AI 中文总结
本文研究二步幂零流形的截面曲率拼挤常数，证明其位于[-3,-3/2]，上界具有刚性，下界非刚性，并给出代数刻画。
AI 中文摘要
我们研究了二步幂零流形类中的经典截面曲率拼挤问题，这类流形必然具有混合曲率。我们证明任意二步幂零流形的拼挤常数位于紧区间 $[-3, -\frac{3}{2}]$ 内。上界 $-\frac{3}{2}$ 由带有Ricci孤子度量的复海森堡群 $\mathrm{Heis}_3(\mathbb{C})$ 取得。上界具有刚性：若单连通的二步幂零流形 $N$ 的拼挤常数为 $-\frac{3}{2}$，则 $N$ 含有一个作为全测地子群的Ricci孤子复海森堡群。另一方面，下界满足非刚性：任意二步幂零李群都承认一个拼挤常数为 $-3$ 的度量。这是通过证明在 $n$ 维二步幂零流形空间中，存在 $\mathrm{Heis}_3(\mathbb{R})\times \mathbb{R}^{n-3}$ 的一个开邻域 $U$，使得在 $U$ 上拼挤常数为 $-3$，且任意二步幂零李群 $N$ 都有一个度量 $g$ 使得 $(N,g)$ 位于 $U$ 中。事实上，若 $N$ 不同构于 $\mathrm{Heis}_3(\mathbb{R})\times \mathbb{R}^{n-3}$，则存在 $N$ 上的度量曲线 $g_t$ 使得 $(N,g_t)\in U$，这表明 $N$ 上存在不可数多个左不变度量，其拼挤常数为 $-3$。我们还给出了承认拼挤常数为 $-\frac{3}{2}$ 的度量的二步幂零李群的一个代数刻画，并计算了多个例子的拼挤常数。
英文摘要
We study the classical problem of sectional curvature pinching in the class of 2-step nilmanifolds, which are necessarily of mixed curvature. We show that the pinching constant of any 2-step nilmanifold lies in the compact interval $[-3, -\frac{3}{2}]$. The upper bound $-\frac{3}{2}$ is achieved by the complex Heisenberg group $\mathrm{Heis}_3(\mathbb{C})$ with a Ricci soliton metric. The upper bound exhibits rigidity: if a simply connected 2-step nilmanifold $N$ has the pinching constant $-\frac{3}{2}$, then $N$ admits a Ricci soliton complex Heisenberg group as a totally geodesic subgroup. On the other hand, the lower bound satisfies non-rigidity: any 2-step nilpotent Lie group admits a metric with pinching constant $-3$. This is derived by showing that there is an open neighborhood $U$ of $\mathrm{Heis}_3(\mathbb{R})\times \mathbb{R}^{n-3}$ in the space of $n$-dimensional 2-step nilmanifolds such that the pinching constant is $-3$ on $U$, and any 2-step nilpotent Lie group $N$ has a metric $g$ such that $(N,g)$ lies in $U$. In fact, if $N$ is not isomorphic to $\mathrm{Heis}_3(\mathbb{R})\times \mathbb{R}^{n-3}$, then there is a curve $g_t$ of metrics on $N$ with $(N,g_t)\in U$, showing that there are uncountably many left-invariant metrics on $N$ such that the pinching constant is $-3$. An algebraic characterization of a 2-step nilpotent Lie group that admits a metric with the pinching constant $-\frac{3}{2}$ is also given, and the pinching constants of various examples are computed.
Comments32 pages
