A natural generalization of regular convex polyhedra

A natural generalization of regular convex polyhedra

Topology and its Applications 219 (2017) 43–54 Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol...

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Topology and its Applications 219 (2017) 43–54

Contents lists available at ScienceDirect

Topology and its Applications www.elsevier.com/locate/topol

A natural generalization of regular convex polyhedra ✩ Jin-ichi Itoh a , Fumiko Ohtsuka b,∗ a b

Faculty of Education, University of Kumamoto, Kumamoto, Kumamoto 860-8555, Japan Faculty of Science, Ibaraki University, Mito, Ibaraki, 310-8512, Japan

a r t i c l e

i n f o

Article history: Received 3 March 2016 Received in revised form 4 January 2017 Accepted 4 January 2017 Available online 6 January 2017 MSC: primary 53C23 secondary 57M20

a b s t r a c t As a natural generalization of surfaces of Platonic solids, we define a class of polyhedra, called simple regular polyhedral BP-complexes, as a class of 2-dimensional polyhedral metric complexes satisfying certain conditions on their vertex sets, and we give a complete classification of such polyhedra. They are either the surface of a Platonic solid, a p-dodecahedron, a p-icosahedron, an m-covered regular n-gon for some m  2 or a complete tripartite polygon. © 2017 Elsevier B.V. All rights reserved.

Keywords: Polyhedron Cell complex Piecewise linear Positive excess

1. Introduction The objects studied in this paper are 2-dimensional polyhedral metric complexes obtained by identifying isometric edges of convex polygons. We focus on their geometric aspects as geodesic metric spaces, in which their cell complex structures play auxiliary roles. Let X be a connected 2-dimensional locally finite cell complex. By a cell of X we always mean an open cell. We call X homogeneous if each 1-cell of X is a proper face of some 2-cell of X. By taking an appropriate subdivision if necessary, we may assume that every homogeneous 2-dimensional cell complex X in question satisfies the following condition: (∗) The number of vertices on the boundary of each 2-cell is at least 3. ✩ The first author was supported partially by Grant-in-aid for Scientific Research(C) No. 26400072, Japan Society for the Promotion of Science. The second author was partially supported by Grant-in-aid for Scientific Research(C) No. 16K05119, Japan Society for the Promotion of Science. * Corresponding author. E-mail addresses: [email protected] (J. Itoh), [email protected] (F. Ohtsuka).

http://dx.doi.org/10.1016/j.topol.2017.01.004 0166-8641/© 2017 Elsevier B.V. All rights reserved.

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A 2-dimensional homogeneous locally finite cell complex X satisfying (∗) is called a polyhedral complex if it is equipped with a piecewise linear metric d satisfying the following property: if e is a 2-cell of X whose boundary has n vertices (n  3), then the closure e¯ of e is isometric to some n-gon in the Euclidean plane R2 with corresponding vertices, with respect to the intrinsic metric induced by d. For a polyhedral complex X, we call the closure of a 2-cell a face, the closure of a 1-cell an edge, and a 0-cell a vertex, respectively. The metric space X is an M0 -polyhedral complex in the sense of [2], but is not necessarily a CAT(0) space. A polyhedral complex X is said to be non-degenerate if, for any two distinct faces E1 and E2 of X, the intersection E1 ∩ E2 is either an edge, a vertex or an empty set. Otherwise, we call X degenerate. For instance, let S be an n-gon for n  3, and let S1 and S2 be its isometric copies. Identifying the corresponding edges of S1 and S2 , we obtain a degenerate polyhedral complex X, which is called the double n-gon. Clearly, no degenerate polyhedral complex can be isometrically embedded into Euclidean spaces of any dimension in such a way that each face is embedded in some plane. Recall that every convex polyhedron P is defined as the surface of a finite intersection of half-spaces in the Euclidean 3-space R3 . Note that an intrinsic metric on P is induced from R3 . Disregarding the extrinsic geometry, we consider a convex polyhedron P as a non-degenerate polyhedral complex satisfying the following topological condition (T) and the convexity condition (C): (T) P is homeomorphic to a 2-sphere S 2 . (C) The sum of interior angles at any vertex v is less than 2π. Also, by a regular convex polyhedron we mean a non-degenerate convex polyhedron P satisfying the following regularity conditions (R1) and (R2): (R1) There is a regular polygon to which every face is congruent. (R2) The space of directions at every vertex is isometric to each other. The definition of the space of directions is given in Section 2. As is well known, every regular non-degenerate convex polyhedron is the surface of a Platonic solid (namely, a tetrahedron, a cube, an octahedron, a dodecahedron or an icosahedron). For any of such polyhedron, three or more faces meet at each vertex. On the other hand, if only two faces meet at each vertex, then it is degenerate and hence is a double regular polygon. It is convenient to regard a double regular polygon as a regular convex polyhedron. The purpose of this paper is to classify all polyhedra in a class of polyhedral complexes satisfying some conditions as a generalization of regular polyhedra. Our primary concern is geometry of piecewise Riemannian spaces which are not manifolds. We are also concerned with their model spaces. To this end, we examine the conditions (T) and (C). First, we assume that a polyhedral complex P satisfies (I) The intersection of any two faces is connected. Such a polyhedral complex is said to be simple. Obviously, regular polyhedra and non-degenerate polyhedral complexes satisfy this condition. Instead of the condition (T), we consider a weaker condition (T ) P is a 2-manifold.

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Then as the classification of polyhedral complexes with conditions (T ), (C), (R1), (R2) and (I), we have Proposition 4.1: other than regular convex polyhedra, there are only two types of polyhedral complexes, namely a p-dodecahedron and a p-icosahedron, which are illustrated in Example 3.1. Note that, under either the condition (T) or (T ), the space of directions Σv at any vertex v is homeomorphic to a standard circle S 1 . However, for a general polyhedral complex, Σv may be complicated. It may have infinite diameter. It should be remarked that, under either the condition (T) or (T ), we have for every vertex v of any polyhedral complex X diam(Σv ) = inj(Σv ), since Σv is a rescaled circle. However, for a general polyhedral complex, they are not necessarily identical. We relax the condition (T ) to (B) diam(Σv ) = inj(Σv ) for any vertex v. A metric graph satisfying (B) is called a Blaschke graph after Blaschke manifolds. Then the condition (C) is replaced to (P) diam(Σv ) < π for any vertex v, which is equivalent to (C) for a polyhedral complex whose space of directions is a rescaled circle. A polyhedral complex satisfying (P) is said to be of positive excess after [5]. Polyhedral complexes satisfying (R1), (R2) and (P) form a large class. Nevertheless, the condition (B) is strong enough to classify them under this condition. Since we relax only the condition (T), the class of polyhedra satisfying the conditions (R1), (R2), (B) and (P) may be thought of a natural generalization of regular convex polyhedra. Such polyhedra other than the surfaces of Platonic solids are illustrated in Examples 3.2 and 3.3, and are called a m-covered regular n-gon for some m  2 and a complete tripartite polyhedron, respectively. Our main classification theorem is stated as follows. Theorem 4.9. Let X be a simple regular polyhedral BP-complex. Then X is either the surface of a Platonic solid, a p-dodecahedron, a p-icosahedron, an m-covered regular n-gon for some m  2 or a complete tripartite polyhedron. The authors would like to express their sincere thanks to the referee and the others for careful reading and kind suggestions. 2. Definition of regular polyhedral BP-complexes In this section, we give definitions relevant to polyhedral complexes. Also we collect several definitions regarding regular polyhedral BP-complexes and illustrate relevant examples and non-examples. Let X be a polyhedral complex with a piecewise linear geodesic metric d defined in the previous section. For x ∈ X and a sufficiently small ε > 0, we set Sε (x) = {p ∈ X | d(p, x) = ε}. The space Sε (x) is equipped with the extended intrinsic metric induced from d/ε, so that the radial projection Sε (x) → Sε (x) is isometric for sufficiently small ε, ε > 0. We define the space of directions Σx to be the set of initial directions of unit speed geodesics emanating from x. Furthermore, we define the distance

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between the directions γ1 (0) and γ2 (0) of two unit speed geodesics γ1 and γ2 to be the distance between γ1 (ε) and γ2 (ε) measured in Sε (x), so that the space of directions Σx is isometric to Sε (x), Σx ∼ = Sε (x), for any sufficiently small ε > 0. Note that Σx is not necessarily connected, and hence two points in different components have infinite distance, which causes Σx to have infinite diameter. Recall that for an extended geodesic metric space (X, d) the diameter diam(X) and the injectivity radius inj(X) of X are defined respectively by diam(X) = sup{ d(x, y) | x, y ∈ X}, inj(X) = inf{ i(x) | x ∈ X}, where i(x) denotes the supremum of non-negative numbers r such that (i) there exists a point p ∈ X satisfying d(p, x) = r, and (ii) for each point y ∈ Br (x) = {y ∈ X | d(x, y) < r} there exists exactly one minimizing geodesic from x to y with unit speed. Definition 2.1. A polyhedral complex X is said to be a polyhedral BP-complex if it satisfies the following: (B) diam(Σv ) = inj(Σv ) for any vertex v. (P) diam(Σv ) < π for any vertex v. Also, a polyhedral complex X is said to be of positive excess if it satisfies (P). In the following, we call a metric graph (X, d) a Blaschke graph if it satisfies diam(X) = inj(X). In this terminology the space of directions Σv at v of a polyhedral BP-complex is a Blaschke graph. Next the regularity of a polyhedral complex is defined as follows. Definition 2.2. A polyhedral complex X is said to be regular if it satisfies the following: (R1) There is a regular polygon Δ such that every face is congruent to Δ. (R2) The space of directions at every vertex is isometric to each other. For a regular polyhedral complex X, ΣX denotes the metric space isometric to any Σv , called the vertex structure of X. Definition 2.3. A polyhedral complex X is said to be simple if it satisfies the following condition (I): (I) The intersection of any two faces is connected. The condition (I) is naturally satisfied by non-degenerate polyhedral complexes. It plays an essential role in the proof of Lemma 4.6 and Lemma 4.8. Here we illustrate several examples and non-examples. Example 2.1. Let Δn be a regular n-gon. Then Δn is a simple regular polyhedral complex of positive excess, consisting of exactly one face. Note that Δn does not satisfy (B), because its vertex structure ΣΔn is isometric to the closed interval [0, (n − 2)π/n], and hence diam(ΣΔn ) = (n − 2)π/n > (n − 2)π/2n = inj(ΣΔn ).

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Example 2.2. (1) Let X be a regular tiling of the Euclidean plane by squares, which is a noncompact simple regular polyhedral complex satisfying (B), but not of positive excess. In fact, ΣX is isometric to a standard circle, and diam(ΣX ) = inj(ΣX ) = π. (2) Let X be the union of the black squares of an infinite chessboard, which is also an noncompact simple regular polyhedral complex satisfying neither (P) nor (B). In fact, ΣX is isometric to the disjoint union of two closed intervals [0, π/2] ∪ [0, π/2], and hence diam(ΣX ) = ∞ > π/4 = inj(ΣX ). Example 2.3. (1) Let X be a regular polyhedral complex consisting of two tetrahedra with four corresponding vertices identified respectively. Note that the intersection of two corresponding faces (triangles) of X is the set of three vertices, and hence X is not simple. Now, for any vertex v ∈ X, the space of directions Σv is the disjoint union of two circles of circumference π, and diam(Σv ) = ∞ > π/2 = inj(Σv ), which shows that X satisfies neither (B) nor (P). (2) Let X be a regular polyhedral complex consisting of two tetrahedra with six corresponding edges identified respectively. Then X is simple, but diam(ΣX ) = π/2 > π/3 = inj(ΣX ). Therefore X satisfies (P), but not (B). A higher dimensional generalization of a polyhedron is called a polytope. We remark that every 2-skeleton P of a higher-dimensional regular polytope is a simple regular polyhedral complex which does not satisfy (B). For exact values of diam(ΣP ) and inj(ΣP ) of such examples, we refer the reader to a forthcoming paper of the second author [7]. A free edge of a polyhedral complex X is an edge of X which is contained in a unique face. The point set union of free edges of X is called the boundary of X and is denoted by BX. The complement X \ BX, denoted by IX, is called the interior of X. Clearly, v is a point in BX if and only if the space of directions Σv has end points. We say that a vertex w of a graph is an end point if it has degree one. The following Lemma follows immediately. Lemma 2.4. Let X be a polyhedral complex. Then the following hold. (1) If X satisfies (R2) and there is a vertex v ∈ IX, then X has no boundary. (2) If X satisfies (B), then X has no boundary. (3) If X satisfies (B), then spaces of directions of X are connected spaces of finite diameter. Proof. (1) If there is a vertex v ∈ IX, Σv has no end point. Since X satisfies (R2), it follows that there are no vertices on BX. Hence X has no boundary. (2) If there is a vertex v ∈ BX, then Σv has an end point w. Let u be the unique vertex adjacent to w, and let l = d(w, u). Then we have i(u)  max{i(w) − l, l}. If l < i(w), then inj(Σv )  i(u) < i(w)  diam(Σv ).

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If l = i(w), then Σv is isometric to the closed interval [0, l], so that inj(Σv ) = diam(Σv )/2. In both cases, X does not satisfy (B). (3) Since X is locally finite, inj(Σv ) is always finite. Hence, from (B), diam(Σv ) is also finite. This implies that Σv is connected. 2 3. Examples of simple regular polyhedral BP-complexes Before classifying the simple regular polyhedral BP-complexes, we illustrate several non-trivial examples in detail. First recall that surfaces of Platonic solids are simple regular polyhedral BP-complexes. We now consider their quotient spaces. Example 3.1. By identifying the antipodal points on a dodecahedron and on an icosahedron, we obtain simple regular polyhedral BP-complexes homeomorphic to a projective plane. We call them a p-dodecahedron and a p-icosahedron, respectively. Note that the above construction does not apply to a tetrahedron since it is not centrally symmetric. Also, for a cube and an octahedron, their quotient spaces are not simple. Indeed, the quotient of a cube consists of 3 squares and the intersection of any two of them consists of two disjoint closed edges. Hence the quotient space of a cube satisfies (B) and (P) but not (I). The same holds for the octahedron. Next we describe examples of simple regular polyhedral BP-complexes whose vertex structures are not necessarily homeomorphic to a circle S 1 . Example 3.2. Let S be a regular n-gon for n  3, and let S1 , · · · , Sm be m isometric copies of S. Identifying the corresponding edges of S1 , · · · , Sm , we obtain a simple regular polyhedral BP-complex Xm,n . We call Xm,n an m-covered n-gon. Note that X2,n is a double n-gon. The graph of the vertex structure ΣXm,n of Xm,n , denoted by θm , consists of two vertices and m edges which connect these vertices. Since the interior angle of a regular n-gon is (n − 2)π/n, the diameter of ΣXm,n is also (n − 2)π/n, which coincides with the injectivity radius. A simple regular polyhedral BP-complex X whose vertex structure is homeomorphic to θm is said to be of type θm . Note that θ2 is homeomorphic to S 1 , and hence all regular polyhedra are of type θ2 . Example 3.3. Let Km,n be a complete bipartite graph of type (m, n), that is a graph having two disjoint classes of vertices consisting of m ( 2) and n ( 2) vertices, which are connected by an edge. It has mn edges in total. π/3 We denote by Km,n the metric graph homeomorphic to Km,n whose edges have the same length π/3. π/3 We now construct a simple regular polyhedral BP-complex whose vertex structure is isometric to Km,m for m  2. Let p be a point, and S a polyhedral complex homeomorphic to the cone p ∗ Km,m with vertex p and base Km,m such that, for every edge e of Km,m , all faces of S corresponding to p ∗ e are congruent to some π/3 regular triangle Δ. Since the interior angle of Δ is π/3, the space of directions Σp is isometric to Km,m . π/3 Note that the boundary of S is exactly the graph Km,m , but is not necessarily isometric to Km,m . Let S1 , · · · , Sm be m isometric copies of S. A simple regular polyhedral BP-complex Ym is constructed by identifying the corresponding boundaries of S1 , · · · , Sm . The polyhedral complex Ym is homeomorphic to a 3-fold join Vm ∗ Vm ∗ Vm of the discrete space Vm of m points. More precisely, Ym is a simple regular polyhedral BP-complex consisting of three classes Vm with m vertices, 3m2 edges which connect any two vertices in different classes, and m3 faces which are triangles with three vertices belonging to each of these classes and are all isometric to Δ.

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We call Ym a complete tripartite polyhedron. Here the space of directions Σv for any vertex v is isometric π/3 to Km,m , and the diameter of Σv is 2π/3, which coincides with the injectivity radius. Note that Y2 is an octahedron. Let Y be a simple regular polyhedral BP-complex whose vertex structure is homeomorphic to Km,n . In what follows, we call such Y to be of type Km,n . Note that K2,2 is homeomorphic to S 1 , and hence being of type K2,2 is identical to being of type θ2 . More generally, being of type Km,2 and being of type K2,m are both identical to being of type θm . 4. Classification of regular polyhedral BP-complexes Let X be a simple regular polyhedral BP-complex. To prove our main classification theorem, we first study a polyhedron whose vertex structure is homeomorphic to a standard circle S 1 . Then X is classified as follows. Proposition 4.1. Let X be a simple regular polyhedral BP-complex. If the vertex structure ΣX of X is homeomorphic to a standard circle S 1 , then X is either the surface of a Platonic solid, a double regular n-gon (n  3), a p-dodecahedron or a p-icosahedron. Proof. Since ΣX is homeomorphic to S 1 , X is a topological 2-manifold. Let k(X) := 2(π − diam ΣX ). Then from the well-known combinatorial Gauss–Bonnet theorem, we see that the Euler characteristic of X is equal to the number of the vertices times k(X)/2π if X is orientable. Since k(X) > 0 from the condition (P), the universal covering of X has positive Euler characteristic, so that it is homeomorphic to S 2 . As a result, if X is simply connected, it is homeomorphic to S 2 and is either the double of the regular n-gon or the surface of a Platonic solid. Otherwise, X is homeomorphic to a projective plane, and hence the double covering of X is one of the above. Note here that, if X is a simple regular polyhedral BP-complex, then so is its double covering. Therefore X is obtained from one of the above by identifying the antipodal points. Hence X is a p-dodecahedron or a p-icosahedron in Example 3.1. Note that there is no other possibility, as mentioned. 2 Next we study the case where the vertex structure ΣX of a regular polyhedral BP-complex X is not homeomorphic to S 1 . The space ΣX is a connected metric graph. We need several observations on connected metric graphs, due to Ballmann and Brin [1]. First we recall the following lemmas. These are the rescaled version of Lemmas 6.1 and 6.3 of [1] and can be proved by the same argument as in [1]. Lemma 4.2. Let Σ be a Blaschke graph of the finite diameter L such that the degree of each vertex is at least 3. Then there is an integer k  1 such that every edge of Σ has length L/k. Lemma 4.3. Let Σ be a Blaschke graph of the finite diameter L such that the degree of each vertex is at least 3. If the length of each edge is L/2, then Σ is a complete bipartite graph. Applying the above lemmas, we have Proposition 4.4. Let X be a simple regular polyhedral BP-complex. If the vertex structure ΣX of X is not homeomorphic to a standard circle S 1 , then X is either of type θm or of type Km,n for m, n  3. Proof. The conditions (B) and (P) imply that ΣX is a Blaschke graph of the finite diameter. Note that ΣX has no end points from Lemma 2.4 (2). Since ΣX is not homeomorphic to S 1 , ΣX is regarded as a graph such that the degree of each vertex is at least 3. Hence we see from Lemma 4.2 that there is an integer

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k  1 such that every edge of ΣX has length L/k. We also note that each edge has length not less than π/3, the interior angle of the regular triangle. Hence it holds that L  π if k  3, from which the condition (P) implies that k  2. If k = 1, then ΣX is homeomorphic to θm for some m  3. If k = 2, then by Lemma 4.3, ΣX is a metric graph having the structure of a complete bipartite graph Km,n for m, n  3. Note that both Km,2 and K2,n have a vertex of degree 2. Recall that being of type Km,2 and being of type K2,m are both identical to being of type θm . Therefore X is either of type θm or of type Km,n for m, n  3. 2 Now we investigate the realization of type θm and of type Km,n for m, n  3. Simple regular polyhedral BP-complexes of type θm are classified in the following manner. Proposition 4.5. Let X be a regular polyhedral BP-complex of type θm for m  3. Then X is an m-covered n-gon for n  3. In particular, X is simple. Proof. Let {x1 , x2 } and {e1 , · · · , em } be the vertices and the edges of the vertex structure ΣX θm , respectively. Since m  3 and ΣX is a Blaschke graph, it follows from Lemma 4.2 that all edges of ΣX have the same length. Take an arbitrary vertex v ∈ X, and fix an isometry ΣX ∼ = Σv . We first show that, for each edge ei of ΣX , being identified with an edge of the space of directions Σv via the above isometry, there corresponds exactly one face Ei of X in such a way that each point of ei is the initial direction of a geodesic in Ei . This is the place where we use the hypothesis m  3. When m = 2, it is not generally the case that each edge of ΣX corresponds to one face. Indeed, assume that an edge e of Σv corresponds to the union of two faces E1 and E2 . Note that each Ei must be a triangle, since the diameter of Σv is identical to the length of e, which is twice the interior angle of a vertex of Ei and less than π. Now, let vi1 and vi2 be the vertices of Ei adjacent to v. Without loss of generality, we may assume that the edges vv11 , vv12 = vv21 and vv22 correspond to x1 , the midpoint of ei , and x2 on Σv , respectively. Since the space of directions at v12 = v21 is also isometric to ΣX , it follows that either of the following cases occurs: (i) The edge v11 v12 is incident with m − 1 faces other than E1 , and simultaneously the edge v12 v22 is incident with m − 1 faces other than E2 . (ii) The edge vv12 is incident with m − 2 faces other than E1 and E2 . In the case (i), since the edge of the space of directions Σv11 (resp. Σv22 ) corresponding to E1 (resp. E2 ) is π/3 in length, neither of Σv11 nor Σv22 is isometric to ΣX , a contradiction. In the case (ii), the space of directions Σv is branched at the midpoint of ei and is no longer isometric to ΣX , a contradiction. Therefore, for any vertex v ∈ X, each edge ei of Σv corresponds to one face Ei of X for each i = 1, · · · , m. Note that Ei is not necessarily a triangle. The set of all faces containing v consists of m elements of n-gons E1 , · · · , Em , and the edge of Ei containing v, which corresponds to the vertex x1 (resp. x2 ) of Σv ∼ = ΣX , are identified to each other. Next we focus on an adjacent vertex v˜ of v. Since Σv˜ ∼ = ΣX , we see that the opposite edge to the edge v˜ v on each Ei must also be identified to each other. Consequently, X is a regular polyhedral BP-complex obtained in the same way as in Example 3.2. 2 Next, we study simple regular polyhedral BP-complexes of type Km,m for m  3. Let Y be a simple regular polyhedral BP-complex of type Km,m for m  3, and α the length of the edges of Y . Since the vertex structure ΣY is homeomorphic to Km,m for m  3 and is a Blaschke graph, it follows from Lemma 4.2 that all edges of ΣY have the same length, say a.

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a We denote by Km,m the graph Km,m with a metric such that each edge has the same length a, so that a ΣY is isometric to Km,m . Since diam(ΣY ) = inj(ΣY ) = 2a, the condition (P) implies that a < π/2. Since the interior angle of a regular n-gon is (n − 2)π/n, we have n = 3 and a = π/3, and hence the face of Y must be isometric to the regular triangle Δ with edges of length α. α , a cone with vertex z and base Km,m admitting Let z be a point, and S a polyhedral complex z ∗ Km,m π/3

a metric such that all faces are isometric to Δ. Note that the space of directions Σz is isometric to Km,m . We will call S a component of Y . For each vertex v ∈ Y , let Star(v) be the subspace of Y consisting of the union of all faces of Y containing v, which is called the closed star at v. Note that Star(v) is also a polyhedral complex having the structure induced from Y . Then we have the following Lemma 4.6. Let Y be a simple regular polyhedral BP-complex of type Km,m for m  3. For any vertex v ∈ Y , the closed star Star(v) at v is isometric to the component S of Y . Proof. Since every face of S and Star(v) is isometric to the regular triangle Δ, and Σz and Σv are both π/3 isometric to Km,m , it is obvious that the interior of Star(v) is isometric to the interior of S. α , there is a natural projection So, we need to check on the boundaries of Star(v) and S. Since S = z ∗Km,m p : BS → Σz such that for x ∈ BS, p(x) ∈ Σz is the initial direction of the geodesic from z to x. Regarding p(x) ∈ Σv via an isometry Σz ∼ = Σv , we define f (x) ∈ BStar(v) as the first intersection of BStar(v) and the geodesic γ with γ  (0) = p(x). It is clear that f : BS → BStar(v) is surjective. Also we see that f is injective by applying the condition (I) as follows. α Let {x1 , · · · , xm } and {y1 , · · · , ym } be two classes of vertices of the metric graph BS = Km,m . For a vertex x on BS, we denote by x ˜ the vertex f (x) on Star(v). For two distinct vertices xi and xj on BS, suppose that x ˜i and x ˜j coincide. Then both of v and x ˜i = x ˜j are contained in the intersection of two faces v˜ xi y˜1 and v˜ xj y˜2 . By the condition (I), two edges v˜ xi and v˜ xj are identified, or v y˜1 ∪ y˜1 x ˜i and v y˜2 ∪ y˜2 x ˜j π/3 ∼ are identified. But both cases contradict that Σv = Km,m . Similarly, we have a contradiction if y˜i and y˜j (i = j) coincide. Also, if x ˜i and y˜j coincide, this contradicts that the face v˜ xi y˜j is isometric to the regular triangle Δ. As a consequence, we have v˜1 = v˜2 for any two distinct vertices v1 , v2 ∈ BS. Hence f is a bijection. This implies that Star(v) is isometric to S with respect to the intrinsic metric induced from Y . 2 Applying this lemma, we obtain Proposition 4.7. Let Y be a simple regular polyhedral BP-complex of type Km,m for m  3. Then Y is a complete tripartite polyhedron, which is obtained in the same way as in Example 3.3. Proof. For a vertex z1 ∈ Y , let S1 be the closed star Star(z1 ). Then it follows from Lemma 4.6 that S1 is isometric to the component S of Y , and hence BS1 has the structure of a complete bipartite graph. Let 1 {x11 , · · · , x1m } and {y11 , · · · , ym } be the two classes of vertices on BS1 as a complete bipartite graph. 1 }. First we focus on the vertex x11 of S1 . Then the vertices of Y adjacent to x11 contains {z1 , y11 , · · · , ym π/3 Since the space of directions Σx11 is isometric to Km,m , there exist other m − 1 vertices {z2 , · · · , zm } ⊂ Y adjacent to x11 . Therefore there are at least m closed stars Si = Star(zi ) for i = 1, · · · , m, which are connected to each other at x11 . We now check how they are connected. To do this, let xij and yji be the vertices on Si , corresponding to xj and yj via an isometry from S to Si for i, j = 1, · · · , m. Without loss of generality, we may assume 1 that xi1 = x11 for any i. Then it is verified that the vertices on BStar(x11 ) are {z1 , · · · , zm } ∪ {y11 , · · · , ym }, 1 1 i i especially {y1 , · · · , ym } = {y1 , · · · , ym } for any i. i Indeed, from Lemma 4.6, any two vertices of {y1i , · · · , ym } are distinct for any i. Hence the vertices 1 i i i 1 i of BStar(x1 = x1 ) are identical to {z1 , · · · , zm } ∪ {y1 , · · · , ym }. Therefore {y11 , · · · , ym } = {y1i , · · · , ym }.

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Changing the indices if necessary, we conclude that for each j, the m vertices yj1 , · · · , yjm must be identified with each other. Similarly, if we look at the vertex y11 (= y1i ), then, in the same way, BStar(y11 ) has the structure as a complete bipartite graph with two classes of vertices {z1 , · · · , zm } and {x11 , · · · , x1m }, which implies that {x11 , · · · , x1m } = {xi1 , · · · , xim } for any i. Indeed, for each i, any two vertices of {xi1 , · · · , xim } are distinct from Lemma 4.6. Hence the vertices of BStar(y11 ) are identical to {z1 , · · · , zm } ∪ {x11 , xi2 , · · · , xim }. (Note that x11 = xi1 is already identified.) Therefore the vertices {xi2 , · · · , xim } must be identified with {x12 , · · · , x1m }. Consequently, changing the indices if necessary, the points xij (resp. yji ) on BSi corresponding to xj (resp. yj ) on BS must be identified with each other. Thus BSi must be identified with each other. This completes the proof. 2 Finally we consider regular polyhedral BP-complexes of type Km,n for m = n (m, n  3). Proposition 4.8. There is no realization of a simple regular polyhedral BP-complex whose space of directions is Km,n for m = n (m, n  3). Proof. We assume that there is a regular polyhedral BP-complex X of type Km,n for m > n  3. Fix a vertex z1 ∈ X, and let S1 = Star(z1 ). Then, by a similar argument in Lemma 4.6, we see that BS1 is a isometric to Km,n , where a is the length of the edges of X. Let {x1 , · · · , xm } and {y1 , · · · , yn } be the two classes of vertices on BS1 as a complete bipartite graph. a Set S = Star(x1 ). Note that BS contains the vertices {z1 , y1 , · · · , yn }. Then, since BS is isometric to Km,n , there exists another set of vertices {z2 , · · · , zm } ⊂ BS. Let Si = Star(zi ) for i = 2, · · · , m. Since BS is a bipartite graph Km,n having two classes of vertices {z1 , · · · , zm } and {y1 , · · · , yn }, it follows that each vertex yi is contained in each of S1 , · · · , Sm . We now look at Star(y1 ). Then on the boundary BStar(y1 ) there are at least 2m vertices {x1 , · · · , xm } ∪ {z1 , · · · , zm }. Applying a similar argument in Lemma 4.6 to BS1 and BS, we see that these vertices are a distinct in Y . Hence there is no contraction on Star(y1 ) such that BStar(y1 ) is isometric to Km,n , which is a contradiction. In consequence, there is no realization of a simple regular polyhedral BP-complex whose space of directions is Km,n for m = n. 2 Summarizing the discussions above, we obtain the following main theorem. Theorem 4.9. Let X be a simple regular polyhedral BP-complex. Then X is either the surface of a Platonic solid, a p-dodecahedron, a p-icosahedron, an m-covered regular n-gon for some m  2 or a complete tripartite polyhedron. Proof. Let X be a simple regular polyhedral BP-complex and L the diameter of the vertex structure ΣX . If ΣX is homeomorphic to a standard circle S 1 , then by Proposition 4.1, X is either the surface of a Platonic solid, a double regular n-gon, a p-dodecahedron or a p-icosahedron. Otherwise, by Proposition 4.4, X is either of type θm or of type Km,n for m, n  3. If X is of type θm , then by Proposition 4.5, X is an m-covered n-gon. If X is of type Km,m , then by Proposition 4.7, X is a complete tripartite polyhedron. Furthermore from Lemma 4.8, there is no realization of a regular polyhedral BP-complex of type Km,n for m = n. This completes the proof. 2

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Appendix We investigated a class of piecewise Riemannian polyhedra in [3] and we were interested in finding model complexes which is typical of them like as space forms on Riemannian manifolds. In a previous paper [6], the second author classified flat piecewise Riemannian 2-polyhedra, and remarked in its Appendix that a simply connected polyhedral complex X satisfying diam(Σv ) = inj(Σv ) = π for any vertex v ∈ X is a Hadamard space such that the diameter of its ideal boundary is π. Thus the classification of such regular polyhedral complexes is obtained in Nagano’s classification of Hadamard 2-spaces in [4]. We are concerned in this paper with polyhedral complexes admitting piecewise linear metrics. A spherical counterpart to those complexes are defined by considering polyhedra modeled by regular spherical polygon, that is, a convex region on a standard sphere bounded by arcs of three or more great circles, whose sides have the same length and whose angles are identical. Let X be a connected 2-dimensional homogeneous locally finite cell complex with piecewise spherical metric satisfying the conditions (I), (B), (P), (R2), and the following (SR1) instead of (R1): (SR1) There is a regular spherical n-gon (n  3) to which every face is congruent. We call such X a simple regular spherical polyhedral BP-complex. Recall that on a standard sphere, the Gauss–Bonnet theorem implies that S + n(π − α) = 2π, where S is the area of a regular geodesic n-gon, and α is its interior angle. An appropriate modification of the proof of Theorem 4.9 combined with the above identity then yields the following classification. Theorem 5.1. Let X be a simple regular spherical polyhedral BP-complex whose faces are congruent to some regular spherical n-gon on a standard sphere, and S the area of a face of X. Then S < 2π and X is classified as follows in accordance with n. (1) (The case n  6) X is an m-covered regular n-gon for m  2. (2) (n = 5) If π/3  S < 2π, then X is an m-covered pentagon for m  2. If S < π/3, then X is an m-covered pentagon, a dodecahedron or a p-dodecahedron. (3) (n = 4) If 2π/3  S < 2π, then X is an m-covered square for m  2. If S < 2π/3, then X is an m-covered square or a cube. (4) (n = 3) If π  S < 2π, then X is an m-covered equilateral triangle for m  2. If π/2  S < π, then X is an m-covered equilateral triangle or a tetrahedron. If π/5  S < π/2, then X is an m-covered equilateral triangle, a tetrahedron or a complete tripartite polyhedron (which may be an octahedron). If S < π/5, then X is an m-covered equilateral triangle, a tetrahedron, a complete tripartite polyhedron, an icosahedron or a p-icosahedron. It should be remarked that there is a regular n-gon Δ on the standard sphere with S  2π. However, such Δ can not be a face of regular BP-complex polyhedron, since the interior angle of Δ is not less than π. In the hyperbolic case, a regular geodesic n-gon Δ on the hyperbolic space of curvature −1 has the interior angle

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α = ((n − 2)π − S)/n, where S is the area of Δ. As a result, the situation is more complicated in this case. For example, a noncompact simple regular polyhedral BP-complex is obtained by replacing the square of a regular tiling of the Euclidean plane with the regular hyperbolic square. Also, when k = 2 in Lemma 4.2, faces of a regular BP-complex polyhedron are not necessarily triangles; they may be squares and so on. Also we might have examples with k  3. Thus the classification of the hyperbolic case needs further investigations. References [1] [2] [3] [4] [5]

W. Ballmann, M. Brin, Orbihedra of nonpositive curvature, Publ. Math. IHÉS 82 (1995) 169–209. M.R. Bridson, A. Haefliger, Metric Spaces of Non-positive Curvature, Springer-Verlag, Berlin, Heidelberg, New York, 1999. J. Itoh, F. Ohtsuka, Total curvature of noncompact piecewise Riemannian 2-polyhedra, Tsukuba J. Math. 29 (2005) 471–493. K. Nagano, Asymptotic rigidity of Hadamard 2-spaces, J. Math. Soc. Jpn. 52 (2000) 699–723. K. Kawamura, F. Ohtsuka, Total excess and Tits metric for piecewise Riemannian 2-manifolds, Topol. Appl. 94 (1999) 173–193. [6] F. Ohtsuka, Erratum to: “Structures of flat piecewise Riemannian 2-polyhedra”, Math. J. Ibaraki Univ. 37 (2005) 107–114. [7] F. Ohtsuka, Structure of 2-skeletons of higher-dimensional regular polytopes, 2017, in preparation.