AI 中文总结
该研究构造了黎曼流形上用于描述度量线性化形变理论的不变微分算子序列，关联了扭曲 de Rham 序列与 Cartan 联络形变理论，提供了无挠 Cartan 几何线性化形变理论的新解释。
AI 中文摘要
我们获得了黎曼流形$(M,g)$上不变微分算子序列的一种新构造，该算子序列用于描述$g$的线性化形变理论。从自然丛$\boldsymbol{\textit{AM}}\to M$上的显式线性联络出发，我们构造了一个扭曲的 de Rham 序列，随后应用了 BGG 序列构造的类似方法。若$g$具有常截面曲率，两个序列均为复形，可计算局部 Killing 场层的上同调，而局部 Killing 场等价于$\boldsymbol{\textit{AM}}$的平行截面。第二步，我们将该构造与$(M,g)$作为（无挠）Cartan 几何$(\boldsymbol{\textit{OM}},\boldsymbol{\textit{\textomega}})$的描述关联起来，其中$\boldsymbol{\textit{OM}}$是$M$的标架丛。这明确建立了扭曲 de Rham 序列与 Cartan 联络$\boldsymbol{\textit{\textomega}}$形变理论的关系（该关系比$g$的形变理论更易处理），BGG 类构造可很好地被视为用基础黎曼度量来解释无挠 Cartan 几何的线性化形变理论。
英文摘要
We obtain a new construction of a sequence of invariant differential operators on a Riemannian manifold $(M,g)$ that governs the linearized deformation theory of $g$. Starting from an explicit linear connection on a natural bundle $\mathcal AM\to M$, we construct a twisted de Rham sequence and then apply an analog of the construction of BGG sequences. If $g$ has constant sectional curvature, both sequences are complexes which compute the cohomology of the sheaf of local Killing fields, which are equivalent to parallel sections of $\mathcal AM$. In a second step, we relate the construction to the description of $(M,g)$ as a (torsion-free) Cartan geometry $(\mathcal OM,ω)$, where $\mathcal OM$ is the orthonormal frame bundle of $M$. This provides a manifest relation of the twisted de Rham sequence to the deformation theory of the Cartan connection $ω$ (which is easier do deal with than the deformation theory of $g$). The BGG-like construction can then be nicely viewed as interpreting the linearized deformation theory of torsion free Cartan geometries in terms of the underlying Riemannian metric.
Comments25 pages, comments are welcome
