The X-Ray Transform for Connections in Negative Curvature
The X-Ray Transform for Connections in Negative Curvature
复制标题
负曲率连接的 X 射线变换
DOI：
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发表时间：
2015
影响因子：
2.4
通讯作者：
G. Uhlmann
中科院分区：
文献类型：
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作者：
C. Guillarmou;G. Paternain;M. Salo;G. Uhlmann
我们考虑具有负截面曲率的流形上的酉联络和斜厄米希格斯场的积分几何反问题。结果适用于任何维度的流形，有或没有边界，也在存在被困测地线。在边界情况下，我们证明了衰减射线变换在具有厄米特丛值的张量场中的内射性（即，向量值情况）。我们还表明，一个连接和Higgs场的厄米特丛上的规范所确定的知识的边界点之间的平行运输沿着所有可能的测地线。主要的工具是一个能量的身份，佩斯托夫身份与酉连接，这是在一个一般的形式，和一个精确的分析时，有被困测地线输运方程的解的奇异性。在闭流形的情形下，我们得到了类似的结果模扭曲共形Killing张量的障碍，我们也研究了这种障碍。
We consider integral geometry inverse problems for unitary connections and skew-Hermitian Higgs fields on manifolds with negative sectional curvature. The results apply to manifolds in any dimension, with or without boundary, and also in the presence of trapped geodesics. In the boundary case, we show injectivity of the attenuated ray transform on tensor fields with values in a Hermitian bundle (i.e., vector valued case). We also show that a connection and Higgs field on a Hermitian bundle are determined up to gauge by the knowledge of the parallel transport between boundary points along all possible geodesics. The main tools are an energy identity, the Pestov identity with a unitary connection, which is presented in a general form, and a precise analysis of the singularities of solutions of transport equations when there are trapped geodesics. In the case of closed manifolds, we obtain similar results modulo the obstruction given by twisted conformal Killing tensors, and we also study this obstruction.