论文标题
连接的通用动力学特性在矢量束上
Generic dynamical properties of connections on vector bundles
论文作者
论文摘要
给定封闭的Riemannian歧管$（M，G）$上平滑的Hermitian Vector Bundle $ \ MATHCAL {E} $，我们在矢量Bundle $ \ Mathcal {e} $上研究了统一连接的通用属性$ \ nabla^{\ nabla^{\ Mathcal {e}} $。首先，我们表明，当$ \ dim（m）\ geq 3 $时，扭曲的保形杀伤量（CKT）通常很琐碎，回答了Guillarmou-Paternain-Salo-uhlmann的一个开放问题。在负曲率中，众所周知，扭曲的CKT的存在是唯一解决可能出现在各种几何问题（例如透明连接研究）中的扭曲的共同体方程的唯一障碍。本文的主要结果表明，这些方程式可以通常解决。作为一个副产品，我们还获得了内态捆绑$ \ mathrm {endrm {endrm {end}（\ nathcal {e}）$在$ ckts $ ckts $（mmathcal and $ g）上， 捆。最终，我们表明，在其他假设下，$（m，g）$是Anosov（即，测量流是单位切线捆绑包上的Anosov），连接通常是$ \ textit {opaque} $，即，没有$ \ \ \ \ \ \ \ \ \ ratecal {e e} $ callially clive clive clive clive cally closess $ clistical calless $ clise cally clistally clistally clistally clise。证明依赖于（伪）差异操作员的引入新的微局部特性，称为$ \ textit {统一发散类型} $的运算符，以及光谱理论的扰动论点（尤其是基于Anosov案中Pollicott-Ruelle共振理论）。
Given a smooth Hermitian vector bundle $\mathcal{E}$ over a closed Riemannian manifold $(M,g)$, we study generic properties of unitary connections $\nabla^{\mathcal{E}}$ on the vector bundle $\mathcal{E}$. First of all, we show that twisted Conformal Killing Tensors (CKTs) are generically trivial when $\dim(M) \geq 3$, answering an open question of Guillarmou-Paternain-Salo-Uhlmann. In negative curvature, it is known that the existence of twisted CKTs is the only obstruction to solving exactly the twisted cohomological equations which may appear in various geometric problems such as the study of transparent connections. The main result of this paper says that these equations can be generically solved. As a by-product, we also obtain that the induced connection $\nabla^{\mathrm{End}(\mathcal{E})}$ on the endomorphism bundle $\mathrm{End}(\mathcal{E})$ has generically trivial CKTs as long as $(M,g)$ has no nontrivial CKTs on its trivial line bundle. Eventually, we show that, under the additional assumption that $(M,g)$ is Anosov (i.e. the geodesic flow is Anosov on the unit tangent bundle), the connections are generically $\textit{opaque}$, namely there are no non-trivial subbundles of $\mathcal{E}$ which are generically preserved by parallel transport along geodesics. The proofs rely on the introduction of a new microlocal property for (pseudo)differential operators called $\textit{operators of uniform divergence type}$, and on perturbative arguments from spectral theory (especially on the theory of Pollicott-Ruelle resonances in the Anosov case).
