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 [ 14]. 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 $\nabla ^{\textrm{End}({\operatorname{{\mathcal{E}}}})}$ on the endomorphism bundle $\textrm{End}({\operatorname{{\mathcal{E}}}})$ 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 opaque, namely that generically there are no non-trivial subbundles of $\mathcal{E}$ that are preserved by parallel transport along geodesics. The proofs rely on the introduction of a new microlocal property for (pseudo)differential operators called operators of uniform divergence type, and on perturbative arguments from spectral theory (especially on the theory of Pollicott–Ruelle resonances in the Anosov case).

中文翻译：

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向量丛上连接的一般动力学性质

给定一个闭黎曼流形 $(M,g)$ 上的光滑 Hermitian 矢量丛 $\mathcal{E}$，我们研究矢量丛 $ 上的酉连接 $\nabla ^{\mathcal{E}}$ 的一般性质\mathcal{E}$。首先，我们证明了当 $\dim (M) \geq 3$ 时，扭曲保形杀戮张量 (CKT) 通常是微不足道的，回答了 Guillarmou-Paternain-Salo-Uhlmann [14] 的一个开放问题。在负曲率中，已知扭曲CKTs的存在是精确求解扭曲上同调方程的唯一障碍，它可能出现在透明连接研究等各种几何问题中。本文的主要结果是这些方程可以通用求解。作为副产品，我们还得到了自同态束 $\textrm{End}({\operatorname{{ \mathcal{E}}}})$ 具有一般平凡的 CKT，只要 $(M,g)$ 在其平凡的线束上没有非平凡的 CKT。最终，我们证明，在 $(M,g)$ 是 Anosov 的附加假设下（即，测地线流是单位切线丛上的 Anosov），连接通常是不透明的，即通常没有非平凡的$\mathcal{E}$ 的子束通过沿测地线的平行传输而保留。证明依赖于为（伪）微分算子引入一个新的微局部属性，称为均匀散度类型的算子，