We consider the problem of identifying a unitary Yang–Mills connection $\nabla$ on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the connection Laplacian $\nabla^*\nabla$ over compact Riemannian manifolds with boundary. We establish uniqueness of the connection up to a gauge equivalence in the case of trivial line bundles in the smooth category and for the higher rank case in the analytic category, by using geometric analysis methods.

Moreover, by using a Runge-type approximation argument along curves to recover holonomy, we are able to uniquely determine both the bundle structure and the connection. Also, we prove that the DNmap is an elliptic pseudodifferential operator of order one on the restriction of the vector bundle to the boundary, whose full symbol determines the complete Taylor series of an arbitrary connection, metric and an associated potential at the boundary.

中文翻译：

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Yang-Mills连接的Calderón问题

我们考虑了一个问题，即从带边界的紧黎曼流形上的拉普拉斯算子\\ nabla ^ * \ nabla $的Dirichlet-to-Neumann（DN）映射的Dirichlet-to-Neumann（DN）映射的Hermitian向量束上识别单一的Yang-Mills联系$ \ nabla $ 。通过使用几何分析方法，在平滑类别中的平凡线束情况下以及在解析类别中的较高级别情况下，我们建立了连接至量规等效的唯一性。

此外，通过沿曲线使用Runge型逼近参数来恢复完整性，我们能够唯一确定束结构和连接。此外，我们证明DNmap是向量束对边界的限制的一阶椭圆伪微分算子，其全符号确定任意连接的完整泰勒级数，度量以及边界处的关联电势。