论文标题
Anosov类型的流形的局部镜头刚度
Local lens rigidity for manifolds of Anosov type
论文作者
论文摘要
带边界的Riemannian歧管的镜头数据是地球学长度的集合，并在边界上端点及其传入和传出向量。我们表明，具有严格凸边界的负面弯曲的Riemannian歧管在以下意义上是刚性的：如果$ g_0 $是这样的度量，那么任何公制的$ g $都足够接近$ g_0 $，并且具有相同的镜头数据是等于$ g_0 $的，可以超过边界范围，以达到边界的差异差异。更普遍地，我们考虑具有严格凸边界（称为Anosov类型的指标）的更广泛类别的更广泛的指标。我们证明，同样的刚度结果以尺寸$ 2 $，并且在任何维度上都具有相同的刚度结果。
The lens data of a Riemannian manifold with boundary is the collection of lengths of geodesics with endpoints on the boundary together with their incoming and outgoing vectors. We show that negatively-curved Riemannian manifolds with strictly convex boundary are locally lens rigid in the following sense: if $g_0$ is such a metric, then any metric $g$ sufficiently close to $g_0$ and with same lens data is isometric to $g_0$, up to a boundary-preserving diffeomorphism. More generally, we consider the same problem for a wider class of metrics with strictly convex boundary, called metrics of Anosov type. We prove that the same rigidity result holds within that class in dimension $2$ and in any dimension, further assuming that the curvature is non-positive.
