We study a generalization of the geodesic spray and give conditions for noncomapct manifolds with a Lie structure at infinity to have positive injectivity radius. We also prove that the geometric operators are generated by the given Lie algebra of vector fields. This is the first one in a series of papers devoted to the study of the analysis of geometric differential operators on manifolds with Lie structure at infinity.2000 Mathematics Subject Classification: 53C21, 53C27, 58J40.1. Introduction. Geometric differential operators on complete noncompact Riemannian manifolds were extensively studied due to their applications to physics, geometry, number theory, and numerical analysis. Still, their properties are not as well understood as those of differential operators on compact manifolds, one of the main reasons being that differential operators on noncompact manifolds do not enjoy some of the most useful properties enjoyed by their counterparts on compact manifolds.For example, elliptic operators on noncompact manifolds are not Fredholm in general. (We use the term "elliptic" in the sense that the principal symbol is invertible outside the zero section.) Also, one does not have a completely satisfactory pseudodifferential calculus on an arbitrary complete noncompact Riemannian manifold, which might allow us to decide whether a given geometric differential operator is bounded, Fredholm, or compact (see however [2] and the references therein).However, if one restricts oneself to certain classes of complete noncompact Riemannian manifolds, one has a chance to obtain more precise results on the analysis of the geometric differential operators on those spaces. This paper is the first in a series of papers devoted to the study of such a class of Riemannian manifolds, the class of Riemannian manifolds with a "Lie structure at infinity" (see Definition 3.1). We stress here that few results on the geometry of these manifolds have a parallel in the literature, although there are a fair number of papers devoted to the analysis on particular classes of such manifolds [12,13,15,16,17,38,39,55,57,64,67,70,74,75,76,78] A manifold M 0 with a Lie structure at infinity has, by definition, a natural compactification to a manifold with corners M = M 0 ∪∂M such that the tangent bundle T M 0 → M 0

We define and study an algebra Ψ ∞ 1,0,V (M 0 ) of pseudodifferential operators canonically associated to a noncompact, Riemannian manifold M 0 whose geometry at infinity is described by a Lie algebra of vector fields V on a compactification M of M 0 to a compact manifold with corners. We show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ ∞ 1,0,V (M 0 ). We also consider the algebra Diff * V (M 0 ) of differential operators on M 0 generated by V and C ∞ (M ), and show that. Our construction solves a problem posed by Melrose in 1990. Finally, we introduce and study semi-classical and "suspended" versions of the algebra Ψ ∞ 1,0,V (M 0 ).

Dedicated to Ivo Babuška on the occasion of his 80th birthday.Abstract. We prove a regularity result for the Poisson problem −∆u = f , u| ∂P = g on a polyhedral domain P ⊂ R 3 using the Babuška-Kondratiev spaces K m a (P). These are weighted Sobolev spaces in which the weight is given by the distance to the set of edges [4,33]. In particular, we show that there is no loss of K m a -regularity for solutions of strongly elliptic systems with smooth coefficients. We also establish a "trace theorem" for the restriction to the boundary of the functions in K m a (P).

We study the behavior of the spectrum of the Dirac operator on collapsing S 1 -bundles. Convergent eigenvalues will exist if and only if the spin structure is projectable.

Let (M, g) be a compact Riemannian manifold of dimension n 3. We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to g and of volume 1. We study when it is attained. As an application, we find nodal solutions of the Yamabe equation.