. Examples show that this fails dramatically in the pseudo-Riemannian case. Here we concentrate on the geodesic orbit property for Lorentz nilmanifolds $G/H$ with $G = N \rtimes H$ and $N$ nilpotent. When the metric is nondegenerate on $[\mathfrak{n},\mathfrak{n}]$, Theorem 4.2 shows that $N$ either is at most $2$-step nilpotent as in the Riemannian situation, or is $4$-step nilpotent, but cannot be $3$-step nilpotent. Examples show that these bounds are the best possible. Surprisingly, Theorem 5.1 shows that $N$ is at most $2$-step nilpotent when the metric is degenerate on $[\mathfrak{n},\mathfrak{n}]$. Both theorems give additional structural information and specialize to naturally reductive and to weakly symmetric Lorentz nilmanifolds. Key Words: Geodesic Orbit Space; Lorentz nilmanifold; Weakly Symmetric Space; Naturally Reductive Space; Pseudo-Riemannian Manifold.

math.DG↗

arXiv2020

Some geometric correspondences for homothetic navigation

In this paper, we provide conceptional explanations for the geodesic and Jacobi field correspondences for homothetic navigation, and then let them guide us to the shortcuts to some well known flag curvature and S-curvature formulas. They also help us directly see the local correspondence between isoparametric functions or isoparametric hypersurfaces, which generalizes the classification works of Q. He and her coworkers for isoparametric hypersurfaces in Randers space forms and Funk spaces.

math.DG↗