Combinatorial identities and Chern numbers of complex flag manifolds

罗宾逊流形与陈-罗宾连接

Robinson manifolds and the Chern-Robinson connection

本文研究了罗宾逊流形的复切包上的陈-罗宾连接及其曲率性质，并探讨了其在几何学中的应用。

Universal curvature identities III

We examine universal curvature identities for pseudo-Riemannian manifolds with boundary. We determine the Euler-Lagrange equations associated to the Chern-Gauss-Bonnet formula and show that they are given solely in terms of curvature {and the second fundamental form and do not involve covariant derivatives thus generalizing a conjecture of Berger to this context.

Spin(7)流形的$dd^Φ$-引理与Bott–Chern型上同调

A $dd^Φ$-Lemma and Bott--Chern-type Cohomology for Spin(7)-Manifolds

该文研究8维无挠Spin(7)-流形上的$dd^Φ$-算子，证明其Hodge分解与类似Kähler几何的$dd^Φ$-引理，定义Bott–Chern型上同调并建立其与模空间、校准几何的关联，同时给出具有独立价值的Spin(7)结构相关恒等式。

Curvature identities for Einstein manifolds of dimension 5 and 6

Patterson discussed the curvature identities on Riemannian manifolds in [14], and a curvature identity for any 6-dimensional Riemannian manifold was independently derived from the Chern-Gauss-Bonnet Theorem [8]. In this paper, we provide the explicit formulae of Patterson's curvature identity that holds on 5-dimensional and 6-dimensional Einstein manifolds. We confirm that the curvature identities on the Einstein manifold from the previous work [8] are the same as the curvature identities deduced from Patterson's result. We also provide examples that support the theorems.

Kähler与拟Kähler旗流形的曲率正性

Curvature positivity for Kähler and quasi-Kähler flag manifolds

本文研究Kähler与拟Kähler旗流形及复射影空间上Chern联络曲率的Griffiths与dual-Nakano正性，证明了相关度量的存在性，给出了半正曲率情形的完整分类，并对非可积近复结构提出猜想。

Flag structures on real 3-manifolds

We define flag structures on a real three manifold M as the choice of two complex lines on the complexified tangent space at each point of M. We suppose that the plane field defined by the complex lines is a contact plane and construct an adapted connection on an appropriate principal bundle. This includes path geometries and CR structures as special cases. We prove that the null curvature models are given by totally real submanifolds in the flag space SL(3, C)/B, where B is the subgroup of upper triangular matrices. We also define a global invariant which is analogous to the Chern-Simons seco…

万能向量丛、推前公式和特征形式的正性

Universal vector bundles, push-forward formulae and positivity of characteristic forms

研究万能向量丛的推前公式及特征形式正性，通过计算Chern曲率验证推前公式的点wise成立，并提供微分形式层面的Jacobi-Trudi恒等式替代版本，部分验证Griffiths猜想。

On Chern minimal surfaces in Hermitian surfaces

In this paper we introduce the Chern minimal surface in Hermitian surfaces by using the Chern connection, and we show that it only has isolated complex and anticomplex points for a generic one (neither holomorphic nor antiholomorphic). For a generic Chern minimal $f$ from compact Riemann surface $Σ$ in a Hermitian surface $M$, we establish two identities which related to the sum of the orders of all complex points, anticomplex points denoted by $P$, $Q$ respectively, the cap product of the pull-back of the first Chern class $f^*c_1(M)$ and $[Σ]$, the Euler characteristic of tangent bundle $χ(T…

The complex geometry of two exceptional flag manifolds

We discuss the complex geometry of two complex five-dimensional Kähler manifolds which are homogeneous under the exceptional Lie group $G_2$. For one of these manifolds rigidity of the complex structure among all Kählerian complex structures was proved by Brieskorn, for the other one we prove it here. We relate the Kähler assumption in Brieskorn's theorem to the question of existence of a complex structure on the six-dimensional sphere, and we compute the Chern numbers of all $G_2$-invariant almost complex structures on these manifolds.

Chern-Ricci流与t-Gauduchon Ricci平坦条件

Chern-Ricci flow and t-Gauduchon Ricci-flat condition

本文研究Chern-Ricci流下的t-Gauduchon Ricci平坦条件，提供了不保持该条件的例子、非平凡新解及几何流的Gromov-Hausdorff极限行为，明确描述了特定主丛上pluriclosed流的极限。

黎曼超曲面的总平均曲率比较公式

Comparison formulas for total mean curvatures of Riemannian hypersurfaces

本研究通过构造陈省身型微分形式，推导黎曼流形嵌套超曲面总平均曲率的比较公式，为研究者与Joel Spruck此前通过Reilly恒等式得到的相关结果提供了更简洁的证明方法。