On the geometry of Riemannian manifolds with a Lie structure at infinity

A manifold with a “Lie structure at infinity” is a non-compact manifold M0 whose geometry is described by a compactification to a manifold with corners M and a Lie algebra V of vector fields on M subject to constraints only on M rM0. This definition recovers several classes of non-compact manifolds that were studied before: manifolds with cylindrical ends, manifolds that are Euclidean at infinity, conformally compact manifolds, and others. It hence provides a unified setting for the study of these classes of manifolds and of their geometric differential operators. The Lie structure at infinity on M0 determines a complete metric on M0 up to bi-Lipschitz equivalence. This leads to the natural problem of understanding the Riemannian geometry of these manifolds, which is the main question addressed in this paper. We prove, for example, that on a manifold with a Lie structure at infinity the curvature tensor and its covariant derivatives are bounded, by extending the Levi-Civita connection to an A-valued connection where the bundle A is uniquely determined by the Lie algebra V . We study a generalization of the geodesic spray and give conditions for these manifolds to have positive injectivity radius. We also prove that the geometric operators are generated by the given Lie algebra of vector fields. An important motivation for our study is to prepare the ground for the investigation of the analysis of geometric operators on manifolds with a Lie structure at infinity. The simplest examples of manifolds with a Lie structure at infinity are the manifolds with cylindrical ends. For these manifolds the corresponding analysis is that of totally characteristic operators on a compact manifold with boundary equipped with a “b-metric.”