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 $M_0$ whose geometry is described by a compactification to a manifold with corners M and a Lie algebra of vector fields on M, subject to constraints only on $M \smallsetminus M_0$. The Lie structure at infinity on $M_0$ determines a metric on $M_0$ up to bi-Lipschitz equivalence. This leads to the natural problem of understanding the Riemannian geometry of these manifolds. We prove, for example, that on a manifold with a Lie structure at infinity the curvature tensor and its covariant derivatives are bounded. We also study a generalization of the geodesic spray and give conditions for these manifolds to have positive injectivity radius. An important motivation for our work is to study the analysis of geometric operators on manifolds with a Lie structure at infinity. For example, a manifold with cylindrical ends is a manifold with a Lie structure at infinity. The relevant analysis in this case is that of totally characteristic operators on a compact manifold with boundary equipped with a ``b-metric.'' The class of conformally compact manifolds, which was recently proved of interest in the study of Einstein's equation, also consists of manifolds with a Lie structure at infinity.


Introduction
Geometric differential operators on complete, non-compact Riemannian manifold 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 reason being that differential operators on non-compact manifolds do not enjoy some of the most useful properties enjoyed by their counterparts on compact manifolds.
For example, elliptic operators on non-compact 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, non-compact 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 within).
However, if one restricts oneself to certain classes of complete, non-compact 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 2.1). We stress here that few results on the geometry of these manifolds have a parallel in the literature, although there is a fair number of papers devoted to the analysis on particular classes of such manifolds [11,12,14,15,16,39,40,57,56,66,69,72,76,78,79,81]. The philosophy of Cordes' comparison algebras [13], Kondratiev's approach to analysis on singular spaces [39], Parenti's work on manifolds that are Euclidean at infinity [66], and Melrose's approach to pseudodifferential analysis on singular spaces [57] have played an important role in the development of this subject.
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 extends to a vector bundle A → M with some additional structure. We assume, for example, that the Lie bracket of vector fields on M 0 defines, by restriction, a Lie algebra structure on the space of sections of A such that the space V := Γ(A) of sections of A identifies with a Lie subalgebra of the Lie algebra of all vector field on M 0 . The pair (M, V) then defines a Lie structure at infinity on M 0 . A simple, non-trivial class of manifolds with a Lie structure at infinity is that of manifolds with cylindrical ends. Let M 0 be a manifold with cylindrical ends. In this case, the compactification M is a manifold with boundary, V consists of all vector fields tangent to the boundary of M . This example plays a prominent role in the analysis of boundary value problems on manifolds with conical points [39,41,59,55,73,74]. See the above references for earlier results.
Let (M, V) be a Lie structure at infinity on M 0 , V = Γ(A). The choice of a fiberwise scalar product on A gives rise to a fiberwise scalar product g on T M 0 , i.e. a Riemannian metric on M 0 . Since M is compact, any two such metrics g 1 and g 2 are equivalent, in the sense that there exists a positive constant C > 0 such that C −1 g 1 ≥ g 2 ≥ Cg 1 . One can thus expect that the properties of the Riemannian manifold (M 0 , g) obtained by the above procedure depend only on the Lie structure at infinity on M 0 and not on the particular choice of a metric on A. However, as shown in the following example, a metric on M 0 does not determine a Lie structure at infinity on M 0 .
Example 0.1. Let us compactify R by including +∞ and −∞: We define ϕ : [−1, +1] → R, ϕ(t) = log(t + 1) − log(1 − t), ϕ(±1) = ±∞. The pullback of the differentiable structure on [−1, 1] defines a differentiable structure on R. On R we consider the Lie algebra of vector fields that vanish at ±∞. The product of these compactifications of R defines a Lie structure on M 0 := R n , in which the compactification M is diffeomorphic to the manifold with corners [−1, 1] n and the sections of A are all vector fields tangent to all hyperfaces. (The resulting Lie structure at infinity is that of the b-calculus (see Example 1.5)). Alternatively, one can consider the radial compactification of R n . The resulting Lie structure at infinity is described in Example 1.6, which is closely related to the so-called scattering calculus [56,66].
We thus see that R n fits into our framework and is in fact a manifold with a Lie structure at infinity for several distinct compactifications M .
Thus, although our motivation for studying manifolds with a Lie structure at infinity comes from analysis, this class of manifolds leads to some interesting questions about their geometry, and this paper (the first one in a series of papers on this subject) is devoted mainly to the issues and constructions that have a strong Riemannian geometric flavor. It is important to mention here that only very few results on the geometry of particular classes of Riemannian manifolds with a Lie structure at infinity were proved before, except some special examples (e.g. compact manifolds and manifolds with cylindrical ends). For example, we prove that M 0 is complete and has bounded curvature, in the sense that the Riemannian curvature R and all its covariant derivatives ∇ k R, with respect to the Levi-Civita connection, are bounded. Also, under some mild assumptions on (M, V), we prove that (M 0 , g) has positive injectivity radius, and hence M 0 has bounded geometry. This is very convenient for the analysis on these manifolds. The main technique is based on generalizing the Levi-Civita connection to an "A * -valued connection" on A. (An A * -valued connection on a bundle E → M is a differential operator ∇ : E → E ⊗ A * that satisfies all the usual properties of a connection, but with A replacing the tangent bundle, see Definition 1.20. This concept was first introduced in a slightly different form in [26] by Evens, Lu, and Weinstein. The right approach to the geometry of manifolds with a Lie structure at infinity requires us to replace the tangent bundle by A. This was noticed before in particular examples, see for instance [11,46,56,57,60]. The Lie structure at infinity on M 0 allows us to define a canonical algebra of differential operators on M 0 , denoted Diff(V), as the algebra of differential operators generated by the vector fields in V = Γ(A) and multiplication by functions in C ∞ (M ). If E 0 , E 1 → M are vector bundles on M , then one can similarly define the spaces Diff(V; E 0 , E 1 ) (algebras if E 0 = E 1 ) of differential operators generated by V and acting on sections of E 0 with values sections of E 1 . All geometric operators on M 0 (de Rham, Laplace, Dirac) will belong to one of the spaces Diff(V; E 0 , E 1 ), for suitable bundles E 0 and E 1 . The proof of this result depends on our extension of the Levi-Civita connection to an A * -valued connection.
Many questions in the analysis on non-compact manifolds or on the asymptotics of various families of operators can be expressed in terms of Diff(V). We refer to [15,17,44,33,43,46,51,56,57,58] for just a few of the many possible examples in the literature. Indeed, let ∆ = d * d ∈ Diff(V) be the scalar Laplace operator on M 0 . Then ∆ is essentially self-adjoint on C ∞ c (M 0 ) by old results of Gaffney [29] and Roelcke [70] from 1951 and 1960. Assume that M 0 has positive injectivity radius, then P (1 + ∆) −m/2 and (1 + ∆) −m/2 P are bounded operators on L 2 (M 0 ), for any differential operator P ∈ Diff(V) of order at most m. Cordes [12,13] defined the comparison algebra A(M, V) as the norm closed algebra generated by the operators P (1 + ∆) −m/2 and (1 + ∆) −m/2 P , with P a differential operator P ∈ Diff(V) of order at most m. The comparison algebra is useful because it leads to criteria for differential and pseudodifferential operators to be compact or Fredholm between suitable Sobolev spaces [43,1,2].
We expect manifolds with a Lie structure at infinity and especially the analytic tools (pseudodifferential and asymptotic analysis) that we have established in [1,2] to play an important role for solving some problems in geometric analysis simultaneously for a large class of manifolds. Indeed, in special cases of manifolds with a Lie structure at infinity the solutions to quite a few interesting problems in geometric analysis rely heavily on those methods. For instance, consider asymptotically Euclidean manifolds, a special case of Example 1.6. In general relativity one is interested in finding solutions to the Einstein equations whose spatial part is asymptotically Euclidean. Integration of the first nontrivial coefficient in the asymptotic development of the metric at infinity yields the so-called "mass" of the solution [7]. The positive mass theorem states that any non-flat asymptotically Euclidean Riemannian manifold with non-negative scalar curvature has positive mass. An elegant proof of the positive mass theorem by Witten [67] uses Sobolev embeddings on such manifolds. The positive mass theorem provides the final step in the proof of the Yamabe conjecture on compact manifolds [71]: Any conformal class on a compact manifold M admits a metric with constant scalar curvature. In order to prove the conjecture in the locally conformally flat case, one replaces the metric g on M by a scalar-flat metric u · g on M \ {p} where u is a function u(x) → ∞ for x → p, and a neighborhood of p provides the asymptotically euclidean end, and one applies the positive mass conjecture to this. On most non-compact manifolds, the Yamabe problem is still unsolved. However, special cases have been solved, e.g. on manifolds with cylindrical ends [3].
Both the geometry and the analysis of asymptotically hyperbolic manifolds have been the subject of articles in general relativity and the analysis of 3-manifolds, see [4,6]. One can prove rigidity theorems [5] for asymptotically hyperbolic ends, or existence results for asymptotically hyperbolic Einstein metrics [48]. Similar rigidity problems for asymptotically complex hyperbolic ends are subject in [9,10,31].
Or take the construction of manifolds with with special holonomy SU (m), Sp(m) and G 2 where the analysis of weighted function spaces on manifolds which are quasi asymtotically locally euclidean [35,36,37] has been used.
In summary, our present program will lead to a unified approach to the analysis on various types of manifolds with a "good" asymptotic behavior at infinity.
We now discuss the contents of each section. In Section 1 we introduce and study structural Lie algebras of vector fields and the equivalent concept of boundary tangential Lie algebroids. A structural Lie algebra of vector fields on a manifold with corners M gives rise to a canonical algebra Diff(V) of differential operators. We include numerous examples.
Then, in Section 2, we specialize to the case that the constraints are only on the boundary. This special case is called a "manifold with a Lie structure at infinity." The Lie structure at infinity defines a Riemannian metric on the interior of the manifold. This metric is unique up to bi-Lipschitz equivalence. Hence, the Lie structures at infinity is a tool for studying a large class of open Riemannian manifolds. We are interested in the analysis on such open manifolds. Section 3 is devoted to the study of the geometry of Riemannian manifolds with a Lie structure at infinity. We will prove that these manifolds are complete and have bounded curvature (together with all its covariant derivatives). This depends on an extension of the Levi-Civita connection to an A * -valued connection, the appropriate notion of connection in this setting. Then we investigate the question of whether a Riemannian manifold with a Lie structure at infinity has positive injectivity radius.
In Section 4 we introduce Dirac and generalized Dirac operators and prove that they belong to Diff(V; W ), where W is a Clifford module. The same property is shared by all geometric operators (Laplace, de Rham, signature) on the open manifold M 0 .
We thank Sergiu Moroianu for several discussions on the subject.

Structural Lie algebras and Lie algebroids
We introduce in this section the concept of structural Lie algebras of vector fields, which is then used to define manifolds with a Lie structure at infinity.
1.1. Projective modules. In this subsection, we recall some well-known facts about projective modules over C ∞ (M ), where M is a compact manifold, possibly with corners.
Let V be a C ∞ (M ) module with module structure C ∞ (M )×V ∋ (f, v) → f v ∈ V . Let x ∈ M and denote by p x the set of functions on M that vanish at x ∈ M . Then p x V is a complex vector subspace of V and V /p x V is called the geometric fiber of V at x. In general, the geometric fibers of V are complex vector spaces of varying dimensions.
A subset S ⊂ V will be called a basis of V if every element v ∈ V can be written uniquely as v = s∈S f s s, with f s ∈ C ∞ (M ), #{s ∈ S | f s = 0} < ∞. (In our applications, S will always be a finite set, so we will not have to worry about this last condition.) A C ∞ (M )-module is called free (with basis S) if it has a basis S. Unlike the general case, the geometric fibers of a free module have constant dimension, equal to the number of elements in the basis S. Note however, that if f : V → W is a morphism of free modules, the induced map between geometric fibers may have non-constant rank. For example, it is possible that f is injective, but the induced map on the geometric fibers is not injective on all fibers. An example is provided by M = [0, 1], V = W = C ∞ ([0, 1]) and f being given by the multiplication with the coordinate function x ∈ [0, 1]. Then f is injective, but the induced map on the geometric fibers at 0 is 0.
A C ∞ (M )-module V is called finitely generated projective if, by definition, there exists another module W such that V ⊕ W is free with a finite basis. We then have the following fundamental theorem of Serre and Swan [38] Theorem 1.1 (Serre-Swan). If V is a finitely generated, projective module over C ∞ (M ), then the set E := ∪ x∈M (V /p x V ) × {x}, the disjoint union of all geometric fibers of V , can be endowed with the structure of a finite-dimensional, smooth vector bundle E → M such that V ≃ Γ(M ; E). The converse is also true: Γ(M ; E) is a finitely generated, projective C ∞ (M )-module for any finite-dimensional, smooth vector bundle E → M .
Suppose now that V is a C ∞ (M )-module and that M is connected. Then V is a finitely generated, projective C ∞ (M )-module if, and only if, there exists k ∈ Z + satisfying the following condition: For any x ∈ M , there exist ϕ ∈ C ∞ (M ), ϕ(x) = 1, and k-elements v 1 , . . . , v k ∈ V with the property that for any w ∈ V we can find and, moreover, the germs of f 1 , . . . , f k at x are uniquely determined. A module V satisfying Condition (1) above is called locally free of rank k, and what we are saying here is that "locally free of rank k, for some k," is equivalent to "finitely-generated, projective." It is crucial here that the number of elements k is the same for any x ∈ M . In case M is not connected, the number k needs only be constant on the connected components of M .
Remark 1.2. The introduction of projective modules over C ∞ (M ) in Partial Differential Operators on non-compact manifolds was pioneered by Melrose [54] in the early 1980s.

1.2.
Manifolds with corners and structural Lie algebras. We now fix our terminology and recall the definitions of the main concepts related to manifolds with corners.
In the following, by a manifold we shall always understand a C ∞ -manifold possibly with corners. In contrast, a smooth manifold is a C ∞ -manifold without corners. By definition, for every point p in a manifold with corners M , there is a coordinate neighborhood U p of p and diffeomorphism ϕ p to [0, ∞) k × R n−k , with ϕ p (p) = 0, such that the transition functions are smooth (including on the boundary). The number k here clearly depends on p, and will be called the boundary depth of p. Hence points in the interior have boundary depth 0, points on the boundary of a manifold without corners have boundary depth 1, etc. Roughly speaking the boundary depth counts the number of boundary faces p is in.
Moreover, we assume that each hyperface H of M is an embedded submanifold and has a defining function, that is, there exists a smooth function x H ≥ 0 on M such that H = {x H = 0} and dx H = 0 on H. This assumption is just a simplifying assumption. We can deal with general manifolds with corners using the constructions from [60]. Note that a priori we do not fix a particular system of defining functions, but only use their existence occasionally.
If  The starting point of our analysis is a Lie algebra of vector fields on a manifold with corners. For reasons that will be clearer later, we prefer to keep this concept as general as possible, even if for the analysis on non-compact manifolds, only certain classes of Lie algebras of vector fields will be used. By (ii) we mean that V is closed for multiplication with functions in C ∞ (M ) and the induced C ∞ (M )-module structure makes it a finitely generated projective Given a structural Lie algebra V of vector fields on a manifold with corners, we call the enveloping algebra Diff(V) of V the algebra of V-differential operators on M . Note that any V-differential operator P ∈ Diff(V) can be realized as a polynomial in vector fields in V with coefficients in C ∞ (M ) acting on the space Let us give some examples for structural Lie algebras of vector fields. Some of these examples can also be found in [56]. We also give descriptions of the structural vector fields in local coordinates, because this will be helpful in the applications of the theory developed here. All of the following examples model the analysis on some non-compact manifold, except for the last one, which models the analysis of adiabatic families.
The following example is the simplest and most studied so far, however, it is quite important for us because it models the geometry of manifolds with cylindrical ends, and hence it is easier to grasp. Example 1.5. Let M be a manifold with corners and Then V b is a structural Lie algebra of vector fields, and any structural Lie algebra of vector fields on M is contained in V b , by condition (iii) of the above definition. A vector field X ∈ V b is called a b-vector field X. Fix x 1 , . . . , x k and y ∈ R n−k local coordinates near a point p on a boundary face of codimension k, with x j defining functions of the hyperfaces through p. Then any b-vector field X is of the form on some neighborhood of p, with the coefficients a j and b j smooth everywhere (including the hyperfaces x j = 0), for all j. This shows that the Lie algebra of b-vector fields is generated in a neighborhood U of p by x∂ x and ∂ y as a C ∞ (M )-module. The differential operators in Diff(V b ) are called Fuchs type operators, totally characteristic, or simply, and perhaps more systematically b-differential operators. The structural Lie algebra V b and the analysis of the corresponding differential and pseudodifferential operators are treated in detail for instance in [20,23,34,49,55,56,75].
Example 1.6. Let M be a compact manifold with boundary and x : M → R + a boundary defining function. Then the Lie algebra V sc := xV b does not depend on the choice of x and the vector fields in V sc are called scattering vector fields; with respect to local coordinates (x, y) near the boundary, scattering vector fields are generated by x 2 ∂ x and x∂ y . An analysis of the scattering structure can be found in [56]. Since this structure models the analysis on asymptotically Euclidean spaces, let us be a little bit more precise and recall some basic definitions. A Riemannian metric g on the interior of M is called a scattering metric if, close to the boundary ∂M , it is of the form g = dx 2 x 4 + h x 2 where h is a smooth, symmetric 2-tensor on M which is non-degenerate when restricted to the boundary. Then scattering vector fields are exactly those smooth vector fields on M that are of bounded length with respect to g, and the corresponding Laplacian ∆ g is an elliptic polynomial in scattering vector fields. As a special case of this setting note that the radial compactification map such that the Euclidean metric lifts to a scattering metric on S N + . The following example is one of the examples that we are interested to use in applications.
Example 1.7. Let M be a manifold with boundary ∂M , which is the total space of a fibration π : ∂M → B of smooth manifolds. We let V e = {X ∈ Γ(T M ) : X is tangent to all fibers of π at the boundary} be the space of edge vector fields. In order to show that this is indeed a structural Lie algebra of vector fields, we have to show that it is closed under Lie brackets.
the commutator is again tangent to the fibers of π. If (x, y, z) are coordinates in a local product decomposition near the boundary, where x corresponds to the boundary defining function, y to a set of variables on the base B lifted through π, and z is a set of variables in the fibers of π, then edge vector fields are generated by x∂ x , x∂ y , and ∂ z . Using this local coordinate description is another way, to see immediately that the space of edge vector fields is in fact a Lie algebra. More importantly, it shows that it is a projective C ∞ (M )-module. The analysis of the Lie algebra V e is partly carried out in [51] and more recently in [46].
A special case of the edge structure is of particular importance for the analysis on hyperbolic space, so it deserves its own name: Example 1.8. Let M be a compact manifold with boundary, and let V 0 be the edge vector fields corresponding to the trivial fibration π = id : ∂M → ∂M , i. e., we have which explains the name 0-vector fields for the elements in V 0 . With respect to local coordinates (x, y) near the boundary, 0-vector fields are generated by x∂ x and x∂ y . Recall that a Riemannian manifold (M 0 , g 0 ) is called conformally compact provided it is isometric to the interior of a compact manifold M with boundary equipped with a metric g = ̺ −2 h in the interior, where h is a smooth metric on M and ̺ : M → R + a boundary defining function. Note that 0-vector fields are the smooth vector fields on M that are of bounded length with respect to g; moreover, the Laplacian ∆ g0 is given as an elliptic polynomial in 0-vector fields. A particular example of conformally compact spaces is of course the hyperbolic space with compactification given by the ball model. Conformally compact spaces arise naturally in questions related to the Einstein equation [4,48,53], and the "AdS/CFT-correspondence." An analysis of 0-vector fields and the associated 0-differential and pseudodifferential operators was carried out for instance in [42,51,68]. Criteria for the Fredholmness of operators in Diff(V 0 ), which is crucial in the approach to the study of Einstein's equations on conformally compact manifolds used in the above mentioned papers, were established for instance in [42,43,45,46,51,52,56,68].
The structural Lie algebra of vector fields in the next example is a slight variation of the Lie algebra of edge vector fields, however, it is worth pointing out that this slight variation leads to a completely different analysis for the associated (pseudo)differential operators. Example 1.9. Let M be as in Example 1.7 and x : M → R + be a boundary defining function. Then V de := xV e is a structural Lie algebra of vector fields; the corresponding structure is called the double-edge structure. With respect to local product coordinates as in Example 1.7, double-edge vector fields are generated by x 2 ∂ x , x 2 ∂ y , and x∂ z . The analysis of the double-edge structure, which is in fact much simpler than the corresponding analysis of the edge structure, can be found for instance in [44].
The following example appears in the analysis of holomorphic functions of several variables.
is a structural Lie algebra of vector fields that is called the Θ-structure. For a local description as well as for an analysis of the Θ-structure we refer to [25].
All the above examples of structural Lie algebras of vector fields model the analysis on certain non-compact manifolds (giving rise to algebras of differential operators that replace the algebra of totally characteristic differential operators) on manifolds with cylindrical ends. The following example, however, models the analysis of a family of an adiabatic differential operators.
Example 1.11. Let N be a closed manifold that is the total space of a locally trivial fibration π : N → B of closed manifolds, let T N/B → N be the vertical tangent bundle, and let M : is a structural Lie algebra of vector fields that is called the adiabatic algebra. If (y, z) are local coordinates on N , where again the set of variables y corresponds to variables on the base B lifted through π, and z are variables in the fibers, then adiabatic vector fields are generated by x∂ y and ∂ z . The adiabatic structure has been studied and used for instance in [62] and [63] We shall sometimes refer to a structural Lie algebra of vector fields simply as Lie algebra of vector fields, when no confusion can arise. Because V is a finitely generated, projective C ∞ (M )-module, using the Serre-Swan theorem [38] (recalled above, see Theorem 1.1) we obtain that there exists a vector bundle naturally as C ∞ (M )-modules. We shall identify from now on V with Γ(A V ). The following proposition is due to Melrose. Remark 1.13. The condition in Definition 1.4 that V has to be projective is essential.
As an example consider M = [0, 1] and let Then V is a C ∞ (M )-module. However, V is not a projective C ∞ (M )-module, as we can see by contradiction. Assume V were projective. Then there is a bundle A over [0, 1] with V = Γ(A). Let s be a trivialization of A, i. e., s(t) = f (t)∂ t with f as above. Hencef (t) = (1/t)f (t) also decays sufficiently fast; however It is convenient for the following discussion to recall the definition of a Lie algebroid. General facts about Lie algebroids can be found in [17,50] (a few basic facts are also summarized in [65]). Definition 1.14. A Lie algebroid A over a manifold M is a vector bundle A over M , together with a Lie algebra structure on the space Γ(A) of smooth sections of A and a bundle map ̺ : We thus see that there exists an equivalence between the concept of a structural Lie algebra of vector fields V = Γ(A V ) and the concept of a boundary tangential Lie algebroid ̺ : In order to shorten our notation, we will write Xf instead of ̺ Γ (X)f for the action of the sections of a Lie algebroid on functions if the meaning is clear from the context.

1.3.
Constructing new Lie algebroids from old ones. Let f : N → M be a submersion of manifolds with corners in the above sense (which implies in particular that any fiber is a smooth manifold). Let A = A V be a boundary tangential Lie algebroid over M .
Projection to the first component yields a surjective linear map f # A p → A f (p) , denoted in the following by f * , and projection onto the second component yields a We obtain the commuting diagram For example f # T M = T N .
Lemma 1. 16. The thick pull-back f # A is a boundary tangential Lie algebroid over N with anchor map given by f # ̺.
Proof. Let Γ vert T N denote the bundle of vertical sections X, i. e., f * X = 0. This bundle coincides by definition with the analogously defined bundle of vertical sections of A. The rows of the following commutative diagram are exact.
The vertical arrows are inclusions. The horizontal arrows of the second row are Lie algebra homomorphisms. The space Γ(A) is by definition a Lie subalgebra of The fact that A is projective [respectively, boundary tangential] immediately implies that Γ(f # A) is also projective [respectively, boundary tangential].
Let g and h be two Lie algebras. Suppose that there is given an action by derivations of g on h: Then we can define the semi-direct sum g ⋉ ϕ h as follows. As a vector space, g ⋉ ϕ h = g ⊕ h, and the Lie bracket is given by for any X 1 , X 2 ∈ g and Y 1 , Y 2 ∈ h. We shall usually omit the index ϕ denoting the action by derivations in the notation for the semi-direct sum. We want to use this construction to obtain new Lie algebroids from old ones. Assume then that we are given two Lie algebroids A, L → M over the same manifold and that Γ(A) acts by derivations on Γ(L). Denote this action by ϕ, as above. We assume that the action of Γ(A) on Γ(L) is compatible with the C ∞ (M )-module structure on Γ(L), in the sense that  In the language of Lie algebroids, the action of Γ(A) on Γ(L) considered above is called a representation of A on L. In a similar way, the action of Γ(A) on Γ(L) by derivation, considered above, deserves to and will be called a representation by derivations of A on L. If A → M is a tangential Lie algebroid, then A ⋉ L → M will also be one.
1.4. Differential operators. We will from now on assume that A denotes the vector bundle determined by the structural Lie algebra V and vice versa. Definition 1.17. Let Diff(V) denote the algebra of differential operators generated by V, where the vector fields are regarded as derivations on functions.
We also want to study differential operators with coefficients in vector bundles. Let E 1 → M and E 2 → M be two vector bundles. Embed E i ֒→ M × C Ni , i = 1, 2.
Denote by e i a projection in M Ni (C ∞ (M )) whose (pointwise) range is E i . Then we define This definition of Diff(V; E 1 , E 2 ) is independent of the choices of the embeddings E i ֒→ M × C Ni and of the choice of e i . Elements of Diff(V; E 0 , E 1 ) will be called differential operators generated by V. In the special case E 1 = E 2 = E we simply write Diff(V; E), the algebra of differential operators on E generated by V.
It is possible to describe the differential operators in Diff(V; E) locally on M as follows. Proof. In a trivialization of E above some open subset, we can assume that e is a constant matrix.
Assume now that A| M0 = T M 0 . The vector bundles Λ q T * M 0 extend to bundles Λ q A * on M . The Cartan formula (e.g. [8]) says that on M 0 the de Rham differential is the de Rham differential of ordinary differential geometry. Definition 1.20. Let E → M be a vector bundle. An A * -valued connection on E is a differential operator such that, for any X ∈ Γ(A), the induced operator D X : Γ(E) → Γ(E) satisfies the usual properties of a connection: It is clear from (i) that the operator D X is of first order. Our definition of an A * -valued connection is only slightly more restrictive than that of A-connection introduced in [26]. (In that paper, Evens, Lu, and Weinstein considered (ii) only up to homotopy.) Clearly if D and D ′ are A * -valued connections on E and, respectively, E ′ , then See also [21,22] 2. Lie structures at infinity In this section we introduce the class of manifolds with a Lie structure at infinity, and we discuss some of their properties. Our definition, Definition 2.1, formalizes some definitions from [56].
In some of the first papers on the analysis on open manifolds using Lie algebras of vector, for example [12,13,15,79], the vector fields considered were required to vanish at infinity. In order to obtain more general results and in agreement with the more recent papers on the subject (for example [14,56,64,80]), we do not make this assumption. As a consequence, the comparison algebras that result from our setting do not have in general the property that the commutators are compact. Here is now an explicit test for a Lie algebra of vector fields V on a compact manifold with corners M to define a Lie structure at infinity on the interior M 0 of M . This characterization of Lie structures at infinity is in the spirit of our discussion of local basis (see Equation (1) and the discussion around it).

Proposition 2.2.
We have that the Lie algebra V ⊂ Γ(M ; T M ) defines a Lie structure at infinity on M 0 if, and only if, the following conditions are satisfied: , and Y is a vector field on M 0 , then there exists X ∈ V, such that X| U = Y | U . (iii) If x ∈ ∂M = M M 0 , then we can find n linearly independent vector fields X 1 , X 2 , . . . , X n ∈ V, n = dim M , defined on a neighborhood U of x, such that for any X ∈ V, there exist smooth functions f 1 , . . . , f n ∈ C ∞ (U ) uniquely determined by f k X k on U .
(iv) There are functions f ijk ∈ C ∞ (U ) (in particular smooth on the boundary ∂M ∩ U ) such that the vector fields X j from (iii) satisfy [X i , X j ] = n k=1 f ijk X k on U .
Proof. The proof is an immediate translation of the definition of a manifold with a Lie structure at infinity using the description of projective CI(M ) module given at the end of Subsection 1.1 (especially Equation (1)).

2.2.
Riemannian manifolds with Lie structures at infinity. We now consider Riemannian metrics on A → M . In particular g defines a Riemannian metric on M 0 . The geometry of these metrics will be the topic of the next section. Note that the metrics on M 0 that we obtain are not restrictions of Riemannian metrics on M . In the following section, we will prove for example that (M 0 , g) is a complete Riemannian metric. Any curve joining a point on the boundary ∂M to the interior M 0 is necessarily of infinite length.

Example 2.4. Manifolds with cylindrical ends.
A manifold M with cylindrical ends is obtained by attaching to a manifold M 1 with boundary ∂M 1 the cylinder (−∞, 0] × ∂M 1 , using a tubular neighborhood of ∂M 1 , where the metric is assumed to be a product metric. The metric on the cylinder is also assumed to be the product metric. Let t be the coordinate of (−∞, 0]. By the change of variables x = e t , we obtain that M is diffeomorphic to the interior of M 1 and V = V b . Other changes of variables lead us to different Lie structures at infinity. Similarly, products of manifolds with cylindrical ends can be modeled by manifolds with corners and the structural Lie algebra of vector fields V b . This applies also to manifolds that are only locally at infinity products of manifolds with cylindrical ends.

2.3.
Bi-Lipschitz equivalence. It turns out that the metric on a manifold with a Lie structure at infinity is essentially unique, namely any two such metrics are bi-Lipschitz equivalent (see the corollary below). Lemma 2.5. We assume that a manifold M 0 which is the interior of a compact manifold with corners M carries two Lie structures at infinity (M, V 1 ) and (M, V 2 ) satisfying V 1 ⊂ V 2 . Furthermore, let g j be Riemannian metrics on A Vj , j = 1, 2. Then there is a constant C such that for all X ∈ T M 0 .
Proof. The pull-back of g 2 to A V1 is a non-negative symmetric two-tensor on A V1 . The statement then follows from the compactness of M .
As a consequence the volume element of g 2 is bounded by a multiple of the volume element of g 1 . Furthermore, we have inclusions of L p -functions: L p (M 0 , g 1 ) ֒→ L p (M 0 , g 2 ).
Corollary 2.6. If two Riemannian metrics g 1 , g 2 on M 0 are Riemannian metrics for the same Lie structure at infinity (M, V), then they are bi-Lipschitz, i. e., there is a constant C > 0 with for all X ∈ T M 0 . In particular, C −1 d 2 (x, y) ≤ d 1 (x, y) ≤ Cd 2 (x, y), where d i is the metric on M 0 associated to g i .
Proof. The first part is clear. The proof of the last statement is obtained by comparing the metrics on a geodesic for one of the two metrics.

Geometry of Riemannian manifolds with Lie structures at infinity
We now discuss some geometric properties of Riemannian manifolds with a Lie structure at infinity. We begin with a simple observation about volumes. Proof. Because of Lemma 2.5, we can assume that Γ(A) are the vector fields tangential to the boundary. For simplicity in notation, let us assume that M is a compact manifold with boundary. Let dvol ′ be the volume element on M associated to some metric on M that is smooth up to the boundary. Then dvol ≥ Cx −1 dvol ′ for any boundary defining function x and a constant C.
So, if f is non-zero on ∂M with defining function x, then

Connections and Curvature.
Most of the natural differential operators between bundles functorially associated to the tangent bundle extend to differential operators generated by V, with the tangent bundle replaced by A. The main example is the Levi-Civita connection.
for all X, Y, Z ∈ Γ(A) and f ∈ C ∞ (M ). Moreover, the above equations uniquely determine ∇.
Proof. Suppose X, Y ∈ V = Γ(A) ⊂ Γ(T M ). We shall define ∇ X Y on M 0 using the usual Levi-Civita connection ∇ on T M 0 . We need to prove that there exists X 1 ∈ Γ(A) whose restriction to M 0 is ∇ X Y .
Recall (for example from [8]), that the formula for ∇ X Y is given by Suppose X, Y, Z ∈ Γ(A) in the above formula. We see then that the function 2 ∇ X Y, Z , which is defined a priori only on M 0 , extends to a smooth function on M . Since the inner product , is the same on A and on T M 0 (where they are both defined), we see that the above equation determines ∇ X Y as a smooth section of A. This completes the proof.
The above lemma has interesting consequences about the geometry of Riemannian manifolds with Lie structures at infinity.
Using the terminology of A * -valued connections (see Definition 1.20), Lemma 3.2 can be formulated as saying that the usual Levi-Civita connection on M 0 extends to an A * -valued connection on A. Similarly, we get A * -valued connections on A * and on all vector bundles obtained functorially from A. We use this remark to obtain a canonical A * -valued connection on the bundles A * ⊗k ⊗ Λ 2 A * ⊗ End(A).
(Here E ⊗k denotes E ⊗ . . . ⊗ E, k-times, as usual. ) We define the Riemannian curvature tensor as usual Once we have obtained that R ∈ Γ(Λ 2 A * ⊗ End(A)), we can apply the A * -valued Levi-Civita connection to obtain The boundedness of ∇ k R follows from the fact that M is compact.
The covariant derivative will be called, by abuse of notation, the A * -valued Levi-Civita connection, for any k and j. Sometimes, when no confusion can arise, we shall call this A * -valued connection ∇ simply the Levi-Civita connection. The fiber of A in p ∈ M is denoted by A p .
We chose the terminology "Ehresmann connection" to honor important work of Ehresmann's on the subject [24]. In fact, let ∇ be an A * -valued connection. Then we obtain the Ehresmann connection as follows: For X 0 ∈ V , p = π(X 0 ) we extend X 0 to a local section X : U → V , where U is a neighborhood of p in M . We define the horizontal lift It is easy to check that this map does not depend on the extension X of X 0 , that H X0 is injective and that we have (π * ) X0 • H X0 = id. Then τ X0 := im H X0 is an Ehresmann connection on V . The associated Ehresmann connection completely characterizes the A * -valued connection. However, there are Ehresmann connections on V that do not come from A * -valued connections (they are not "compatible" with the vector space structure).
(b) If the A * -valued connection is metric with respect to a chosen metric on V , then the Ehresmann connection is tangential to the sphere bundle in V with respect to that metric.
3.4. Geodesic flow. For any boundary tangential Lie algebroid A equipped with a metric, let S(A) be the unit tangent sphere in A, The canonical projection map π : S(A) → M is a submersion of manifolds with corners. Let π # A be the thick pull-back of A.
The manifold (with corners) S(A) carries an Ehresmann connection and a horizontal lift H given by the Levi-Civita-connection on A.
Definition 3.7. The geodesic spray is defined to be the map which defines a section S of π # A → S(A). The flow of this vector field is called the geodesic flow.
Restricted to the interior of the manifold, these concepts recover the analogous concepts of ordinary Riemannian geometry.
By definition, the image of S through the anchor map is a vector field along S(A) that is tangential to all the boundary faces of S(A). These boundary faces are preimages of the boundary faces of M under π.
Lemma 3.8. Let A be a boundary tangential Lie algebroid and let X ∈ Γ(A). Then X is complete in the sense that the flow lines ϕ t of X are defined on R. The flow ϕ t preserves the boundary depth. In particular, flow lines emanating from N 0 := N ∂N stay in N 0 .
Proof. For any boundary defining function x H one has d dt t=0 x H (ϕ t ) = dx H (X) = 0, hence the flow preserves the boundary depth. In particular, the flow preserves the boundary. Let I = (a, b) be a maximal open interval on which one particular flow-line is defined. Let t i → b. Assume that b < ∞. Since N is compact, after passing to a suitable subsequence, we can assume that ϕ ti converges to p ∈ N . In a neighborhood of p, the flow exists, which contradicts the maximality of b. Hence b = ∞. The proof for a = −∞ is similar.
By applying this lemma to the geodesic flow on N = S(A), we obtain two corollaries.  3.5. Positive injectivity radius. We already know that Riemannian manifolds with a Lie structure at infinity are complete and have bounded sectional curvature. For many analytic statements it is very helpful if we also know that the injectivity radius inj(M 0 , g) = inf p∈M0 inj p is positive. For example see [40], where a "uniform bounded calculus of pseudodifferential operators" was defined on a manifold with bounded geometry. Hebey [30,Corollary 3.19] proves Sobolev embeddings for manifolds with bounded geometry, i. e., complete Riemannian manifolds with positive injectivity radius and bounded covariant derivatives ∇ k R of the Riemannian curvature tensor R. We will say more about this in a future paper. Conjecture 3.11. All Riemannian manifolds with Lie structures at infinity have positive injectivity radius.
We now introduce two conditions on a Riemannian manifold M 0 with a Lie structure at infinity (M, V), see Definitions 3.12 and 3.16, and prove that if any of these conditions holds, then the injectivity radius of M 0 is positive. Definition 3.12. A manifold M 0 with a Lie structure at infinity (M, V) is said to satisfy the local closed extension property for 1-forms if any p ∈ ∂M has a small neighborhood U ⊂ M such that any α p ∈ A * p extends to a closed one-form on U . Example 3.13. For the b-calculus, the local closed extension property holds, because in the notation of Example 1.5 the locally defined closed 1-forms dx j /x j and dy k span A * p = (T b p M ) * for any p ∈ ∂M .
Theorem 3.14. Let M 0 be a manifold with a Lie structure at infinity (M, V) which satisfies the local closed extension property. Then for any Riemannian metric g on A the injectivity radius of (M 0 , g) is positive.
Proof. We prove the theorem by contradiction. If the injectivity radius is zero, then, as the curvature is bounded, there is a sequence of geodesics loops c i : [0, a i ] → M 0 , parametrized by arc-length, with a i → 0. Because of the compactness of S(A) we can choose a subsequence such thatċ i (0) converges to a vector v ∈ S(A).
Obviously, the base point of v has to be in ∂M . By the local closed extension property, there is a closed 1-form α on a small neighborhood of the base-point of v such that α(v) = 0. On the other hand, because of the closedness of α we get for sufficiently large i where ϕ t : S(A) → S(A) denotes the geodesic flow. As i → ∞, the integrand converges uniformly to α(v), thus we obtain the contradiction α(v) = 0.
In the remainder of this subsection we will prove another sufficient criterion.
be a local parametrization of M , i. e., ϕ −1 is a coordinate chart. Then for v ∈ R n , the local vector field ϕ * (v), i. e., the image of a constant vector field v on R n is called a coordinate vector field with respect to ϕ. Definition 3.16. A manifold M 0 with a Lie structure at infinity (M, V) is said to satisfy the coordinate vector field extension property if A V carries a Riemannian metric g such that for any p ∈ ∂M there is a parametrization ϕ : [0, ∞) n−k × R k → U of a neighborhood U of p such that (i) for any v ∈ R n \ {0} the normalized coordinate vector field which a priori is only defined on U ∩ M 0 extends to a section of A| U , (ii) for linearly independent vectors v and w, X v (p) and X w (p) are linearly independent.
Note that Property (i) is equivalent to claiming that extends to a smooth function on M .
Theorem 3.17. Let M 0 be a manifold with a Lie structure at infinity (M, V) that satisfies the coordinate vector field extension property. Then for any Riemannian metric g on A, the injectivity radius of (M 0 , g) is positive.
The theorem will follow right away from Proposition 3.19, Lemma 3.23, and Lemma 3.24, which we proceed to state and prove after the following definition.
In the following, balls in euclidean R n will be called flat balls or sometimes even just ǫ-ball. Definition 3.18. For C ≥ 1 and ε > 0, we say that (M 0 , g) is locally C-bi-Lipschitz to an ε-ball if each point p ∈ M 0 has a neighborhood that is bi-Lipschitz diffeomorphic to a flat ball of radius ε with bi-Lipschitz constant C.
Proposition 3.19. Let (M 0 , g) be a complete Riemannian manifold with bounded sectional curvature. Then the following conditions are equivalent: (1) (M 0 , g) has positive injectivity radius.
(2) There are numbers δ 1 > 0 and C > 0 such that any loop of length δ ≤ δ 1 is the boundary of a disk of diameter ≤ C · δ. (3) There is C > 0 and ε > 0 such that (M 0 , g) is locally C-bi-Lipschitz to a ball of radius ε. Proof.
(3)=⇒ (2): Under the condition of (3) any loop of length ≤ 2ε/C based in p lies completely inside B ε/C (p). On the other hand B ε/C (p) is contained in a neighborhood U of p which is C-bi-Lipschitz to a flat ball of radius ε. Inside a flat ball any short loop is the boundary of a disk of small diameter. Hence (2) follows from (3) with δ 1 := 2ε/C and with the same C in (2) as in (3).
(2)=⇒(1): Because sectional curvature is bounded from above, there is a ̺ > 0 such that there are no conjugate points along curves of length smaller than ̺. For each p ∈ M , the exponential map is a local diffeomorphism from B ̺ (p) into M . We want to show that the exponential map is injective on any ball of radius ε := min{δ 1 /2, ̺/(4C)}. For this we assume that exp p (q 1 ) = exp p (q 2 ), q 1 , is a closed loop of length ≤ 2ε. Because of the conditions in the proposition, this loop is the boundary of a disk of diameter ≤ ̺/2. Because the exponential map is a local diffeomorphism, it is not difficult to see that such disks lift to T p M . Hence q 1 = q 2 which yields injectivity.
As a consequence of this proposition, the property of having positive injectivity radius is a bi-Lipschitz invariant inside the class of complete Riemannian manifolds with bounded curvature.
Corollary 3.20. Let g 1 and g 2 be two complete metrics on M 0 , such that there is Then (M 0 , g 1 ) has positive injectivity radius if, and only if, (M 0 , g 2 ) has positive injectivity radius.
Proof. Proposition 3.19 gives necessary and sufficient criteria for positive injectivity radius, that are bi-Lipschitz equivalent.
Together with Corollary 2.6 we obtain. at infinity. We say that the Lie structure at infinity is controlled if for all p ∈ ∂M there is a parametrization ϕ : [0, ∞) n−k × R k → U around p, a δ > 0 and a constant C > 0 such that for all x ∈ M 0 ∩ U and all v ∈ R n the inequality holds. Here B δ (x) denotes the ball of radius δ around x with respect to the metric g. Proof. Let ϕ be a parametrization of a neighborhood U of p ∈ ∂M and let ϕ * (v) and ϕ * (w) be arbitrary coordinate vector fields. Because of Property (i) in Definition 3.16, the (local) functions f v := 1/ g(ϕ * (v), ϕ * (v)) and f w := 1/ g(ϕ * (w), ϕ * (w)) extend to the boundary, and X v := f v ϕ * (v) and X w := f w ϕ * (w) are sections of A. For linearly independent v and w we calculate This is again a section of A, and because of Property (ii) in Definition 3.16, X v and X w are even linearly independent on the boundary ∂M . As a consequence, is also bounded. Hence, in both cases: (13) |X v (log g(ϕ * (w), ϕ * (w)))| ≤ C.
By summing up, one immediately sees that C can be chosen independently from the choice of v. Hence for fixed w, (13) holds uniformly for any unit vector X.
We take two arbitrary points y 1 , y 2 ∈ B δ (x), x ∈ U ∩ M 0 . We can assume that B δ (x) ⊂ U . We join y 1 and y 2 by a path c : [0,δ] → M 0 , parametrized by arc-length,δ ≤ 2δ. We estimate Hence, the quotient is bounded by an expression that depends only on p, U, δ and global data. Thus, (M, V, g) is controlled.
Lemma 3.24. If the boundary tangential Lie algebroid is controlled, then there is C > 0 and ε > 0 such that (M 0 , g) is locally C-bi-Lipschitz to an ε-ball.
Proof. On the ball B δ (x) we regard two metrics: the original metric g and the metricg with g x =g x andg x is constant in the local coordinate chart. These metrics are bi-Lipschitz on B with a bi-Lipschitz constant C 1 . Thus, the ballB x of radius δ/C 1 around x with respect to the metricg is a flat ball. Hence, (B x ) x∈M0 are neighborhoods that are uniformly bi-Lipschitz to δ/C 1 -balls.
This completes the proof of Theorem 3.17. We continue with some examples and applications.   ∋ (t, x, y). The vector fields X = t 2 ∂ t , Y = e −C/t ∂ x and Z = e −C/t ∂ y span a free C ∞ (M )-module that is closed under Lie-brackets. Hence, it is a structural Lie algebra of vector fields V. We define a metric g by claiming that X, Y, Z are orthonormal, then (M, g, V) satisfies the coordinate vector field extension property.
x, y). The vector fields X = t 2 ∂ t , Y = e −C/t sin(1/t)∂ x + cos(1/t)∂ y and Z = e −C/t cos(1/t)∂ x − sin(1/t)∂ y span a free C ∞ (M )-module that is closed under Lie-brackets. Hence it is a structural Lie algebra of vector fields V. We define a metric by claiming that X, Y, Z are orthonormal. In this example the normalized coordinate vector fields e −C/t ∂ x and e −C/t ∂ y are not contained in V. Hence (M, V) does not satisfy the coordinate vector field extension property. However, (M 0 := M \∂M, g) is isometric to the interior of the previous example. Corollary 3.20 together with Theorem 3.17 will show that (M 0 , g) has positive injectivity radius, although the conditions in Theorem 3.17 are not satisfied for (M, V) directly.
3.6. Adjoints of differential operators. We shall fix in what follows a metric on A, thus we obtain a Riemannian manifold (M 0 , g) with a Lie structure at infinity, which will remain fixed throughout this section.
Let us now discuss adjoints of operators in Diff(V). The metric on M 0 defines a natural volume element µ on M 0 , and hence it defines also a Hilbert space L 2 (M 0 , dµ) with inner product (g 1 , g 2 ) := M0 g 1 g 2 dµ. The formal adjoint D ♯ of a differential operator D is then defined by the formula (14) (Dg 1 , g 2 ) = (g 1 , D ♯ g 2 ), ∀g 1 , g 2 ∈ C ∞ c (M 0 ). We would like to prove that D ♯ ∈ Diff(V) provided that D ∈ Diff(V). To check this, we first need a Lemma. Fix a local orthonormal basis X 1 , . . . , X n of A (on some open subset of M ). Then ∇ Xi X = c ij (X)X j , for some smooth functions c ij (X). Then div(X) := − c jj (X) is well defined and gives rise to a smooth function on M . See [27], Chapter IV. A.
In particular, the formal adjoint of X is X ♯ = −X + div(X) ∈ Diff(V).
The divergence theorem (e.g. [27], Chapter IV.A.) states for X ∈ Γ(A) and compactly supported functions f Now, if we set f = g 1 g 2 , we see directly that This implies the formula for the adjoint of X. Proof. The formal adjoint of a vector field X ∈ V, when regarded as a differential operator on M 0 , is given by X ♯ = −X + div(X). The adjoint of f ∈ C ∞ (M ) is given by f ♯ = f . Since Diff(V) is generated as an algebra by operators of the form X and f , with X and f as above, and (D 1 D 2 ) ♯ = D ♯ 2 D ♯ 1 , this proves that Diff(V) is closed under taking adjoints.
If E is a hermitian vector bundle, then we can choose the embedding E → M × C N to preserve the metric. Then the projection e onto the range of E is a selfadjoint projection in M N (Diff(V)). The equation e * = e satisfied by e guarantees that Diff(V; E) := eM N (Diff(V))e is also closed under taking formal adjoints.
Similarly, we obtain the following easy consequence. Proof. Write E := E 0 ⊕ E 1 and use the resulting natural matrix notation for operators in Diff(V; E).

Geometric operators
In this section we will see that the Hodge Laplacian (d + d * ) 2 on forms and the classical Dirac operator on a Riemannian (spin) manifold M 0 with a Lie structure at infinity (M, V) are differential operators generated by V = Γ(A). (See also [46] for some similar results).
Both the classical Dirac operator and d + d * are generalized Dirac operators. We will show that any generalized Dirac operator is a differential operator generated by V. Our approach follows closely that in [28].
that is, it is a differential operator generated by V.
Proof. This follows directly from Corollary 3.30 and the construction in Example 1.19.

4.2.
Principal bundles and connection-1-forms. Let E → M be a vector bundle of rank k carrying a metric and an orientation. In this subsection, we will show that giving a metric A * -valued connection on E is equivalent to giving a A * -valued connection-1-form on the frame bundle of E. Our approach generalizes the case of Riemannian manifolds (see e.g. [47, II, §4]), hence we will omit some details. For simplicity, we will assume from now on that the vector bundle A → M is orientable.
Let P be a principal G-bundle. We denote the Lie algebra of G with g. The most important example will be the bundle of oriented orthonormal frames of the bundle E, denoted by π P : P SO (E) → M , which is a principal SO(k)-bundle. Differentiating the action of G gives rise to the canonical map Definition 4.2. An A * -valued connection-1-form ω is an g ⊗ A * -valued 1-form on P SO (A) satisfying the compatibility conditions ω( V ) = V and g * ω = Ad g −1 ω for all V ∈ so(g).
If g ⊂ so(k), we write ω = (ω ij ) with respect to the standard basis of so(k). In particular, the ω ij are A * -valued 1-forms on P SO (A) satisfying ω ij = −ω ji .
Here "A * -valued" is in the sense of Definition 3.4. Any A * -valued connection-1-form on P gives rise to a G-invariant Ehresmann connection on the bundle P via τ = {V ∈ π P # A | ω(V ) = 0}. It is easy to check that this yields a one-toone correspondence between G-invariant Ehresmann connections and connection-1-forms.
where E = (e 1 , . . . , e n ) is a local section of P SO (E). Conversely, any metric connection on E arises from such an A * -valued connection-1-form.
Note that E * ω ij is a well-defined A * -valued 1-form on M . The proof is straightforward and runs completely analogous to [47, II, Proposition 4.4] with ordinary 1-forms replaced by A * -valued 1-forms. As a result, we conclude that the Levi-Civita connection on A determines an SO(n)-invariant Ehresmann connection and an A * -valued connection-1-form on P SO (A).  The (thick) pull-back of any SO(n)-invariant Ehresmann connection on the principal SO(n) bundle P SO (A) → M with respect to A defines a Spin(n)-invariant Ehresmann connection on P Spin → M with respect to A. Similarly, by using the standard identification of the Lie algebra of SO(n) with the Lie algebra of Spin(n), any A * -valued connection-1-form on P SO (A) pulls back to an A * -valued connection-1-form on P Spin . Now, let σ n : Spin(n) → SU(Σ n ) be the complex spinor representation, e.g. the restriction of an odd irreducible complex representation of the Clifford algebra on n-dimensional space [47]. The complex dimension of Σ n is d n := 2 [n/2] . Definition 4.6. Let M 0 be an n-dimensional Riemannian manifold with a Lie structure at infinity (M, V, g) carrying a spin structure P Spin (A) → P SO (A). The spinor bundle is the associated vector bundle ΣM := P Spin (A) × σn Σ n on M .
Any metric A * -valued connection on A gives rise to an A * -valued connection on ΣM as follows: Proposition 4.3 defines an A * -valued connection-1-form on P SO (A) which can be pull-backed to P Spin (A). With Definition 4.5 applied to ̺ = σ n : Spin(n) → SU(d n ) ⊂ SO(2d n ) we obtain an A * -valued connection-1form on P Spin (A) × σn SO(2d n ) compatible with complex multiplication. Another application of Proposition 4.3 yields a complex A * -valued connection on ΣM .
In particular, the Levi-Civita-connection on A defines then a metric connection on ΣM , the so-called Levi-Civita-connection.
satisfying (16) X · Y + Y · X + 2g(X, Y ) · ϕ = 0 for all X, Y ∈ R n and all ϕ ∈ Σ n , the so-called Clifford multiplication relations. By forming the associated bundles this gives rise to a bundle map A ⊗ ΣM → ΣM , called Clifford multiplication. Equation (16) is satisfied for all X, Y ∈ A, ϕ ∈ ΣM in the same base point.

Generalized Dirac operators.
We now discuss Clifford modules in our setting.
The principal symbol of any generalized Dirac operator is elliptic, as for any non zero vector X, Clifford multiplication X· is an invertible element of End(ΣM ).
Example 4.8. For any p ∈ M , we define the Clifford algebra Cl(A p ) as the universal commutative algebra generated by A p subject to the relation X · Y + Y · X + 2g(X, Y )1 = 0.
Let Cl(A) be the Clifford-bundle of (A, g), i. e., the bundle whose fibers at the point p ∈ M is the Clifford algebra Cl(A p ). The A * -valued connection on A extends to an A * -valued connection on Cl(A). Let W = Cl(A), equipped with the module structure given by left multiplication. After identifying with the canonical isomorphism Cl(A) ∼ = Λ * (A), e i1 · . . . · e i k → e b i1 ∧ . . . ∧ e b i1 for an orthonormal basis (e i ) with dual (e b i ), the generalized Dirac operator on this bundle is the de Rham operator d + d * . Example 4.9. If M is spin then the spinor bundle from Definition 4.6 is also a Clifford module. The corresponding Dirac operator is called the (classical) Dirac operator.  where V is isomorphic to the homomorphisms from ΣM to W that are Cl(A) equivariant. V carries a compatible metric. After choosing any metric A * -valued connection ∇ V on V , the product connection on W satisfies (2) and (4).
If M is not spin, the connection can be constructed locally on a open covering in the same way, and the connection can then be glued together by using a partition of unity, hence we obtain the statement. This expression is not always defined. However, it is well-defined scalar product on generalized L 2 -spinor fields, i. e., generalized spinor fields with M σ i , σ i < ∞. It is also well-defined, if one of the sections s i or s j has compact support and the other is locally L 2 .
For the benefit of the reader, let us recall the following basic result (see for example [28]). if at least one of the sections σ 1 or σ 2 has compact support, and the maximal and minimal extension of D coincide, hence D extends uniquely to a self-adjoint operator densely defined on the L 2 -sections of W .
For any choice of a connection as in the above theorem, the resulting Dirac operator is generated by V. Proof. The Dirac operator is the composition of Clifford multiplication and the A * -valued connection ∇ W on W . Clifford multiplication is a zero order differential operator generated by V. The A * -valued connection ∇ W on W is a first order differential operator generated by V. Hence the Dirac operator is also a first order differential operator generated by V.