NNoname manuscript No. (will be inserted by the editor)
Closed Affine Manifolds with an Invariant Line
Charles Daly
Received: date / Accepted: date
Abstract
A closed affine manifold is a closed manifold with coordinate patchesinto affine space whose transition maps are restrictions of affine automor-phisms. Such a structure gives rise to a local diffeomorphism from the uni-versal cover of the manifold to affine space that is equivariant with respectto a homomorphism from the fundamental group to the group of affine auto-morphisms. The local diffeomorphism and homomorphism are referred to asthe developing map and holonomy respectively. In the case where the linearholonomy preserves a common vector, certain ‘large’ open subsets upon whichthe developing map is a diffeomorphism onto its image are constructed. Amodified proof of the fact that a radiant manifold cannot have its fixed pointin the developing image is presented. Combining these results, this paper ad-dresses the non-existence of certain closed affine manifolds whose holonomyleaves invariant an affine line. Specifically, if the affine holonomy acts purelyby translations on the invariant line, then the developing image cannot meetthis line.
Keywords affine · manifolds · holonomy · invariant · parallel · radiant The author gratefully acknowledges research support from NSF Grant Award Number1709791Charles Daly4176 Campus Dr, College Park, MD 20742E-mail: [email protected] a r X i v : . [ m a t h . DG ] S e p Charles Daly
This section is largely dedicated to preliminary notions regarding closed affinemanifolds. Specifically, this section provides examples, notation, and some ba-sic 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 diffeomor-phism onto an open subset of affine n -space such that the restriction of eachtransition map on each connected component of U α ∩ U β is an affine automor-phism. Explicitly, this says for each pair of charts ( U α , Φ α ) and ( U β , Φ β ) andeach 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 VΦ α ( V ) Φ β ( V ) Φ α Φ β Φ β ◦ Φ − α = A βα | V 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 mani-fold 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 thetwo-torus. Example 1
Let T be a rank two free abelian subgroup of Aff(2 , R ) which actson the affine plane by translations. Let M = A /T . Since T acts properly andfreely on A , the associated quotient map is a smooth covering and the chartson M are naturally diffeomorphisms onto open subsets of A . The transitionmaps are given by elements of T , and thus M inherits an affine structure. Infact, quite a bit more can be said. Since the group of translations preserve thestandard euclidean metric, M inherits a riemannian structure. This provides M a riemannian metric locally isometric to the euclidean plane. Manifoldsarising as quotients of affine space by discrete subgroups of Isom( n, R ) areknown as euclidean manifolds. Bieberach showed that such manifolds are finitecoverings of euclidean tori.This construction of finding discrete subgroups of the affine group that actproperly and freely on affine space yield an entire class of affine manifolds. In-variants of the group such as vector fields, covector fields, and metrics descendto 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 puncturedplane C × by positive dilations centered at the origin. Say that D is generatedby some λ >
0, so D acts properly and freely on C × as in the previousexample, and thus the quotient N = C × /D inherits an affine structure. Thegroup D preserves the lorrentzian metric m = ( dx + dy ) / ( x + y ) and thus losed Affine Manifolds with an Invariant Line 3 Fig. 1
The quadrilaterals Q and Q (cid:48) are labeled with their corresponding edges in a counter-clockwise fashion. Note the edges β and δ (cid:48) are identified in the quotient. The quadrilateral Q serves as a fundamental domain for the corresponding group action. [Made in Mathematica12 Student Version] descends to a lorrentzian metric on N . In contrast to the previous riemannianexample, this metric is incomplete. In fact, geodesics on C × pointed towardsthe origin will descend to geodesics on N that become undefined in finite time.This should be contrasted with the Hopf-Rinow of riemannian geometry whichstates that every closed riemannian manifold is geodesically complete. Example 3
One may generalize the construction in Example 1 in the followingfashion. Pick a quadrilateral Q in the affine plane. From the bottom left vertexand reading counterclockwise, label the edges α, β, γ and δ . Construct a newquadrilateral in the following fashion. Rotate and scale Q at the bottom leftvertex 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 (cid:48) with edges α (cid:48) , β (cid:48) , γ (cid:48) , and δ (cid:48) . This process glues β of Q to δ (cid:48) of Q (cid:48) . Repeat this processwith each pair of edges indefinitely. Figure 1 above shows an example of thisprocess where the edges are no longer simple translates of each other. Thecase where both pairs of opposite edges of Q are parallel is the method ofidentification 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, thismethod provides an inequivalent structure on the torus known as a similarity Charles Daly structure which are affine structures whose holonomy lies within the groupof similarity transforms of affine space. In fact, Figure 1 was generated bythe quadrilateral Q = { (0 , , (2 , , (1 , , (0 , } and the group of similaritytransformations generated by a = (cid:18) / / (cid:19) (cid:18) (cid:19) and b = (cid:18) −
11 1 (cid:19) (cid:18) (cid:19) These three examples serve to illustrate the complexities that arise once onedeparts from the riemannian case to the affine case. For those interested inlearning more about the affine structures supported by the two torus, OliverBaues provides an excellent treatment about different types of affine structuressupported on the two torus [1].Given an affine structure on a manifold there is a natural associated localdiffeomorphism from the universal cover of M to affine space and a homo-morphism from π ( M, p ) −→ Aff( n, R ). The local diffeomorphism is called thedeveloping map, the group homomorphism is called the holonomy, and thetwo together are called a developing pair. A brief description of these maps isprovided below, but further details about their construction may be found inWilliam Goldman’s Geometric Structure on Manifolds [3].Base the fundamental group at a point p ∈ M . Let p ∈ ( U, Φ ) be an affinecoordinate patch about p . Each path γ : [0 , −→ M beginning at p maybe 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( U i , Φ i ) where ( U , Φ ) = ( U, Φ ). Pick a mesh of times 0 = t < t < . . . 1] so that each γ ( t i ) ∈ U i − ∩ U i for i = 1 . . . k . Let γ i bethe restriction of γ to [ t i , t i +1 ] for each i = 0 . . . k .Inductively define paths in affine space in the following fashion. Let α = Φ ◦ γ . Let V be the connected component of U ∩ U containing γ ( t ) and g , be the affine automorphism so that g , | V = Φ ◦ Φ − : Φ ( V ) −→ Φ ( V ).Define α = g , ( Φ ◦ γ ). Note that the initial point of α is the terminal pointof α . Let V be the connected component of U ∩ U containing γ ( t ) and g , be the affine automorphism so that g , | V = Φ ◦ Φ − : Φ ( V ) −→ Φ ( V ).Define α = g , g , ( Φ ◦ γ ). Note the initial point of α is the terminal pointof α . Continue this process inductively to obtain ( k + 1)-paths into affinespace and concatenate them to obtain the path α · α · . . . · α k = ( Φ ◦ γ ) · ( g , ( Φ ◦ γ )) · ( g , g , ( Φ ◦ γ )) · . . . · ( g , g , . . . g k − ,k ( Φ k ◦ γ k )) (1)The developing map is defined as the terminal point of this path. Figure 2illustrates this construction with three charts. Several facts need to be verifiedabout this assignment. One must show that this map is independent of thechoice of charts after the initial chart about p ∈ ( U, Φ ) is chosen. In addition, losed Affine Manifolds with an Invariant Line 5 Fig. 2 Three charts U , U , and U cover a path γ based at p . The path is separated intothree pieces γ , γ and γ in red, green, and blue respectively. The charts are shown to theright with their images in affine space. The pink arrows represent the affine transformationstaking one affine image to the next, i.e. g , and g , . For example, the blue path in thethird patch is mapped to the to blue path in the second patch by g , . This constructionyields the concatenation α · α · α in the top right whose terminal point is the developingmap. one must show that the map is well-defined up to homotopy of paths basedat p . After these technical details are established, this assignment induces alocal diffeomorphism from the universal cover of M based at p to affine spacewhich is denoted dev : (cid:102) M −→ A n .Let γ be a path based at p contained in a chart ( U, Φ ) as above, and let[ β ] ∈ π ( M, p ). As the developing map is defined in terms of homotopy classesof paths based at p , it is natural to consider how the developing map behavesby precomposition of loops based at p . That is, one may consider how Equa-tion 1 changes by considering the path β · γ where β is a representative ofthe homotopy class [ β ]. As β begins and ends at p , one may take the initialand terminal charts covering β to both be ( U, Φ ) with charts ( V i , Θ i ) cover-ing the remainder of β . Say that the corresponding construction applied to β yields paths δ , δ , . . . , δ j in affine space with change of coordinate elements h , , h , , . . . , h j − ,j . Then the developing map applied to the concatenation β · γ yields δ · . . . · δ j · γ · . . . · γ k =( Φ ◦ β ) · ( h , ( Θ ◦ β )) · ( h , h , ( Θ ◦ β )) · . . . · ( h , h , . . . h j − ,j ( Θ j ◦ β j )) · γ · . . . · γ k As the right hand side product of the γ i ’s is left unchanged, and the developingmap is defined as the terminal point of the constructed path above, one cansee that( δ · . . . · δ j · γ · . . . · γ k ) (1) = h , h , . . . h j − ,j ( γ · . . . · γ k ) (1) (2) Charles Daly Since the path γ was arbitrary, Equation 2 holds for all such paths based at p , so precomposition with an element of [ β ] ∈ π ( M, p ) yields a differencein the developing map by the element of the affine group h , h , . . . h j − ,j which corresponds to [ β ]. This element is known as the holonomy of [ β ], de-noted hol[ β ], and defines a homomorphism, known as the holonomy map, from π ( M, p ) −→ Aff( n, R ). Equation 2 is the statement that the developing mapis equivariant with respect to the holonomy homomorphism in the sense of thefollowing commutative diagram. (cid:102) M (cid:102) M A n A n [ β ]dev devhol[ β ] (3)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 broadercontext 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. The purpose of this section is establish more specialized notation, preliminaryobservations about the consequences of having an affine structure whose affineholonomy admits an invariant affine line, and to prove a theorem about thenon-existence of certain affine structures.Let l ⊂ A n +1 be a line and G ≤ Aff( n + 1 , R ) be the group of affine au-tomorphisms 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 identifieswith the first factor of R in R × R n . Up to conjugation G is isomorphic to G = (cid:40)(cid:18) r w A (cid:19) (cid:18) d (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r (cid:54) = 0 , d ∈ R , w T ∈ R n , A ∈ GL( n, R ) (cid:41) (4)For this purpose of this paper the coordinates x and y , . . . , y n are reservedfor the first and second factors of R × R n respectively. In an abuse of notation, R will frequently denote the invariant line R × ⊂ R × R n . The instances inwhich this occurs will be clear throughout the paper. Before stating one of thetheorems 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. losed Affine Manifolds with an Invariant Line 7 Since both the universal cover of M and A n are simply connected, this defi-nition implies that the developing map is a diffeomorphism onto A n . In thiscase, Corollary 3 in Section 7 yields that the holonomy group, hol( Γ ), acts bothproperly and freely on affine space, and M is diffeomorphic to A n / hol( Γ ). Inthis context the manifold M can be recovered from the image of the holonomyhomomorphism. It is a non-trivial problem to construct inequivalent struc-tures on a manifold with the same holonomy groups. The interested reader isreferred to Projective Structure with Fuchsian Holonomy [2]. Theorem 1 For n ≥ , there are no complete affine structures on closed ( n + 1) -manifolds whose holonomy lies in the group G of line preserving affineautomorphisms. This proof is broken into two pieces. Let H ≤ G denote a subgroup of G asin Equation 4 that acts both properly and freely on R × R n with a compactquotient. The proof begins by showing that H must act purely by transla-tions on the invariant line, otherwise, there will be an accumulation point ofthe group action on the invariant line, contradicting properness. After thatis established, H will be shown to be cylic, whereby the quotient manifoldwill 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 acontradiction. Proof Let H ≤ G act both properly and freely on R × R n with a compact quo-tient. Without loss of generality the scaling factor, r (cid:54) = 0, as in Equation 4 maybe taken to be non-negative. The subgroup of G + ≤ G whose scale factors r onthe invariant line are positive form an index two subgroup of G . Consequentlythe quotient R × R n /G + is a double cover of the quotient by G , and thuspreserves both an affine structure and compactness. In addition one may liftto 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 (cid:54) = 1, then there is a solu-tion to the equation rx + d = x . Conjugate by translation along the invariantline to assume h is of the form h = (cid:18) r w A (cid:19) (cid:18) (cid:19) Note that h acts linearly on R × R n and fixes the origin. That said, the orbit ofthe origin and say for example (1 , ∈ 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 closeto (1 , 0) on the invariant line that is not. The cyclic group generated by h fixesthe origin, whereas h n (1 , 0) will tend arbitrarily close to the origin along theinvariant line for sufficiently large positive or negative values of n dependingon whether r is less than or bigger than one. The orbits of the origin and (1 , H . Charles Daly Now assume that H acts by pure translations on the invariant line R . Sincethe action is proper, it follows that map T : H −→ R defined by T (cid:18)(cid:18) w A (cid:19) (cid:18) d (cid:19)(cid:19) = d 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. Sincethe action of H on R × R n is free, this homomorphism is injective. For if twoelements h, h (cid:48) ∈ H yield the same translational part, then h − h (cid:48) fixes theorigin in R × R n , and by freeness h = h (cid:48) . Thus H is generated by some h = (cid:18) w A (cid:19) (cid:18) d (cid:19) where d ∈ R , w T ∈ R n and A ∈ GL( n, R )Note if w = 0, then R × R n quotiented by the cyclic group H is homeomorphicto the mapping cylinder of the linear map A : R n −→ R n as claimed. If w (cid:54) = 0,then conjugating h by a sheer along R yields (cid:18) v I n (cid:19) (cid:18) w A (cid:19) (cid:18) d (cid:19) (cid:18) − v I n (cid:19) = (cid:18) w + v ( A − I n )0 A (cid:19) (cid:18) d (cid:19) (5)If ( A − I n ) is invertible then w + v ( A − I n ) can be chosen to be zero, thus pro-viding the desired homeomorphism. Such a choice of v is available if ( A − I n )is invertible, which is to say that λ = 1 is not an eigenvalue of A .Assume w (cid:54) = 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 ifnot let k ∈ R , and then (cid:18) w A (cid:19) (cid:18) d (cid:19) (cid:18) ku (cid:19) = (cid:18) k ( wu ) + dku (cid:19) where A ( ku ) = ku as u is an eigenvector of A with eigenvalue 1. Since wu (cid:54) = 0as w T and u are not perpendicular, there is a choice of k for which k ( wu )+ d = 0and 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 thoughis a contradiction as U/H is diffeomorphic to S × R , and is therefore non-compact. Since w was assumed to be non-zero, this necessitates λ = 1 is notan eigenvalue of h .Since λ = 1 is not an eigenvalue of h , one may conjugate by translation alongthe invariant line as in Equation 5 to assume h is a matrix of the form h = (cid:18) A (cid:19) (cid:18) d (cid:19) losed Affine Manifolds with an Invariant Line 9 Thus the quotient R × R n /H is homeomorphic to M A , the mapping torus ofthe linear map A : R n −→ R n . A standard result in topology [5] provides thelong exact sequence of homology groups of a mapping torus is given by . . . −→ H n +1 ( R n ) −→ H n +1 ( M A ) −→ H n ( R n ) −→ . . . 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 homol-ogy is necessarily non-trivial. This shows that there are no complete structureson a compact ( n + 1)-dimensional affine manifold whose affine holonomy pre-serves an affine line.It is worth noting that although Theorem 1 forbids the existence of a completestructure on a closed manifold M whose holonomy preserves an invariant line,this theorem says nothing about the existence of non-complete structures. Infact, there are plenty of examples of non-complete structures in which thedeveloping map fails to be a covering onto all of R × R n . Example 4 Pick a λ > M be R × C × /H where H is the subgroup ofaffine transformations generated by a = and b = λ 00 0 λ M defines an affine structure on the three-torus whose affine holonomy pre-serves the invariant line defined by the x -axis. In this case the induced actionon the invariant line R is purely translational, yet the invariant line lies entirelyoutside the developing image of this affine structure.The fact that the invariant line lies outside the developing image is no coin-cidence in the case where the affine holonomy acts purely by translations onthe invariant line. The remainder of this paper is dedicated to showing this isalways the case.The basic strategy is to show that if the affine holonomy does indeed actby translations on the invariant line and the developing image meets the in-variant line, the affine structure is complete thus yielding a contradiction toTheorem 1. To show the developing map is a diffeomorphism, two main tech-niques will be employed.The first is show that if the affine manifold admits a parallel flow, one canconstruct ‘large’ open submanifolds of the universal cover upon which the re-stricted developing map is a diffeomorphism. The second is to show that if themanifold admits a so called ‘cylindrical’ flow, these ‘large’ open sets can betaken to be arbitrarily large. Once these two facts are established, the prooffollows immediately as a consequence of Theorem 1. Let M be a closed affine ( n + 1)-dimensional manifold with a developing pair(dev , hol) and fundamental group Γ = π ( M, p ) acting on the universal cover (cid:102) 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 , andidentify A n +1 with R × R n and Aff( n + 1 , R ) with GL( n + 1 , R ) (cid:110) R n +1 . By thenatural identification of parallel vector fields V on R × R n with the tangentspace of R × R n at a point, the statement that the affine holonomy preservesa parallel vector field is equivalent to the statement that there exists a v ∈ T ( 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 thesubgroup of affine automorphisms given by P = (cid:40)(cid:18) w A (cid:19) (cid:18) dv (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ∈ R , w T , v ∈ R n , A ∈ GL( n, R ) (cid:41) (6)The non-vanishing parallel H -invariant vector field ∂/∂x lifts to a non-vanishingparallel Γ -invariant vector field (cid:101) V which descends to a non-vanishing parallelvector field V on M . As M is compact, the flow associated to V is complete,and so too is the flow associated to (cid:101) V . Denote this flow on (cid:102) M by (cid:101) Θ t . As (cid:101) V isrelated to ∂/∂x by the developing map, so too are their corresponding flows.Denoting T t as the translational flow corresponding to ∂/∂x , one obtains thecommutative diagram (cid:102) M (cid:102) M R × R n R × R n (cid:101) Θ t dev dev T t As T t is the flow associated the vector field ∂/∂x which is invariant under theholonomy, this necessitates that T t commutes with each element of the holon-omy as the holonomy will take flow lines to flow lines. The same statementholds for the commutativity of (cid:101) T t and Γ .It is clear that the R -action on R × R n given by translation T t is both freeand proper, consequently, so too is the R -action on (cid:102) M by Lemma 2. Thus (cid:102) M is a principal R -bundle over the quotient manifold (cid:102) M / R . Denote this quotientmanifold by N and the associated quotient map by q : (cid:102) M −→ N . Since R iscontractible, the principal R -bundle structure on (cid:102) M is isomorphic to the trivialbundle p : R × N −→ N where p is factor projection onto the second factor.By triviality of the principal bundle, there is an R -equivariant diffeomorphism losed Affine Manifolds with an Invariant Line 11 Φ : (cid:102) M −→ R × N for which the below diagrams commute. (cid:102) M R × NN q Φ p and (cid:102) M (cid:102) M R × N R × N (cid:101) Θ t Φ Φ (cid:101) T t (7)A standard result in the theory principal bundles states that a principal bun-dle is trivial if and only if the bundle admits a global section, and thus N maybe thought of as an embedded submanifold of N ⊂ (cid:102) M for which the saturationof N by the R -action yields all of (cid:102) M .The identification of Φ : (cid:102) M −→ R × N provides a Γ -action on R × N viaconjugation by Φ . Specifically, [ γ ]( t, n ) = ( Φ ◦ [ γ ] ◦ Φ − )( t, n ). The Γ -actionon R × N commutes with the R -action on R × N as per consequence of thedefinition of the Γ -action on R × N and Equation 7. In addition, one obtainsthe commutative diagram R × N R × N (cid:102) M (cid:102) M R × R n R × R n [ γ ] Φ − Φ − [ γ ]dev devhol[ γ ] (8)The composition of dev ◦ Φ − provides a local diffeomorphism of R × N into R × R n which is equivariant with respect to the Γ -action on R × N . In thestandard abuse of notation, this composition is also denoted dev : R × N −→ R × R n .By the R -equivariance of Φ as in Equation 7, one obtains the commutativediagram R × N R × N R × R n R × R n dev (cid:101) T t dev T t (9)Before continuing, it is worth noting that Equation 9 is more or less the state-ment that there exists a transverse submanifold N that generates the R -actionon the universal cover, R × N , in such a fashion that that the developing maptakes flow lines to flow lines. As both R -actions are free and proper, one maypass 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 [ γ ] and hol[ γ ]. Specifically, one has the commutativesquare N N R n R n dev [ γ ] devhol[ γ ] (10)where dev : N −→ R n is a local diffeomorphism, and the holonomy hol[ γ ] isacting 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 andproper, which is not necessarily the case. The following example below illus-trates possible obstructions when passing the quotient. Example 5 Let D be a closed unit disk centered at the origin in R . Let Γ be the cyclic group acting on R × D generated by translation along R androtation by an irrational angle θ . For example let Γ be generated by a = θ ) − sin( θ )0 sin( θ ) cos( θ ) Let M = R × D/Γ which is a compact three-dimensional manifold with bound-ary as the action of Γ on R × D is both free and proper. In fact, M is themapping torus of the map a : D −→ D . Since the holonomy Γ preserves thevector field ∂/∂x , Γ maps flow lines to flow lines whose images are R × p foreach p ∈ D .The induced action of Γ on D is neither free no proper. In particular theinduced action of a on the unit disk preserves the flow line of the origin, and isconsequently not free. In addition, the orbit of each point p ∈ D with a radius0 < r ≤ r ,and consequently the induced action of Γ on D is not proper. Though this isexample with boundary, it nevertheless conveys the fragility of proper actions.Proper actions in general do not pass to proper actions on quotients. Thisshould be contrasted with case where a proper action is lifted, see for exampleLemma 2.As mentioned towards the end of Section 2, the goal of this section is to provethe existence of ‘large’ open subsets upon which the developing map whenrestricted to them are diffeomorphisms. As all the necessary terminology is inplace, the theorem may be stated. Theorem 2 Let M be a closed affine ( n + 1) -dimensional manifold whoseaffine holonomy admits a parallel vector field. There exists a complete parallel losed Affine Manifolds with an Invariant Line 13 flow on the universal cover of M equivariant with respect to the parallel flowinduced by the parallel vector field on affine space. Additionally, there existsopen subsets of the universal cover invariant under the parallel flow for whichthe restricted developing map is a diffeomorphism onto its image.Proof The previous paragraphs establish the existence of a complete parallelflow satisfying the equivariance condition in the statement of the theorem. Tofinish the proof of Theorem 2, it suffices to show the existence of the opensubset of R × N invariant under the flow so that the restricted developing mapis a diffeomorphism onto its image.Form the commutative squares as in Equation 8 and Equation 10, and picka point y in the image of dev( N ). As dev is a local diffeomorphism, the fibreover y is a discrete subset of N . Pick a point n ∈ N so that dev( n ) = y , andlet n ∈ U be an open set so that dev | U is a diffeomorphism onto its imagedev( U ). Let (cid:101) U = p − ( U ) = R × U ⊂ R × N .Since (cid:101) U is open in R × N , to show that dev | (cid:101) U is a diffeomorphism onto itsimage, it suffices to show that the developing map is injective on (cid:101) U . To thisend, let ( t, m ) and ( s, m (cid:48) ) be points in (cid:101) U so that dev( t, m ) = dev( s, m (cid:48) ). Thisimplies that (dev ◦ (cid:101) T t )(0 , m ) = (dev ◦ (cid:101) T s )(0 , m (cid:48) ) and by the definition of devand Equation 9, one obtains that dev( m ) = dev( m (cid:48) ). Since m, m (cid:48) ∈ U , byconstruction, this necessitates that m = m (cid:48) . Consequently ( t, m ) and ( s, m (cid:48) )are in the same R -orbit. By freeness of the Equation 9 and freeness of the R -actions, this necessitates that s = t so ( t, m ) = ( s, m (cid:48) ). Consequently thedeveloping map restricted to (cid:101) U is diffeomorphism.To summarize, the basic idea of Theorem 2 is that there exists a codimensionone submanifold N ⊂ (cid:102) M transverse to the parallel flow on (cid:102) M that gener-ates the R -action on (cid:102) M . In addition, the developing map sends flow lines toflow lines. The deck transformations, holonomy, and developing map all factorthrough the corresponding R -actions to yield a local diffeomorphism on N thatprovide coordinate charts on N . Since N ⊂ (cid:102) M , these transverse coordinatecharts may be saturated by the parallel flow and the saturations are still diffeo-morphisms as the flow lines are sent to flow lines, which never loop around andthus provide ‘large’ open subsets so that the restricted developing map is a dif-feomorphism 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 argumentworks on any subset U ⊂ N for which dev | U is a diffeomorphism onto itsimage. That said, the failure of the original developing map to be a diffeo-morphism onto its image is entirely determined by the failure of dev to be adiffeomorphism. Fig. 3 Here U is an open subset in N labeled in red for which dev | U is a diffeomorphismonto its image. The map dev is the induced map on the collapsed blue flow lines. Saturating U by the R -action yields the open subset R × U ⊂ (cid:102) M such that the developing map restrictedto it is a diffeomorphism. Similar to Section 3, let M be a closed affine n -dimensional manifold withfundamental group Γ = π ( M, p ) acting on the universal cover by deck trans-formations. Pick a developing pair (dev , hol) for the affine structure on M . Asopposed to the previous section, instead of assuming the linear holonomy fixesa common vector, this section explores some consequences of when the affineholonomy fixes a point in A n . Pick said fixed point as the origin and make thestandard identification of A n with R n . Up to conjugation, one may assumethe holonomy lies inside the group of linear transformations GL( n, R ). Theseclass 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 holon-omy fixes a point in A n . This is equivalent to the condition that the affineholonomy is conjugate to a subgroup of GL( n, R ). Example 6 The affine structure on the torus given in Example 2 provides thetorus with a radiant structure, whereas the structure in Example 1 is notradiant. The structure given in Example 3, specifically from Figure 1, alsoprovides a radiant structure on the torus. This structure is inequivalent to theone in Example 2, as the holonomy in Example 3 is non-cyclic. losed Affine Manifolds with an Invariant Line 15 Fig. 4 An illustration of a three-dimensional hopf-manifold. The solid space between twoconcentric spheres is drawn above with an equator in blue. The action of the homothetyidentifies the inner sphere with the outer sphere via a dilation. The curve is red is projectedto a circle in the quotient. Example 7 The structure in Example 2 can be generalized in the followingfashion. Consider the puncture euclidean space R n \ 0. Let H be a group gen-erated by a positive homothety induced by some λ > 0. Then R n /H is readilyseen to be diffeomorphic to S × S n − . These manifolds, known classically ashopf manifolds, provide a class of examples of radiant manifolds with cyclicholonomy. An illustration of the identification is provided in Figure 4.A standard result of in the theory of geometric structures states that a closedradiant manifold cannot have its fixed point as an element of the developingimage [4]. Below is a modified version of the standard argument that is suitedfor the content of this paper. Theorem 3 Let M be a closed radiant manifold. The developing image cannotmeet the fixed point of the radiant structure.Proof Fix a developing pair dev : (cid:102) M −→ A n and hol : Γ −→ GL( n, R ). Let R = − y i ∂/∂y i be the attractive radial vector field on R n . Routine calculationshows that R is invariant under general linear group, so it may be lifted bythe developing map to a Γ -invariant vector field (cid:101) R on (cid:102) M . The vector field (cid:101) R descends to a vector field on M , which is complete by compactness, and thusthe corresponding flow on (cid:102) M is also complete. Denote the flow on (cid:102) M by (cid:101) R t and the corresponding radial flow on R n by R t . These flows are related by the commutative diagram below (cid:102) M (cid:102) M R n R n (cid:101) R t dev dev R t Now, assume that the origin is an element of the developing image. Thendev − { } ⊂ (cid:102) M is a discrete subset of stationary points of (cid:101) R . Choose a collec-tion of pairwise disjoint open sets U i about each element u i ∈ dev − { } . Sincedev is a local diffeomorphism, one may shrink each U i if necessary to assumethe developing map restricted to each U i is a diffeomorphism onto an openball centered at the origin. For each t ≥ 0, one has that R t dev( U i ) ⊆ dev( U i )and thus (cid:101) R t U i ⊆ U i .For each U i , let (cid:101) R ∞ U i denote the forward and backward saturation of U i with respect to the radial flow on the universal cover. Explicitly, (cid:101) R ∞ U i = (cid:91) t ∈ R (cid:101) R t U i (11)As (cid:101) R ∞ U i is union of open subsets in (cid:102) M , it is itself an open submanifold of (cid:102) M .Additionally, the developing map restricted to each (cid:101) R ∞ U i is a diffeomorphismonto its image. To prove this, it suffices to show that the developing map isinjective when restricted to each (cid:101) R ∞ U i .Let u, v ∈ (cid:101) R ∞ U i so that dev( u ) = dev( v ). There exists times t, s ∈ R andpoints u i , v i ∈ U i so that dev( (cid:101) R t u i ) = dev( (cid:101) 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 (cid:101) R t − s U i ⊆ U i so R t − s dev( U i ) ⊆ dev( U i ). Becausedev( (cid:101) R t − s u i ) = dev( v i ) and (cid:101) R t − s U i ⊆ U i on which the developing map is adiffeomorphism, one has (cid:101) R t − s u i = v i so u = v as claimed.The above paragraphs show that the developing map when restricted to any (cid:101) R ∞ U i is a diffeomorphism onto its image, which is the radial saturation of anopen ball about the origin, and consequently a diffeomorphism onto all of R n .Lemma 3 shows that (cid:101) R ∞ U i is closed, and consequently by connectedness of (cid:102) M , is equal to the universal cover. Thus the developing map is a diffeomor-phism onto R n and defines a complete radial structure.As per consequence there exists a subgroup H ≤ GL( n, R ) acting both prop-erly and freely on R n so that R n /H is diffeomorphic to M . Since the origin isa 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 anelement of the developing image. losed Affine Manifolds with an Invariant Line 17 As mentioned previously, the above proof yields a corollary that assists theproof 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 alocal diffeomorphism where ∈ F ( N ) . If the radial action on R n can be liftedto a complete action on N , then F is a diffeormophism onto R n . 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 whoseholonomy leaves invariant an affine line where n ≥ . If the holonomy acts bypure translations on the invariant line, then the developing image cannot meetthe 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 onit, one may assume that the holonomy lies in the subgroup of affine automor-phisms of the form defined by G = (cid:40)(cid:18) w A (cid:19) (cid:18) d (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ∈ R , w T ∈ R n , A ∈ GL( n, R ) (cid:41) (12)As per consequence, there is a parallel flow on the universal cover as in The-orem 2. One can then form the commutative diagram as in Equation 10 toobtain the local diffeomorphism dev : N −→ R n . Because the induced holon-omy in Equation 10 acts linearly by the matrices A in Equation 12, there isa complete flow on N that lifts the radial flow on R n . If the developing im-age meets the invariant line then 0 ∈ dev( N ), and by Corollary 1, dev willdefine a global diffeomorphism from N onto R n . Saturating N by the parallelflow yields that the developing map is diffeomorphism onto all of R × R n thusdefining a complete structure whose affine holonomy leaves invariant an affineline thus contradicting Theorem 1. The details of this argument are providedbelow in the proof. Proof As mentioned in the preceding paragraph, assume the holonomy lies in-side the group defined by Equation 12. By Theorem 2, there exists a completeparallel flow on the universal cover of M . Form the commutative diagram de-fined by Equation 10 with the induced developing map dev : N −→ R n andthe corresponding induced actions of Γ and the holonomy on N and R n re-spectively.Since the holonomy of M lies inside of G , the induced holonomy action inEquation 10 acts linearly on R n , and thus preserves the attractive radial vectorfield R = − y i ∂/∂y i as in Section 4. This vector field lifts via dev : N −→ R n Fig. 5 An illustration of the cylindrical vector field on R × N whose vectors are in red.Several leaves are drawn transverse to the blue parallel flow lines. The red inward pointingarrows represent the radial vector field whose flow preserves each leaf x × N . The radialvector field on R × N projects to the vector field (cid:101) R on N via factor projection p . Thisvector field on N is complete, as the cylindrical vector field on R × N is complete. The greenparallel flow line in R × N is left invariant by the radial flow and represents the parallel flowof a point n ∈ N so that dev( n ) = 0. to a Γ -invariant vector field (cid:101) R on N . Identifying the tangent bundle T ( R × N )with T R ⊕ T N yields a natural lift of (cid:101) R to a vector field on R × N , wherebyconstruction, this vector field is also Γ -invariant. The Γ -invariant vector fieldon R × N descends to a vector field on M which is complete by compact-ness. The flow on M lifts to a complete flow on R × N which leaves each leaf x × N of R × N invariant. This flow may be thought of as a cylindrical flowwhich is radial on each leaf. Above is a figure that illustrates this construction.Since the flow of the lift of (cid:101) R to R × N is complete, by construction theflow of (cid:101) 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, then0 ∈ dev( N ). Corollary 1 implies that dev : N −→ R n is a diffeomorphism, andthe remark after Theorem 2 yields that dev : R × N −→ R × R n is a diffeo-morphism. Hence M admits a complete affine structure whose holonomy liesinside 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 thefollowing corollary. Corollary 2 Let M be a closed ( n + 1) -dimensional affine manifold whoseholonomy leaves invariant an affine line where n ≥ . If the holonomy acts losed Affine Manifolds with an Invariant Line 19 by pure translations and reflections on the invariant line, then the developingimage cannot meet the invariant line.Proof This is an immediate consequence of the fact that the group defined byEquation 12 is an index two subgroup of (cid:40)(cid:18) ± w A (cid:19) (cid:18) d (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ∈ R , w T ∈ R n , A ∈ GL( n, R ) (cid:41) Passing to the double cover M and applying Theorem 4 yields the desiredresult. A natural follow up to Theorem 4 would be to analyze the situation where theholonomy lies in the extension of the group G as defined in Equation 4. Theauthor suspects that such manifolds are radiant. Loosely the idea at hand isthe following. If there is indeed an element of the holonomy of the formhol[ γ ] = (cid:18) r w A (cid:19) (cid:18) d (cid:19) where r (cid:54) = 1, then without loss of generality, one may conjugate to assumethis element of the holonomy acts on the invariant line by scaling, and conse-quently admits a fixed point, which can be taken as the origin. It seems likelythat the developing image cannot meet this point, much like in the exampleof a hopf circle. If that were so, then this would impose certain restrictionsabout having a non-trivial translational part in the holonomy. The difficultyin showing this point does not meet the developing image is that [ γ ] ∈ Γ neednot stabilize the components of the inverse image of the invariant line underthe developing map. This difficulty is lost in the case where the fundamentalgroup is abelian, but seems like an excessive and unnecessary hypothesis.The proof of Theorem 4 relied largely on the existence of a parallel flow anda ‘cylindrical’ flow on the universal cover of M . These flows can coexist oncertain manifolds such as the product of a euclidean circle and a hopf torus asin Example 4. It is of great interest to the author as to whether or not paralleland radial flows can coexist on compact affine manifolds. The dynamics ofsuch flows would certainly lead to very interesting examples. Here is a collection of some lemmas used throughout the paper. In this context,all topological spaces are assumed to be smooth manifolds, as is the concern ofthis paper. That said, some of these propositions hold in more general contextssuch as metric spaces. 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 adiffeomorphism F : X −→ Y equivariant with respect to φ in the sense thatthe below diagram commutes for all g ∈ G . X XY Y gF Fφ ( g ) (13) If G acts properly and freely on X then so too does φ ( G ) on Y . In this case, onemay 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 , andthus by freeness of G on X , g = 1, so φ ( g ) = 1. A similar argument shows thehomomorphism 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 proper-ness it suffices to show a subsequence of φ ( g i ) converges, as is stated in JohnM. Lee’s Smooth Manifolds [6]. Since F is a diffeomorphism, there’s a uniquesequence 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 subse-quence 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 thusthe 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 mayform their smooth quotient manifolds X/G and Y /φ ( G ). Denote their corre-sponding projections by p : X −→ X/G and q : Y −→ Y /φ ( G ). One may thenform the commutative square below X YX/G Y /φ ( G ) Fp qF (14)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 dif-feomorphism, u and v are in the same orbit, thus Gu = Gv , so F is a bijection. losed Affine Manifolds with an Invariant Line 21 Since X and Y are the same dimension, as are G and φ ( G ), the map F isa smooth bijective local diffeomorphism, and thus a diffeomorphism.Lemma 1 provides the following result frequently used in the study of geometricstructures. Corollary 3 Let M be a complete affine n -dimensional manifold with fun-damental group Γ = π ( M, p ) . Fix a developing pair dev : (cid:102) M −→ A n andhol : Γ −→ Aff ( n, R ) . Then M is diffeomorphic to A n /H where H is theimage of the holonomy homomorphism.Proof Since the developing map is a diffeomorphism onto A n and M is diffeo-morphic (cid:102) M /Γ , where Γ is the group of deck transformations that acts bothproperly and freely on the universal cover, Lemma 1, applied the developingmap and holonomy homomorphism yield that (cid:102) M /Γ and therefore M , are bothdiffeomorphic to A n /H .The following statement has a proof similar to that of Lemma 1, and is usedin the construction of the quotient manifolds in Section 3. Its proof is omittedas 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 theaction of G on Y is free and proper, then so too is the action of G on X . Inthis case one may form the quotients X/G and Y /G for which F descends toa smooth map F : X/G −→ Y /G . Lemma 3 Let N and P be smooth manifolds and F : N −→ P be an openmap where N is connected. If there exists an open submanifold U ⊆ N forwhich F | U : U −→ P is a diffeomorphism, then N = U . This following lemma finds it use in the proof of Theorem 3, to show the opensubmanifold (cid:101) R ∞ U i defined by Equation 11 is equal to all of (cid:102) M . In this casethe developing map fulfills the role of the open map as stated in the lemma. 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 aunique 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 followsthat u k ∈ V for k ≥ K . Thus u k converges to both u and n , and by uniquenessof limits, u = n . Consequently, U is a non-empty closed and open subset of N , and by connectedness U = N . Acknowledgements I would like to thank Dr. William Goldman for his input and as-sistance throughout the formation of this paper. As always, I thank my wonderful friendswhose creativity and good deeds inspire my day to day life. Finally, a very special thankyou to Yon Hui and Chaz Daly, and Rose of Sharon and Anatoly Bourov-Daly for a list ofkindnesses several times the length of this paper.