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Closed Affine Manifolds with an Invariant Line

A closed affine manifold is a closed manifold with coordinate patches into affine space whose transition maps are restrictions of affine automorphisms. Such a structure gives rise to a local diffeomorphism from the universal cover of the manifold to affine space that is equivariant with respect to a homomorphism from the fundamental group to the group of affine automorphisms. The local diffeomorphism and homomorphism are referred to as the developing map and holonomy respectively. In the case where the linear holonomy preserves a common vector, certain `large' open subsets upon which the developing map is a diffeomorphism onto its image are constructed. A modified proof of the fact that a radiant manifold cannot have its fixed point in the developing image is presented. Combining these results, this paper addresses the non-existence of certain closed affine manifolds whose holonomy leaves invariant an affine line. Specifically, if the affine holonomy acts purely by translations on the invariant line, then the developing image cannot meet this line.


Affine Structures Preliminaries
This section is largely dedicated to preliminary notions regarding closed affine manifolds. Specifically, this section provides examples, notation, and some basic results regarding the developing map and holonomy.
Definition 1 An n-dimensional affine manifold is a n-dimensional manifold M equipped with charts (U α , Φ α ) where each Φ α : U α −→ A n is a diffeomorphism onto an open subset of affine n-space such that the restriction of each transition map on each connected component of U α ∩ U β is an affine automorphism. Explicitly, this says for each pair of charts (U α , Φ α ) and (U β , Φ β ) and each connected component V ⊂ U α ∩ U β , there exists an affine automorphism A βα,V : A n −→ A n so that the following equality holds This definition lends itself to a natural generalization of what is known as a (G, X)-manifold, where G is a lie group acting strongly effectively on a manifold X. The study of affine geometry is the study of (Aff(n, R), A n )-manifolds.
Here are some standard examples in the literature of affine structures on the two-torus.
Example 1 Let T be a rank two free abelian subgroup of Aff(2, R) which acts on the affine plane by translations. Let M = A 2 /T . Since T acts properly and freely on A 2 , the associated quotient map is a smooth covering and the charts on M are naturally diffeomorphisms onto open subsets of A 2 . The transition maps are given by elements of T , and thus M inherits an affine structure. In fact, quite a bit more can be said. Since the group of translations preserve the standard euclidean metric, M inherits a riemannian structure. This provides M a riemannian metric locally isometric to the euclidean plane. Manifolds arising as quotients of affine space by discrete subgroups of Isom(n, R) are known as euclidean manifolds. Bieberach showed that such manifolds are finite coverings of euclidean tori.
This construction of finding discrete subgroups of the affine group that act properly and freely on affine space yield an entire class of affine manifolds. Invariants of the group such as vector fields, covector fields, and metrics descend to the quotient and are studied extensively in the field of geometric structure.
Example 2 Let D be a cyclic subgroup of Aff(2, R) which acts on the punctured plane C × by positive dilations centered at the origin. Say that D is generated by some λ > 0, so D acts properly and freely on C × as in the previous example, and thus the quotient N = C × /D inherits an affine structure. The group D preserves the lorrentzian metric m = (dx 2 + dy 2 )/(x 2 + y 2 ) and thus descends to a lorrentzian metric on N . In contrast to the previous riemannian example, this metric is incomplete. In fact, geodesics on C × pointed towards the origin will descend to geodesics on N that become undefined in finite time. This should be contrasted with the Hopf-Rinow of riemannian geometry which states that every closed riemannian manifold is geodesically complete.
Example 3 One may generalize the construction in Example 1 in the following fashion. Pick a quadrilateral Q in the affine plane. From the bottom left vertex and reading counterclockwise, label the edges α, β, γ and δ. Construct a new quadrilateral in the following fashion. Rotate and scale Q at the bottom left vertex in such a fashion that the rotated δ differs from β by translation along α. Translate the result along α to yield a new quadrilateral Q with edges α , β , γ , and δ . This process glues β of Q to δ of Q . Repeat this process with each pair of edges indefinitely. Figure 1 above shows an example of this process where the edges are no longer simple translates of each other. The case where both pairs of opposite edges of Q are parallel is the method of identification as in Example 1 and defines a euclidean structure on the torus. In the case where pairs of opposite edges of Q are not necessarily parallel, this method provides an inequivalent structure on the torus known as a similarity structure which are affine structures whose holonomy lies within the group of similarity transforms of affine space. In fact, Figure 1 was generated by the quadrilateral Q = {(0, 0), (2, 0), (1, 1), (0, 1)} and the group of similarity transformations generated by a = 1/2 0 0 1/2 These three examples serve to illustrate the complexities that arise once one departs from the riemannian case to the affine case. For those interested in learning more about the affine structures supported by the two torus, Oliver Baues provides an excellent treatment about different types of affine structures supported on the two torus [1].
Given an affine structure on a manifold there is a natural associated local diffeomorphism from the universal cover of M to affine space and a homomorphism from π 1 (M, p) −→ Aff(n, R). The local diffeomorphism is called the developing map, the group homomorphism is called the holonomy, and the two together are called a developing pair. A brief description of these maps is provided below, but further details about their construction may be found in William Goldman's Geometric Structure on Manifolds [3].
Base the fundamental group at a point p ∈ M . Let p ∈ (U, Φ) be an affine coordinate patch about p. Each path γ : [0, 1] −→ M beginning at p may be assigned a point in affine space in the following fashion. Cover the path γ by (k + 1)-coordinate patches beginning with (U, Φ). Label these patches by Inductively define paths in affine space in the following fashion. Let α 0 = Φ 0 • γ 0 . Let V 1 be the connected component of U 0 ∩ U 1 containing γ(t 1 ) and g 0,1 be the affine automorphism so that g 0, . Note that the initial point of α 1 is the terminal point of α 0 . Let V 2 be the connected component of U 1 ∩ U 2 containing γ(t 2 ) and g 1,2 be the affine automorphism so that g 1, . Note the initial point of α 2 is the terminal point of α 1 . Continue this process inductively to obtain (k + 1)-paths into affine space and concatenate them to obtain the path The developing map is defined as the terminal point of this path. Figure 2 illustrates this construction with three charts. Several facts need to be verified about this assignment. One must show that this map is independent of the choice of charts after the initial chart about p ∈ (U, Φ) is chosen. In addition, Fig. 2 Three charts U 0 , U 1 , and U 2 cover a path γ based at p. The path is separated into three pieces γ 0 , γ 1 and γ 2 in red, green, and blue respectively. The charts are shown to the right with their images in affine space. The pink arrows represent the affine transformations taking one affine image to the next, i.e. g 0,1 and g 1,2 . For example, the blue path in the third patch is mapped to the to blue path in the second patch by g 1,2 . This construction yields the concatenation α 1 · α 2 · α 3 in the top right whose terminal point is the developing map.
one must show that the map is well-defined up to homotopy of paths based at p. After these technical details are established, this assignment induces a local diffeomorphism from the universal cover of M based at p to affine space which is denoted dev : M −→ A n .
Let γ be a path based at p contained in a chart (U, Φ) as above, and let [β] ∈ π 1 (M, p). As the developing map is defined in terms of homotopy classes of paths based at p, it is natural to consider how the developing map behaves by precomposition of loops based at p. That is, one may consider how Equation 1 changes by considering the path β · γ where β is a representative of the homotopy class [β]. As β begins and ends at p, one may take the initial and terminal charts covering β to both be (U, Φ) with charts (V i , Θ i ) covering the remainder of β. Say that the corresponding construction applied to β yields paths δ 1 , δ 2 , . . . , δ j in affine space with change of coordinate elements h 0,1 , h 1,2 , . . . , h j−1,j . Then the developing map applied to the concatenation β · γ yields As the right hand side product of the γ i 's is left unchanged, and the developing map is defined as the terminal point of the constructed path above, one can see that The pair (dev, hol) is known as a developing pair for the affine structure on M . The construction of the developing map above carries over to the broader context of (G, X)-structures on manifolds wherein one assumes that a lie group G acts strongly effectively on a manifold X. A very nice exposition about (G, X)-structures may be found in Stephan Schmitt's Geometric Manifolds [7] whereas [3] provides a very thorough general reference.

Affine Structures with an Invariant Line
The purpose of this section is establish more specialized notation, preliminary observations about the consequences of having an affine structure whose affine holonomy admits an invariant affine line, and to prove a theorem about the non-existence of certain affine structures.
Let l ⊂ A n+1 be a line and G ≤ Aff(n + 1, R) be the group of affine automorphisms preserving l. Pick an origin on l and identify A n+1 with R n+1 . Rotate about the origin so that l aligns with the x-axis in R n+1 and l identifies with the first factor of R in R × R n . Up to conjugation G is isomorphic to For this purpose of this paper the coordinates x and y 1 , . . . , y n are reserved for the first and second factors of R × R n respectively. In an abuse of notation, R will frequently denote the invariant line R × 0 ⊂ R × R n . The instances in which this occurs will be clear throughout the paper. Before stating one of the theorems of this paper, a definition is in order.
Definition 2 Let M be a closed affine manifold with a developing pair (dev, hol). If the developing map is a covering onto A n , then M is called complete.
Since both the universal cover of M and A n are simply connected, this definition implies that the developing map is a diffeomorphism onto A n . In this case, Corollary 3 in Section 7 yields that the holonomy group, hol(Γ ), acts both properly and freely on affine space, and M is diffeomorphic to A n /hol(Γ ). In this context the manifold M can be recovered from the image of the holonomy homomorphism. It is a non-trivial problem to construct inequivalent structures on a manifold with the same holonomy groups. The interested reader is referred to Projective Structure with Fuchsian Holonomy [2].
Theorem 1 For n ≥ 1, there are no complete affine structures on closed (n + 1)-manifolds whose holonomy lies in the group G of line preserving affine automorphisms.
This proof is broken into two pieces. Let H ≤ G denote a subgroup of G as in Equation 4 that acts both properly and freely on R × R n with a compact quotient. The proof begins by showing that H must act purely by translations on the invariant line, otherwise, there will be an accumulation point of the group action on the invariant line, contradicting properness. After that is established, H will be shown to be cylic, whereby the quotient manifold will be shown to be a mapping torus, M A , of a linear map A : R n −→ R n . Certain topological obstructions will prevent this from occurring and yield a contradiction.
Proof Let H ≤ G act both properly and freely on R × R n with a compact quotient. Without loss of generality the scaling factor, r = 0, as in Equation 4 may be taken to be non-negative. The subgroup of G + ≤ G whose scale factors r on the invariant line are positive form an index two subgroup of G. Consequently the quotient R × R n /G + is a double cover of the quotient by G, and thus preserves both an affine structure and compactness. In addition one may lift to the orientable double cover to assume the manifold R×R n /G + is orientable.
If there is indeed an element h ∈ H so that r = 1, then there is a solution to the equation rx + d = x. Conjugate by translation along the invariant line to assume h is of the form Note that h acts linearly on R × R n and fixes the origin. That said, the orbit of the origin and say for example (1, 0) ∈ R × R n are inseparable by open sets. If (1, 0) in indeed in the orbit of the origin, one may pick a point arbitrarily close to (1, 0) on the invariant line that is not. The cyclic group generated by h fixes the origin, whereas h n (1, 0) will tend arbitrarily close to the origin along the invariant line for sufficiently large positive or negative values of n depending on whether r is less than or bigger than one. The orbits of the origin and (1, 0) are inseparable so the action is not proper, and thus elements that fail to act by pure translation on the invariant line are not in H.

Now assume that H acts by pure translations on the invariant line R.
Since the action is proper, it follows that map T : H −→ R defined by is a homomorphism. By properness of the action of H, the image of T is cyclic, as a dense image would yield an accumulation point on the invariant line. Since the action of H on R × R n is free, this homomorphism is injective. For if two elements h, h ∈ H yield the same translational part, then h −1 h fixes the origin in R × R n , and by freeness h = h . Thus H is generated by some Note if w = 0, then R × R n quotiented by the cyclic group H is homeomorphic to the mapping cylinder of the linear map A : R n −→ R n as claimed. If w = 0, then conjugating h by a sheer along R yields is invertible, which is to say that λ = 1 is not an eigenvalue of A.
Assume w = 0 and λ = 1 is an eigenvalue of A. If so, then there is a u ∈ R n for which Au = u. Moreover, w T and u are necessarily perpendicular. For if not let k ∈ R, and then where A(ku) = ku as u is an eigenvector of A with eigenvalue 1. Since wu = 0 as w T and u are not perpendicular, there is a choice of k for which k(wu)+d = 0 and thus (0, ku) is fixed by a generator of H contradicting the fact that H acts freely on R × R n . Thus, w T and u are perpendicular as claimed.
Let U denote the plane in R × R n spanned by (1, 0) and (0, u). Since w T and u are orthogonal, U is a closed subspace invariant under the action of H, so its quotient U/H is a compact submanifold of R × R n /H. This though is a contradiction as U/H is diffeomorphic to S 1 × R, and is therefore noncompact. Since w was assumed to be non-zero, this necessitates λ = 1 is not an eigenvalue of h.
Since λ = 1 is not an eigenvalue of h, one may conjugate by translation along the invariant line as in Equation 5 to assume h is a matrix of the form Thus the quotient R × R n /H is homeomorphic to M A , the mapping torus of the linear map A : R n −→ R n . A standard result in topology [5] provides the long exact sequence of homology groups of a mapping torus is given by Since R n is contractible and n ≥ 1, this necessitates that H n+1 (M A ) is trivial.
Since M A is an (n+1)-dimensional, compact, oriented manifold, its top homology is necessarily non-trivial. This shows that there are no complete structures on a compact (n + 1)-dimensional affine manifold whose affine holonomy preserves an affine line.
It is worth noting that although Theorem 1 forbids the existence of a complete structure on a closed manifold M whose holonomy preserves The basic strategy is to show that if the affine holonomy does indeed act by translations on the invariant line and the developing image meets the invariant line, the affine structure is complete thus yielding a contradiction to Theorem 1. To show the developing map is a diffeomorphism, two main techniques will be employed.
The first is show that if the affine manifold admits a parallel flow, one can construct 'large' open submanifolds of the universal cover upon which the restricted developing map is a diffeomorphism. The second is to show that if the manifold admits a so called 'cylindrical' flow, these 'large' open sets can be taken to be arbitrarily large. Once these two facts are established, the proof follows immediately as a consequence of Theorem 1.

Parallel Flow
Let M be a closed affine (n + 1)-dimensional manifold with a developing pair (dev, hol) and fundamental group Γ = π 1 (M, p) acting on the universal cover M satisfying Equation 3. Assume the affine holonomy group, H = hol(Γ ) ≤ Aff(n + 1, R), preserves a parallel vector field V . Pick an origin in A n+1 , and identify A n+1 with R × R n and Aff(n + 1, R) with GL(n + 1, R) R n+1 . By the natural identification of parallel vector fields V on R × R n with the tangent space of R × R n at a point, the statement that the affine holonomy preserves a parallel vector field is equivalent to the statement that there exists a v ∈ T 0 (R × R n ) so that for each h ∈ H, v is an eigenvector of the linear part of h. Thus, up to conjugation, one may assume the affine holonomy lies inside the subgroup of affine automorphisms given by The non-vanishing parallel H-invariant vector field ∂/∂x lifts to a non-vanishing parallel Γ -invariant vector field V which descends to a non-vanishing parallel vector field V on M . As M is compact, the flow associated to V is complete, and so too is the flow associated to V . Denote this flow on M by Θ t . As V is related to ∂/∂x by the developing map, so too are their corresponding flows. Denoting T t as the translational flow corresponding to ∂/∂x, one obtains the commutative diagram As T t is the flow associated the vector field ∂/∂x which is invariant under the holonomy, this necessitates that T t commutes with each element of the holonomy as the holonomy will take flow lines to flow lines. The same statement holds for the commutativity of T t and Γ .
It is clear that the R-action on R × R n given by translation T t is both free and proper, consequently, so too is the R-action on M by Lemma 2. Thus M is a principal R-bundle over the quotient manifold M /R. Denote this quotient manifold by N and the associated quotient map by q : M −→ N . Since R is contractible, the principal R-bundle structure on M is isomorphic to the trivial bundle p 2 : R × N −→ N where p 2 is factor projection onto the second factor. By triviality of the principal bundle, there is an R-equivariant diffeomorphism Φ : M −→ R × N for which the below diagrams commute.
A standard result in the theory principal bundles states that a principal bundle is trivial if and only if the bundle admits a global section, and thus N may be thought of as an embedded submanifold of N ⊂ M for which the saturation of N by the R-action yields all of M . n). The Γ -action on R × N commutes with the R-action on R × N as per consequence of the definition of the Γ -action on R × N and Equation 7. In addition, one obtains the commutative diagram The composition of dev • Φ −1 provides a local diffeomorphism of R × N into R × R n which is equivariant with respect to the Γ -action on R × N . In the standard abuse of notation, this composition is also denoted dev : By the R-equivariance of Φ as in Equation 7, one obtains the commutative diagram Before continuing, it is worth noting that Equation 9 is more or less the statement that there exists a transverse submanifold N that generates the R-action on the universal cover, R × N , in such a fashion that that the developing map takes flow lines to flow lines. As both R-actions are free and proper, one may pass the vertices of Equation 9 to their quotients. As both the Γ -action on R × N and the holonomy on R × R n commute with their respective R-actions, the actions of Γ and the holonomy group pass to their quotients, which are also abusively denoted by [ where dev : N −→ R n is a local diffeomorphism, and the holonomy hol[γ] is acting affinely on R n by the affine action induced by the matrices and vectors A ∈ GL(n, R) and v ∈ R n as in Equation 6.
Equation 10 looks deceptively as though it defines an affine structure on N/Γ . This though assumes that the induced action of Γ on N is both free and proper, which is not necessarily the case. The following example below illustrates possible obstructions when passing the quotient. The induced action of Γ on D is neither free no proper. In particular the induced action of a on the unit disk preserves the flow line of the origin, and is consequently not free. In addition, the orbit of each point p ∈ D with a radius 0 < r ≤ 1 has an orbit that is dense on the corresponding circle of radius r, and consequently the induced action of Γ on D is not proper. Though this is example with boundary, it nevertheless conveys the fragility of proper actions. Proper actions in general do not pass to proper actions on quotients. This should be contrasted with case where a proper action is lifted, see for example Lemma 2.
As mentioned towards the end of Section 2, the goal of this section is to prove the existence of 'large' open subsets upon which the developing map when restricted to them are diffeomorphisms. As all the necessary terminology is in place, the theorem may be stated. Form the commutative squares as in Equation 8 and Equation 10, and pick a point y in the image of dev(N ). As dev is a local diffeomorphism, the fibre over y is a discrete subset of N . Pick a point n ∈ N so that dev(n) = y, and let n ∈ U be an open set so that dev| U is a diffeomorphism onto its image dev(U ).
Since U is open in R × N , to show that dev| U is a diffeomorphism onto its image, it suffices to show that the developing map is injective on U . To To summarize, the basic idea of Theorem 2 is that there exists a codimension one submanifold N ⊂ M transverse to the parallel flow on M that generates the R-action on M . In addition, the developing map sends flow lines to flow lines. The deck transformations, holonomy, and developing map all factor through the corresponding R-actions to yield a local diffeomorphism on N that provide coordinate charts on N . Since N ⊂ M , these transverse coordinate charts may be saturated by the parallel flow and the saturations are still diffeomorphisms as the flow lines are sent to flow lines, which never loop around and thus provide 'large' open subsets so that the restricted developing map is a diffeomorphism onto its image. Figure 3 provides an illustration of this argument.
In fact, as one can see in the proof of Theorem 2, the injectivity argument works on any subset U ⊂ N for which dev| U is a diffeomorphism onto its image. That said, the failure of the original developing map to be a diffeomorphism onto its image is entirely determined by the failure of dev to be a diffeomorphism.

Radial Flow
Similar to Section 3, let M be a closed affine n-dimensional manifold with fundamental group Γ = π 1 (M, p) acting on the universal cover by deck transformations. Pick a developing pair (dev, hol) for the affine structure on M . As opposed to the previous section, instead of assuming the linear holonomy fixes a common vector, this section explores some consequences of when the affine holonomy fixes a point in A n . Pick said fixed point as the origin and make the standard identification of A n with R n . Up to conjugation, one may assume the holonomy lies inside the group of linear transformations GL(n, R). These class of manifolds are of special interest in the study of geometric structures, so much so that they are provided their own name.
Definition 3 A radiant manifold M is an affine manifold whose affine holonomy fixes a point in A n . This is equivalent to the condition that the affine holonomy is conjugate to a subgroup of GL(n, R).

Example 6
The affine structure on the torus given in Example 2 provides the torus with a radiant structure, whereas the structure in Example 1 is not radiant. The structure given in Example 3, specifically from Figure 1, also provides a radiant structure on the torus. This structure is inequivalent to the one in Example 2, as the holonomy in Example 3 is non-cyclic. Fig. 4 An illustration of a three-dimensional hopf-manifold. The solid space between two concentric spheres is drawn above with an equator in blue. The action of the homothety identifies the inner sphere with the outer sphere via a dilation. The curve is red is projected to a circle in the quotient.

Example 7
The structure in Example 2 can be generalized in the following fashion. Consider the puncture euclidean space R n \ 0. Let H be a group generated by a positive homothety induced by some λ > 0. Then R n /H is readily seen to be diffeomorphic to S 1 × S n−1 . These manifolds, known classically as hopf manifolds, provide a class of examples of radiant manifolds with cyclic holonomy. An illustration of the identification is provided in Figure 4.
A standard result of in the theory of geometric structures states that a closed radiant manifold cannot have its fixed point as an element of the developing image [4]. Below is a modified version of the standard argument that is suited for the content of this paper.
For each U i , let R ∞ U i denote the forward and backward saturation of U i with respect to the radial flow on the universal cover. Explicitly, Additionally, the developing map restricted to each R ∞ U i is a diffeomorphism onto its image. To prove this, it suffices to show that the developing map is injective when restricted to each R ∞ U i .
Let u, v ∈ R ∞ U i so that dev(u) = dev(v). There exists times t, s ∈ R and points u i , v i ∈ U i so that dev( R t u i ) = dev( R s v i ). Without loss of generality, let t − s ≥ 0. By equivariance of the radial actions, R t−s dev(u i ) = dev(v i ).
Since t − s ≥ 0, one has that R t−s U i ⊆ U i so R t−s dev(U i ) ⊆ dev(U i ). Because dev( R t−s u i ) = dev(v i ) and R t−s U i ⊆ U i on which the developing map is a diffeomorphism, one has R t−s u i = v i so u = v as claimed.
The above paragraphs show that the developing map when restricted to any R ∞ U i is a diffeomorphism onto its image, which is the radial saturation of an open ball about the origin, and consequently a diffeomorphism onto all of R n . Lemma 3 shows that R ∞ U i is closed, and consequently by connectedness of M , is equal to the universal cover. Thus the developing map is a diffeomorphism onto R n and defines a complete radial structure.
As per consequence there exists a subgroup H ≤ GL(n, R) acting both properly and freely on R n so that R n /H is diffeomorphic to M . Since the origin is a fixed point of each element of GL(n, R) and H acts freely, H must be trivial. This contradicts the fact that M is compact, and thus the origin is not an element of the developing image.
As mentioned previously, the above proof yields a corollary that assists the proof of a later theorem. It is stated here for reference later.
Corollary 1 Let N be a connected smooth manifold and F : N −→ R n be a local diffeomorphism where 0 ∈ F (N ). If the radial action on R n can be lifted to a complete action on N , then F is a diffeormophism onto R n .

Holonomy Acting by Translations on The Invariant Line
In this section of this paper, a mild generalization of Theorem 1 is provided. The goal of this section is to prove the following theorem.
Theorem 4 Let M be a closed (n + 1)-dimensional affine manifold whose holonomy leaves invariant an affine line where n ≥ 1. If the holonomy acts by pure translations on the invariant line, then the developing image cannot meet the invariant line.
Before beginning the proof it is worth explaining the technical ideas. As M admits an invariant affline line and the holonomy acts by pure translations on it, one may assume that the holonomy lies in the subgroup of affine automorphisms of the form defined by As per consequence, there is a parallel flow on the universal cover as in Theorem 2. One can then form the commutative diagram as in Equation 10 to obtain the local diffeomorphism dev : N −→ R n . Because the induced holonomy in Equation 10 acts linearly by the matrices A in Equation 12, there is a complete flow on N that lifts the radial flow on R n . If the developing image meets the invariant line then 0 ∈ dev(N ), and by Corollary 1, dev will define a global diffeomorphism from N onto R n . Saturating N by the parallel flow yields that the developing map is diffeomorphism onto all of R × R n thus defining a complete structure whose affine holonomy leaves invariant an affine line thus contradicting Theorem 1. The details of this argument are provided below in the proof.
Proof As mentioned in the preceding paragraph, assume the holonomy lies inside the group defined by Equation 12. By Theorem 2, there exists a complete parallel flow on the universal cover of M . Form the commutative diagram defined by Equation 10 with the induced developing map dev : N −→ R n and the corresponding induced actions of Γ and the holonomy on N and R n respectively.
Since the holonomy of M lies inside of G, the induced holonomy action in Equation 10 acts linearly on R n , and thus preserves the attractive radial vector field R = −y i ∂/∂y i as in Section 4. This vector field lifts via dev : N −→ R n Since the flow of the lift of R to R × N is complete, by construction the flow of R is itself complete and thus serves as a lift of the radial flow R on R n through dev. If the developing image dev(R×N ) meets the invariant line, then 0 ∈ dev(N ). Corollary 1 implies that dev : N −→ R n is a diffeomorphism, and the remark after Theorem 2 yields that dev : R × N −→ R × R n is a diffeomorphism. Hence M admits a complete affine structure whose holonomy lies inside the group G defined in Equation 12, contradicting Theorem 1. Thus, the developing image cannot meet the invariant line.
As an immediate consequence to the proof of Theorem 4 one obtains the following corollary. Proof This is an immediate consequence of the fact that the group defined by Equation 12 is an index two subgroup of Passing to the double cover M and applying Theorem 4 yields the desired result. Lemma 1 Let φ : G −→ H be a homomorphism of lie groups and let G and H act on the smooth manifolds X and Y respectively. Assume there exists a diffeomorphism F : X −→ Y equivariant with respect to φ in the sense that the below diagram commutes for all g ∈ G.

Concluding Remarks
If G acts properly and freely on X then so too does φ(G) on Y . In this case, one may form the quotients X/G and Y /φ(G), which are in turn diffeomorphic.
Proof Begin by assuming that G acts freely on X. Let φ(g)y = y for some g ∈ G and y ∈ Y . Since F is a diffeomorphism, there's an x ∈ X so that F (x) = y and so φ(g)F (x) = F (x) which by equivariance is equivalent to F (gx) = F (x). Since G is a diffeomorphism, this necessitates that gx = x, and thus by freeness of G on X, g = 1, so φ(g) = 1. A similar argument shows the homomorphism φ is injective, so G and φ(G) are diffeomorphic as manifolds.
Let G act properly on X. Pick sequences φ(g i ) ∈ φ(G) and y i ∈ Y so that φ(g i )y i converges to a point q ∈ Y and y i converges to y ∈ Y . To show properness it suffices to show a subsequence of φ(g i ) converges, as is stated in John M. Lee's Smooth Manifolds [6]. Since F is a diffeomorphism, there's a unique sequence of x i ∈ X converging to an x ∈ X so that F (x i ) = y i and F (x) = y. Equivariance yields that φ(g i )y i = φ(g i )F (x i ) = F (g i x i ) which converges to q ∈ Y , and thus g i x i converges to some p ∈ X. Since g i x i converges to p ∈ X and x i converges to x ∈ X, properness of G on X yields a convergent subsequence of g i to g ∈ G. Continuity of the lie group homomorphism φ : G −→ H provides a convergent subsequence φ(g i ) converging to φ(g) ∈ φ(G), and thus the action of φ(G) on Y is proper.
As both actions of G on X and φ(G) on Y are free and proper, one may form their smooth quotient manifolds X/G and Y /φ(G). Denote their corresponding projections by p : X −→ X/G and q : Y −→ Y /φ(G). One may then form the commutative square below The map F is well defined as Equation 13 ensures F maps orbits to orbits. As q • F is surjective, so too is F . It is injective because if F (Gu) = F (Gv) for some Gu, Gv ∈ X/G, then there exists u, v ∈ X and a g ∈ G so that φ(g)F (u) = F (v). Equivariance implies F (gu) = F (v) and since F is a diffeomorphism, u and v are in the same orbit, thus Gu = Gv, so F is a bijection.
Since X and Y are the same dimension, as are G and φ(G), the map F is a smooth bijective local diffeomorphism, and thus a diffeomorphism.
Lemma 1 provides the following result frequently used in the study of geometric structures.
Corollary 3 Let M be a complete affine n-dimensional manifold with fundamental group Γ = π 1 (M, p). Fix a developing pair dev : M −→ A n and hol : Γ −→ Aff(n, R). Then M is diffeomorphic to A n /H where H is the image of the holonomy homomorphism.
Proof Since the developing map is a diffeomorphism onto A n and M is diffeomorphic M /Γ , where Γ is the group of deck transformations that acts both properly and freely on the universal cover, Lemma 1, applied the developing map and holonomy homomorphism yield that M /Γ and therefore M , are both diffeomorphic to A n /H.
The following statement has a proof similar to that of Lemma 1, and is used in the construction of the quotient manifolds in Section 3. Its proof is omitted as it is nearly identical to that of the previous lemma.
Lemma 2 Let G be a lie group acting on smooth manifolds X and Y and let F : X −→ Y be a smooth map equivariant with respect to the G-actions. If the action of G on Y is free and proper, then so too is the action of G on X. In this case one may form the quotients X/G and Y /G for which F descends to a smooth map F : X/G −→ Y /G. Proof It suffices to show that U is closed. Let u k be a sequence of points in U converging to some point n ∈ N . By continuity of F , the sequence F (u k ) converges to F (n) ∈ P . Since F | U : U −→ P is a diffeomorphism, there's a unique u ∈ U so that F (n) = F (u). The claim is that n = u.
Let u ∈ V be an open neighborhood in N about u.
As U is open in N , one may shrink V sufficiently small so that u ∈ V ⊆ U . Because the sequence F (u k ) converges to F (n) = F (u) and F (V ) is an open subset of P about F (u), there exists a sufficiently large K ∈ N so that F (u k ) ∈ F (V ) for all k ≥ K.
Since F | U : U −→ P is a diffeomorphism, each u k ∈ U , and V ⊂ U , it follows that u k ∈ V for k ≥ K. Thus u k converges to both u and n, and by uniqueness of limits, u = n. Consequently, U is a non-empty closed and open subset of N , and by connectedness U = N .