A convexity theorem for real projective structures
Given a finite collection of convex -polytopes in (), we consider a real projective manifold which is obtained by gluing together the polytopes in along their facets in such a way that the union of any two adjacent polytopes sharing a common facet is convex. We prove that the real projective structure on is
convex if contains no triangular polytope, and
properly convex if, in addition, contains a polytope whose dual polytope is thick.
Triangular polytopes and polytopes with thick duals are defined as analogues of triangles and polygons with at least five edges, respectively.
- 1 Introduction
- 2 Preliminaries
- 3 Main objects
- 4 Convexity
- 5 Proper convexity
- 6 Applications to real projective structures
Consider a planar domain , an open connected subset of . Suppose that admits a tessellation by (a necessarily infinite number of) convex polygons. One may ask if there are any local conditions on the tessellation which can guarantee convexity of the domain . One reasonable such condition we investigate in this paper is the following:
the union of two adjacent polygons sharing a common edge is convex.
See Figure 1.1. This condition was first introduced by Kapovich  and we call tessellations with this property residually convex. It turns out that, under the residual convexity condition, one can prove the following:
If contains no triangle then the domain is a convex subset of .
If, in addition, contains a polygon with at least edges then the convex domain contains no infinite line.
Figure 1.2 illustrates the above assertions: (a) exhibits a generic shape of a convex domain which admits a residually convex tessellation without triangles, (b) shows that a domain containing an infinite line may admit a residually convex tessellation without polygons with at least edges, and (c) shows that a domain with residually convex tessellation containing a pentagon but no triangles is bounded.
On the other hand, Figure 1.3 (b) shows that a non-convex domain may admit a residually convex tessellation if triangles are allowed. Figure 1.3 (a) motivated the definition of residual convexity because it clearly exhibits one way in which a non-convex domain may be tessellated by convex polygons. Both examples are due to Yves Benoist.
Our contribution in this paper is to prove the assertions similar to (I) and (II) above in every dimension – by defining appropriate analogues of triangles and polygons with at least edges. The former is called a triangular polytope and the latter has a thick polytope as its dual. For precise definitions see Definition 4.11 and Definition 5.2. As a matter of fact, we prove these results in a more general context so that they give rise to convexity criteria for certain real projective structures. From now on, to the end of the paper, we assume except those cases which are trivially exceptional (like the one in the next paragraph).
A real projective structure on manifolds is a geometric structure which is locally modelled on projective geometry . If is a convex domain and is a discrete subgroup of acting freely and properly discontinuously on , then the induced real projective structure on the quotient manifold is said to be convex. If, moreover, the closure of the convex domain does not contain any projective line, then the structure is called properly convex. See Section 6.1 for more details. One of the basic references for real projective structures is the lecture notes of Goldman .
Convex real projective structures can be regarded as analogues of complete Riemannian metrics, and properly convex real projective structures are expected to share some nice properties with non-positively curved metrics (see, for example,  and ). For this reason, given a real projective structure, one natural question to ask is whether the structure is (properly) convex. More precisely, let be a finite family of convex -dimensional polytopes in . Suppose that is a real projective -manifold obtained by gluing together copies of via projective facet-pairing transformations. Then there is an associated developing map of the universal cover of , which is a projective isomorphism on each cell of . One now asks:
When is the map an isomorphism onto a (properly) convex domain in ?
The Tits–Vinberg fundamental domain theorem  for discrete linear groups generated by reflections provides a rather restricted but very constructive solution to this question. Recently Kapovich  proved another convexity theorem when the are non-compact polyhedra. See Remark 6.3 for a more detailed discussion. In the present paper, we deal with complementary cases which are not covered by the aforementioned results. Our main theorem is as follows (see also Theorem 6.2):
Let be a finite family of compact convex -dimensional polytopes in . Let be a set of projective facet-pairing transformations for indexed by the collection of all facets of the polytopes in . Let be a real projective -manifold obtained by gluing together the polytopes in by . Assume the following condition:
for each facet of , if is a facet of such that , then the union is a convex subset of .
Then the following assertions are true:
If contains no triangular polytope, then the developing map is an isomorphism onto a convex domain which is not equal to ;
If, in addition, contains a polytope whose dual is thick, then the map is an isomorphism onto a properly convex domain.
An interesting related question is whether every convex real projective structures have convex fundamental domains and how common residually convex structures are. In  we provide partial answer by showing that all properly convex real projective structures have convex fundamental domains.
We sketch our approach to assertion (I) of Theorem A. The details are the contents of Section 3 and Section 4. Let denote the universal covering space of . We consider the lift of the developing map to the sphere , the two-fold cover of . Regarding then as the standard Riemannian sphere, we pull back the Riemannian metric to via so that is locally isometric to . Then the simply-connected manifold becomes a spherical polyhedral complex.
In fact, we define such a spherical polyhedral complex admitting a developing map into in an abstract way (-complex), so that in general the complex does not necessarily admit a cocompact group action (see Definition 3.1). We call a subset convex if it is mapped by injectively onto a convex subset of .
We then place on the residual convexity condition, that is, we require that, for every two -polytopes and in sharing a common facet, their union be convex (see Definition 4.2).
We fix a polytope of and consider the iterated stars of so that they exhaust the whole complex (see Definition 3.5 (1)). Our plan is to show inductively that
each star is convex and its image under is not equal to .
Then this would imply that is an isometric embedding onto a convex proper domain in (see Theorem 4.8).
Projecting down back to we get the desired convexity result on the real projective structure on .
A considerable portion of the present paper is devoted to step (3) of the above plan. We now explain how the induction argument goes:
We next want to show that the star is locally convex. Because of its polyhedral structure, the local convexity of can be drawn from its local convexity near codimension- cells (ridges) in the boundary (see Lemma 3.4).
Let be a codimension- cell in the boundary of the star . The local geometry of near is determined by the union of -cells in which contain and which intersect . Thus we need to find conditions which imply that the union is convex. Interestingly, there is a local condition for this.
Indeed, we consider a small neighborhood (residue) of which consists of those -cells in which contain (see Definition 3.5 (2)). Residual convexity implies that is convex (see Lemma 4.1 (3)). Because the star is also assumed to be convex and because and intersect along their boundaries, their intersection is a convex subset in the boundary of . Then the union can be described as the union of -cells in which intersect .
The condition, which we call strong residual convexity, requires that, for all , the set be always convex regardless of convex subsets in the boundary of (see Definition 4.4 and Definition 4.6). Figure 1.4 illustrates the case where strong residual convexity fails. In conclusion, under the assumption of strong residual convexity, we can show that the star is locally convex near codimension- cells in its boundary (see Lemma 4.7).
Finally, once the local convexity is established, we may regard the star as an Alexandrov space of curvature and then deduce its global convexity using a well-known local-to-global theorem for such spaces (see Corollary 2.5). All induction steps are complete.
To summarize, we have the following convexity theorem:
Let be an -complex. If is strongly residually convex, then is isometric to a convex proper domain in . In particular, is contractible.
As can be seen in steps (iii)-(vi) above, the codimension- phenomena in polyhedral complexes enables us to go from dimension to arbitrary dimensions. This is a rather common trick which can be found, for example, in the proof of the Poincaré fundamental polyhedron theorem for constant curvature spaces (see, for example,  and ). However, we find it worthwhile to develop this trick into a form which is suitable for our present purpose. Hence the most of Section 2 is devoted to the study of geometric links of faces of various dimensions in convex polytopes.
Although strong residual convexity is entirely a local condition, for practical reasons, it is desirable to have simple combinatorial conditions under which residual convexity becomes strong residual convexity. Observe that triangles caused the failure of strong residual convexity in Figure 1.4. See also Figure 4.2. Using the codimension- phenomena once again, we define triangular polytopes and show that without presence of triangular polytopes residual convexity implies strong residual convexity (see Theorem 4.12). Combining this result with Theorem B we obtain the following corollary, which again implies assertion (I) of Theorem A.
Let be a residually convex complex. If contains no triangular polytopes, then is isometric to a convex domain which is not .
1.2 Proper convexity
We now outline our approach to assertion (II) of Theorem A. The details are explained in Section 5. The starting point is the above Corollary C. That is, we assume that our complex is residually convex and contains no triangular polytopes. Then is isometric to a convex domain in . Thus from now on we regard as a convex subset of and find conditions implying proper convexity of .
Our eventual plan is to find supporting hyperplanes of that are in general position. Then is contained in the -simplex which is determined by these hyperplanes. Because -simplices are properly convex, the conclusion then follows. Fortunately, there is a natural way to find supporting hyperplanes of provided that contains no triangular polytope. Thus we need to find further conditions under which there are such in general position.
For example, if is -dimensional and contains no triangle, all polygons in have at least four edges and this enables us to construct the following objects in . We fix a polygon in . Given an edge of , consider the polygon that is adjacent to along the common edge . Then we can choose an edge of which is disjoint from . We then consider the polygon adjacent to along . Choose an edge of which is disjoint from , and so on. This process defines an infinite sequence (directed gallery) of adjacent polygons in (see Figure 1.2 (b) and Definition 5.9). One can then show that the limit of the lines spanned by the edges is a supporting line to . Now, if the polygon is, say, a pentagon then we have five such supporting lines constructed from the edges of as above. It is easy to see that two supporting lines coming from two nearby edges of may coincide but those coming from disjoint edges of never coincide. Because , this implies that there are at least three supporting lines of which are in general position so that they bound a triangle (see Figure 1.2 (c)).
We now explain how the previous arguments in dimension can be generalized to higher dimensions:
To be able to define directed galleries, we need the analogues of polygons with at least four edges. For this, we re-interpret triangles and define cone-like polytopes (see Definition 5.6). If none of the polytopes in is cone-like then we can define directed galleries in . It turns out that non-triangular polytopes are not cone-like (see Lemma 5.7).
Fix a polytope in . Each directed gallery associated to a facet of defines a supporting hyperplane of . Because every -polytope has at least facets, we have at least such supporting hyperplanes.
Such simple counting as above does not work in higher dimensions, where both combinatorial and geometric arguments are necessary. To deal with the arrangement of supporting hyperplanes, we consider the dual of and points dual to the halfspaces which contain and which are bounded by the supporting hyperplanes . On the other hand, the vertices of are dual to the halfspaces which contain and which are bounded by the hyperplanes spanned by facets of .
Finally, we prove that if is thick then there always exist such points in general position, which again implies that there always exist supporting hyperplanes of in general position (see Lemma 5.14).
Let be a residually convex -complex such that none of the -cells of are triangular. If has an -cell whose dual is thick, then is a properly convex domain in .
In the final Section 6 we discuss real projective structures in more detail and explain how all these results are applied to give convexity theorem for certain real projective structures.
It should be noted that we introduce metric to prove Theorem A, which does not involve any metric-dependent notion. There are two main reasons for using metric in our discussion:
When we consider links of polytopes and argue inductively, we can embed links of various dimension in a single space so that our presentation gains more convenience and geometric flavor. However, this is not an essential ingredient in our proof and there is a more natural way of defining links without using metric (see Remark 2.2).
We can use a local-to-global theorem for Alexandrov spaces of curvature bounded below (see Theorem 2.4). We do not know how to draw global convexity of spherical domains from their local convexity without using this theorem.
My advisor Misha Kapovich recommended me to investigate the property of residual convexity. I am grateful to him for this and I deeply appreciate his encouragement and patience during my work. I also thank Yves Benoist and Damian Osajda for helpful discussions. During this work I was partially supported by the NSF grants DMS-04-05180 and DMS-05-54349.
Let be the -dimensional Euclidean vector space. We denote the origin by and the standard inner product by . Given a linear subspace its orthogonal complement is denoted . For two subsets and their sum is the set of all points for and .
Let be a subset of whose closure contains the origin . The smallest linear subspace containing is denoted . The (linear) dimension of is defined to be the dimension of this subspace. We say that is open if it is open relative to . A point is called an interior (resp. boundary) point of if is an interior (resp. boundary) point of relative to .
2.1 Convex cones
A subset is said to be convex if for every and for every such that the point is in , that is, the affine line segment joining and is in . One can show that if is convex then its closure is also convex. The convex hull of a subset is the smallest convex subset containing . A cone is a subset of such that if and then . Thus cones are invariant under positive homotheties of . Note that for any cone its closure necessarily contains the origin .
A convex cone is a cone which is convex. Linear subspaces and halfspaces bounded by codimension- linear subspaces are convex cones; these examples contain a complete affine line. A convex cone is called line-free if it contains no complete affine line. Given a convex cone we denote by the largest linear subspace contained in . The following lemma says that a closed convex cone decomposes into a linear part and a line-free part; compare with  and . See also Figure 2.1(a).
Lemma 2.1 (Decomposition Theorem).
Let be a convex cone in . Then if and only if is line-free. If then decomposes into
and is a line-free convex cone, where denotes the orthogonal complement of .
Let and be two points in . We first claim that contains the complete affine line passing through in the direction of if and only if it contains the parallel line passing through . Suppose first that contains the line . Then for any and , the point
is on the affine segment joining and . Because is convex the point is in . As goes to infinity, however, converges to . Since is closed, this shows that contains the line . Since and play the equivalent roles, this completes the proof of the claim.
Recall that contains the origin . Then the above claim says that contains a complete affine line if and only if it contains a -dimensional subspace. Therefore, if and only if is line-free.
So from now on we suppose that . Because and any translate of intersects , it follows from the above claim that decomposes into . Since both and are convex cones, their intersection is also a convex cone. Suppose by way of contradiction that contains a complete affine line. The above claim then shows that it also contains a -dimensional subspace . But the subspace properly contains and is contained in ; this is contradictory to the definition of . The proof of lemma is complete. ∎
We can avoid using metric and state Lemma 2.1 in terms of quotient space instead of orthogonal complement. Namely, let be the natural projection onto . Then is a line-free convex cone in such that . We may consider as the line-free part of and use this to define links of polyhedral cones and polytopes in the following discussion. While we can proceed in this more natural way, we prefer using metric for the sake of presentational convenience.
A hyperplane is an -dimensional linear subspace of . Let be a convex cone. We say that a hyperplane supports if is contained in one of the closed halfspaces bounded by ; this halfspace is denoted by (and the other one by ) and is also said to support . In fact, it can be shown that if then is contained in some halfspace of (see for example ). A non-empty subset is called a face of if there is a supporting hyperplane of such that . Obviously, faces of are also convex cones.
2.2 Polyhedral cones
A subset is called a polyhedral cone if it is the intersection of a finite family of closed halfspaces of . Clearly, polyhedral cones are closed convex cones. A polyhedral cone is polytopal if it is line-free, that is, .
Let be a polyhedral cone in . It is known that if is a face of then faces of are also faces of . A maximal face of is called a facet of . A ridge of is a facet of a facet of . Let where the are halfspaces bounded by hyperplanes . We further assume that the family is irredundant, that is,
for each . The irredundancy condition implies the following properties of faces of (see ):
If is -dimensional, a facet of is of the form for some ;
The boundary of is the union of all facets of ;
Each ridge of is a non-empty intersection of two facets of ;
Every face of is a non-empty intersection of facets of .
Thus the number of faces of is finite. If is -dimensional then its facets are -dimensional and ridges are -dimensional.
2.3 Links in polyhedral cones
Let be a polyhedral cone in . Let be a face of . If is -dimensional then we may assume without loss of generality that is the intersection of facets of for some , that is,
Because any sufficiently small neighborhood of an interior point of intersects only those hyperplanes which contain , the local geometry of near an interior point of is the same as the local geometry near the origin of the polyhedral cone determined by the corresponding halfspaces . We denote this polyhedral cone by
By Lemma 2.1, the polyhedral cone decomposes into
However, the linear part is just the intersection , which is again equal to the smallest linear subspace containing . Thus we have
Now the link of in is defined to be the line-free part of :
See Figure 2.1 (b).
If has dimension then is -dimensional and is -dimensional. Because has full-dimension in , is also full-dimensional in . It follows that the link is an -dimensional polytopal cone in with its defining halfspaces being .
We defined the link under the assumption that is an -dimensional polyhedron in . If is -dimensional with , however, we just consider the smallest linear subspace containing and define the link with respect to in the same manner as above. Thus if is -dimensional, its link is an -dimensional polytopal cone in .
Let be an -dimensional polyhedral cone in . Let be a face of and a face of . We define a subset of the link as:
The lemma below says that is a face of the polytopal cone , whose link in is equal to the link . Thus the link of has all the information about the links of those faces which contain ; this fact enables us to use inductive arguments on links later on.
Let be an -dimensional polyhedral cone in . Let be a face of and a face of . Then is a face of the polytopal cone . If is a facet of then is also a facet of . Furthermore, we have the following identity between the two links involved:
We write for an irredundant family of halfspaces of bounded by . We may assume that for some the faces and are expressed as
If we set, as before,
then the links of and are by definition
Because and , we then have
Since and the defining halfspaces of are , this shows that is a face of the polytopal cone . If is a facet of then and . Therefore, is a facet of .
To see the claimed equality we first note that, because has non-empty interior in ,
Because and hence , we then have
Finally, unraveling all the definitions, we see that
2.4 Spherical polytopes
Let be the unit sphere in . To any subset we associate the cone over defined by
For a subset and a cone , it is clear that
A subset is an -plane provided that the cone over is an -dimensional linear subspace of . The orthogonal complement of an -plane is defined to be .
Let be a subset of . The smallest -plane containing is denoted and is clearly equal to . The dimension of is defined to be the dimension of this plane. We call open if it is open relative to . Likewise, a point is called an interior (resp. boundary) point of if is an interior (resp. boundary) point of relative to . We also denote by the set of interior points of .
A subset is convex (resp. properly convex) if the cone over is a convex cone (resp. line-free convex cone). It is clear that is convex if and only if for any two points in the (spherical) geodesic connecting them is in . A subset is locally convex if every point of has a neighborhood in which is a convex subset of . The convex hull of a subset is the smallest convex subset containing . Finally, a subset is a noun if the cone over is a noun in , where the noun stands for hyperplane, halfspace, support or face. Note that if is convex then is a convex cone and is contained in a halfspace of . Thus every convex subset not equal to is contained in a halfspace of and hence has diameter at most .
A subset is a polyhedron (resp. polytope) if the cone over is a polyhedral cone (resp. polytopal cone) in . If a polyhedron has dimension we call an -polyhedron and similarly for polytopes. A maximal face of is called a facet of . A ridge of is a facet of a facet of . A vertex (resp. edge) of is a -dimensional (resp. -dimensional) face of . Let where the are halfspaces bounded by hyperplanes , that is, . Under the same irredundancy condition on the family as in Section 2.2, the same properties of faces of as listed therein hold.
Let be a polyhedron and a face of . The link of in is by definition
See Figure 2.2.
Because is a polytopal cone, the link is a polytope in . If is an -polyhedron and is an -face then the link is an -polytope. Let be a face of and define a subset of by
It then follows from Lemma 2.3 that is a face of the polytope and the following identity holds between the two links involved:
Let be the dual vector space of . It is equipped with the standard inner product coming from that of . Denote by the unit sphere in .
Let be a cone in . The dual cone of is defined by
It is easy to see that is a closed convex cone in . If is an -dimensional linear subspace of then is an -dimensional linear subspace of . If is a halfspace bounded by a hyperplane then is a ray in . We have the following well-known facts (compare with  and ):
If is a closed convex cone then (under the natural identification ) and
If and are closed convex cones then
If is a polyhedral cone then so too is .
If is an -dimensional polytopal cone then so too is .
Let be a subset of . The dual of is defined by
Thus the dual of is always a closed convex subset of . If is an -plane then is an -plane. In particular, the dual of a hyperplane is a pair of antipodal points. The dual of a halfspace is a single point; if then . The analogous properties for cones as listed above also hold for subsets of . In particular, if is an -polytope then so too is its dual ; if is expressed as
where each becomes a vertex of the dual polytope .
2.6 Alexandrov spaces of curvature bounded below
The main reference for this subsection is . Fix a real number . Let be the -dimensional complete simply-connected Riemannian manifold of constant curvature , and denote for and for . Thus, for example, we have and . We denote by the induced path metric on .
Let be a metric space. Given three points satisfying
there is a comparison triangle in , namely, three points such that
We define to be the angle at the vertex of the triangle .
Let be a path metric space, that is, a metric space where the distance between each pair of points is equal to the infimum of the length of rectifiable curves joining them. Then is said to be provided that for any four distinct points and in we have the inequality
(If is a -dimensional manifold and , then we require in addition that its diameter be at most .) The path metric space is said to be locally , or more commonly, an Alexandrov space of curvature , if each point has a neighborhood which is .
Examples of locally spaces include Riemannian manifolds without boundary or with locally convex boundary whose sectional curvatures are . (Locally) convex subsets of such Riemannian manifolds are also locally . We shall be interested mostly in the case when and – locally convex subsets of are locally .
The following is a local-to-global theorem for spaces which is analogous to the Cartan-Hadamard theorem for spaces with (see for example ). Unlike the Cartan-Hadamard theorem, however, we do not place any topological restriction on the space in this theorem:
Theorem 2.4 (Globalization Theorem).
If a complete path metric space is locally , then it is and has diameter .
For its proof we refer to . As a corollary of the globalization theorem, we have the following criterion for locally convex subsets of to be convex. Note that if , geodesics in have length at most .
Let be a locally convex connected subset of . If , we assume in addition that is not a -dimensional manifold. If is complete and locally compact with respect to the induced path metric, then is convex in .
Because is locally convex in (and is not a -dimensional manifold in case ), is locally . If is complete with respect to the induced length metric, the globalization theorem tells us that is an space of diameter . Let and be two points of . Because is connected, complete and locally compact with respect to the induced path metric, satisfies the assumption of the Hopf-Rinow Theorem (see for example ) and hence there is a geodesic in joining and . As is locally convex, however, this curve has to be a local geodesic in . Since has diameter , the length of is at most . It follows from the simple-connectedness of that is a (global) geodesic in . ∎
3 Main objects
We define metric polyhedral complexes which are locally isometric to . Our presentation follows that of –polyhedral complexes in , where in our case. We consider subcomplexes of such polyhedral complexes that embed isometrically into as topological balls, and present a convexity criterion for them. We also study special subcomplexes called stars and residues.
Definition 3.1 (-complexes).
Given a family of -polytopes in , let be a connected -manifold (possibly with non-empty boundary ) which is obtained by gluing together members of along their respective facets by isometries. We denote by the equivalence relation on the disjoint union induced by this gluing so that
Let be the natural projection and denote . We call the manifold a spherical polytopal -complex (-complex, for short) provided that
the family is locally finite;
it is endowed with the quotient metric associated to the projection ;
its interior is locally isometric to ;
it is simply-connected.
For each -complex the conditions (3) and (4) guarantee that there is an associated developing map
which is a local isometry on the interior of and which extends naturally to the boundary of . The developing map is well-defined up to post-composition with an isometry of .
Whenever we mention an