$p$-Harmonic and Complex Isoparametric Functions on the Lie Groups $\mathbb{R}^m \ltimes \mathbb{R}^n$ and $\mathbb{R}^m \ltimes \mathrm{H}^{2n+1}$

Minkowski norm and Hessian isometry induced by an isoparametric foliation on the unit sphere

Let $M_t$ be an isoparametric foliation on the unit sphere $(S^{n-1}(1),g^{\mathrm{st}})$ with $d$ principal curvature values. Using the spherical coordinates induced by $M_t$, we construct a Minkowski norm with the presentation $F=r\sqrt{2f(t)}$, which generalizes the notions of $(α,β)$-norm and $(α_1,α_2)$-norm. Using the technique of spherical local frame, we give an exact and explicit answer for the question when $F=r\sqrt{2f(t)}$ really defines a Minkowski norm. Using the similar technique, we study the Hessian isometry $Φ$ between two Minkowski norms induced by $M_t$, which preserves the…

Tangent Lie groups are Riemannian naturally reductive spaces

Given a compact Lie group $G$ with Lie algebra $\mathfrak{g}$, we consider its tangent Lie group $TG\cong G\ltimes_{\mathrm{Ad}} \mathfrak{g}$. In this short note, we prove that $TG$ admits a left-invariant naturally reductive Riemannian metric $g$ and a metric connection with skew torsion $\nabla$ such that $(TG,g,\nabla)$ is naturally reductive. An alternative spinorial description of the same connection on the direct product $G\times \mathfrak{g}$ generalizes in a rather subtle way to $TS^7$, which is in many senses almost a tangent Lie group.

缠绕在轨道流形上的膜及其引力块

Branes wrapped on orbifolds and their gravitational blocks

本文构造了D=6规范超引力中新的超对称AdS2×M4解，对应D4/D8膜缠绕在轨道流形上，通过粘合引力块极值化熵函数重现熵，并推广到离壳中心荷。

可除凸锥的仿射形变与仿射时空

Affine Deformations of Divisible Convex Cones and Affine Spacetimes

本文研究由可除凸锥的无挠离散子群添加平移部分所构造的仿射时空，证明其商空间是极大全局双曲且具有局部一致凸紧致 Cauchy 曲面的仿射时空，并证明这是此类时空的全部例子，从而推广了 Mess、Barbot 和 Bonsante 关于均匀格仿射形变的经典结果。

拟可分凸锥的仿射变形

Affine deformations of quasi-divisible convex cones

本文研究了有限体积凸射影曲面基本群的仿射变形，证明了群作用存在正则凸不连续区域并对其分类，揭示了其商空间为由常仿射高斯曲率凸曲面叶化的仿射流形，且模空间构成相应向量丛。

不满足 hard-Lefschetz 条件的辛可解流形

Symplectic solvmanifolds not satisfying the hard-Lefschetz condition

本文研究 $\R\ltimes_\phi\R^m$ 型完全可解李群，证明作用非半单时其辛可解流形必在次数1或2上违反 hard-Lefschetz 条件，并构造格给出实例。