Invariant Measure and the Euler Characteristic of Projectively Flat Manifolds

关于 $G_2$-轨道空间上平坦联络的计数

On Counting Flat Connections over $G_2$-Orbifolds

本文研究无挠 $G_2$-轨道空间上平坦丛的 $G_2$-瞬子模空间，通过引入和乐扰动证明其紧致与光滑性，定义了在 $C^0$-形变下不变的整数值不变量，并计算了 Joyce 构造中特定轨道空间的该不变量。

非平凡容许 $U(2)$ 丛的 Klein 四群联络与 Casson 不变量

Klein-four connections and the Casson invariant for non-trivial admissible $U(2)$ bundles

本文研究非平凡容许 $U(2)$ 丛的 Casson 不变量，通过模2上同调环及 Klein 四群完整群平坦联络的计数公式，刻画其偶性来源。

On powers of the Euler class for flat circle bundles

Apparently a lost theorem of Thurston states that the cube of the Euler class $e^3\in H^6(BDiff^δ_ω(S^1);\mathbb{Q})$ is zero where $Diff^δ_ω(S^1)$ is the analytic orientation preserving diffeomorphisms of the circle with the discrete topology. This is in contrast with Morita's theorem that the powers of the Euler class are nonzero in $H^*(BDiff^δ(S^1);\mathbb{Q})$ where $Diff^δ(S^1)$ is the orientation preserving $C^{\infty}$- diffeomorphisms of the circle with the discrete topology. The purpose of this short note is to prove that the powers of the Euler class $e^k \in H^*(BDiff^δ_ω(S^1);\mat…

Canonical decomposition of manifolds with flat real projective structure into (n-1)-convex manifolds and concave affine manifolds

We try to understand the geometric properties of $n$-manifolds ($n\geq 2$) with geometric structures modeled on $(\bR P^n, \PGL(n+1, \bR))$, i.e., $n$-manifolds with projectively flat torsion free affine connections. We define the notion of $i$-convexity of such manifolds due to Carriére for integers $i$, $1 \leq i \leq n-1$, which are generalization of convexity. Given a real projective $n$-manifold $M$, we show that the failure of an $(n-1)$-convexity of $M$ implies an existence of a certain geometric object, $n$-crescent, in the completion $\che M$ of the universal cover $\tilde M$ of $M$. …

链环的Seifert形式与切片欧拉示性数

Seifert forms and slice Euler characteristic of links

本文定义了链环的Witt余指标作为协变不变量，给出切片欧拉示性数的上界，并刻画余指标为1的形式，用于证明两分支链环不界定局部平坦圆柱。

Milnor-Wood inequalities for manifolds locally isometric to a product of hyperbolic planes

This note describes sharp Milnor--Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not admit an affine structure, confirming Chern--Sullivan's conjecture in this case. The manifolds under consideration are of particular interest, since in contrary to many other locally symmetric spaces they do admit flat vector bundle of the corresponding dimension. When the manifold is irreducible and of higher rank, it is shown that flat oriented vector bundles …