Estimating distances in the Internet has been studied in the recent years due to its ability to improve the performance of many applications, e.g., in the peer-to-peer realm. One scalable approach to estimate distances between nodes is to embed the nodes in some d dimensional geometric space and to use the pair distances in this space as the estimate for the real distances. Several algorithms were suggested in the past to do this in low dimensional Euclidean spaces. It was noted in recent years that the Internet structure has a highly connected core and long stretched tendrils, and that most of the routing paths between nodes in the tendrils pass through the core. Therefore, we suggest in this work, to embed the Internet distance metric in a hyperbolic space where routes are bent toward the center. We found that if the curvature, that defines the extend of the bending, is selected in the adequate range, the accuracy of Internet distance embedding can be improved. We demonstrate the strength of our hyperbolic embedding with two applications: selecting the closest server and building an application level multicast tree. For the latter, we present a distributed algorithm for building geometric multicast trees that achieve good trade-offs between delay (stretch) and load (stress).We also present a new efficient centralized embedding algorithm that enables the accurate embedding of short distances, something that have never been done before.