Single-view 3D object reconstruction is a challenging fundamental problem in machine perception, largely due to the morphological diversity of objects in the natural world. In particular, high curvature regions are not always represented accurately by methods trained with common set-based loss functions such as Chamfer Distance, resulting in reconstructions short-circuiting the surface or "cutting corners." To address this issue, we propose an approach to 3D reconstruction that embeds points on the surface of an object into a higher-dimensional space that captures both the original 3D surface as well as geodesic distances between points on the surface of the object. The precise specification of these additional "lifted" coordinates ultimately yields useful surface information without requiring excessive additional computation during either training or testing, in comparison with existing approaches. Our experiments show that taking advantage of these learned lifted coordinates yields better performance for estimating surface normals and generating surfaces than using point cloud reconstructions alone. Further, we find that this learned geodesic embedding space provides useful information for applications such as unsupervised object decomposition.