SMALL COVER, INFRA-SOLVMANIFOLD AND CURVATURE
SHINTARÔ KUROKI† , MIKIYA MASUDA‡ , AND LI YU*
Abstract. It is shown that a small cover (resp. real moment-angle manifold)over a simple polytope is an infra-solvmanifold if and only if it is diﬀeomorphic
to a real Bott manifold (resp. ﬂat torus). Moreover, we obtain several equivalent conditions for a small cover being homeomorphic to a real Bott manifold.
In addition, we study Riemannian metrics on small covers and real momentangle manifolds with certain conditions on the Ricci or sectional curvature.
We will see that these curvature conditions put very strong restrictions on the
topology of the corresponding small covers and real moment-angle manifolds
and the combinatorial structure of the underlying simple polytopes.

1. Introduction
The notion of small cover was ﬁrst introduced by Davis and Januszkiewicz [11]as an analogue of smooth projective toric variety in the category of closed manifolds with Z2 -torus actions. An n-dimensional small cover M n is a closed nmanifold with a locally standard (Z2 )n action whose orbit space can be identiﬁed
with a simple convex polytope P n in Rn . The (Z2 )n -action on M n determinesa (Z2 )n -valued characteristic function λM n on the facets of P n , which encodes
the information of isotropy subgroups of the non-free orbits. Conversely, we canrecover M n and the (Z2 )n -action, up to equivariant homeomorphism, by gluing
2n copies of P n according to the function λM n . It is shown in [11] that many
important topological invariants of M n can be easily computed in terms of the
combinatorial structure of P n and the λM n . For example, the fundamental groupof M n is a ﬁnite index subgroup of a right-angled Coxeter group WP n , where
WP n is canonically determined by P n . Note that not all simple convex polytopes
admit small covers over them. But for any simple convex polytope P n , we can
Key words and phrases. Small cover, real moment-angle manifold, real Bott manifold, Ricci
curvature, infra-solvmanifold, infra-nilmanifold, Coxeter group.
2010 Mathematics Subject Classiﬁcation. 57N16, 57S17, 57S25, 53C25, 51H30‡
The author is partially supported by Grant-in-Aid for Scientiﬁc Research 22540094.
*The author is partially supported by the Japanese Society for the Promotion of Sciences (JSPS
grant no. P10018) and Natural Science Foundation of China (grant no.11001120). This work isalso funded by the PAPD (priority academic program development) of Jiangsu higher education
institutions.
12

SHINTARÔ KUROKI† , MIKIYA MASUDA‡ , AND LI YU*

canonically associate a closed manifold RZP n to P n called real moment-angle
manifold (see [11] and [3]).
In this paper, we will mainly use fundamental groups to study diﬀerent kindsof geometric structures on small covers and real moment-angle manifolds. The
following are some results proved in this paper.
Theorem 1.1. If the real moment-angle manifold RZP n of an n-dimensional
simple convex polytope P n is homeomorphic to an infra-solvmanifold, then P n
is an n-cube and hence RZP n is homeomorphic to an n-dimensional torus. In
particular, if a small cover is homeomorphic to an infra-solvmanifold, it must be
homeomorphic to a real Bott manifold (see Corollary 3.7).
Theorem 1.2. Let RZP n be the real moment-angle manifold of an n-dimensional
simple convex polytope P n .
(i) RZP n admits a Riemannian metric with positive constant sectional curvature if and only if P n is an n-simplex (see Theorem 5.4).
(ii) RZP n admits a ﬂat Riemannian metric if and only if P n is an n-cube(see Corollary 3.7).
(iii) If RZP n admits a Riemannian metric with negative (not necessarily constant) sectional curvature, then no 2-face of P n can be a 3-gon or a 4-gon(see Proposition 5.7).
The paper is organized as follows. In section 2, we study when the fundamental group of a real moment-angle manifold (or a small cover) over a simple
convex polytope is virtually nilpotent or ﬁnite (Corollary 2.5 and Corollary 2.6).
Then we introduce a special class of small covers called generalized real Bottmanifolds which are the main examples related to our study in this paper. In
section 3, we will study when a small cover or real moment-angle manifold is an
infra-solvmanifold. It turns out that such a small cover (or real moment-angle
manifold) must be a real Bott manifold (or ﬂat torus) (Corollary 3.7). The proof
essentially uses a nice result in [12] that describes the asphericality of small covers
in terms of ﬂagness of the underlying simple polytope. In section 4, we obtainseveral equivalent conditions for a small cover being homeomorphic to a real Bottmanifold (Theorem 4.3). In section 5, we study Riemannian metrics on small
covers and real moment-angle manifolds with various conditions on the Ricci or
sectional curvature. Most of the results obtained in this section follow from thestudy of fundamental groups in section 2. In addition, some problems are proposed for the future study.

3

2. Right-angled Coxeter group and fundamental group
Suppose P n is an n-dimensional simple convex polytope in the Euclidean space
Rn . Here the word “simple” means that any vertex of P n is the intersection of
exactly n diﬀerent facets of P n . Let F(P n ) denote the set of all facets of P n . Let
WP n be a right-angled Coxeter group with one generator for each facet of P n andrelations s2 = 1, ∀ s ∈ F(P n ), and (st)2 = 1 whenever s, t are adjacent facets of
P n.
Remark: Although we call WP n a right-angled Coxeter group, the dihedral
angle between two adjacent facets of P n may not be a right angle in reality. So
generally speaking, WP n is not the group generated by the reﬂections of Rn about
the hyperplanes passing the facets of P n .
Suppose F1 , · · · , Fr are all the facets of P n . Let e1 , · · · , er be a basis of (Z2 )r .
Then we deﬁne a function λ0 : F(P n ) → (Z2 )r by
λ0 (Fi ) = ei , 1 ≤ i ≤ r.

(1)

For any proper face f of P n , let Gf denote the subgroup of (Z2 )r generated by
the set {λ0 (Fi ) | f ⊂ Fi }. For any point p ∈ P n , let f (p) denote the unique faceof P n that contains p in its relative interior. In [11, Construction 4.1], the real
moment-angle manifold RZP n of P n is deﬁned to be the following quotient space
RZP n := P n × (Z2 )r / ∼(2)where (p, g) ∼ (p0 , g 0 ) if and only if p = p0 and g −1 g 0 ∈ Gf (p) . It is shownin [11] that the fundamental group of RZP n is the kernel of the abelianization
Ab : WP n → WPabn ∼= (Z2 )r , that is, there is an exact sequenceAb

1 −→ π1 (RZP n ) −→ WP n −→ (Z2 )r −→ 1,(3)

so we have
π1 (RZP n ) = ker(Ab) = [WP n , WP n ] (the commutator subgroup of WP n ).
A small cover M n over P n is a closed n-manifold with a locally standard(Z2 )n -action so that its orbit space is homeomorphic to P n . Here “locally standard” means that any point in M n has an (Z2 )n -invariant open neighborhood
which is equivariantly homeomorphic to an (Z2 )n -invariant open subset in an ndimensional faithful linear representation space of (Z2 )n . Let π : M n → P n be
the quotient map. For any facet Fi of P n , the isotropy subgroup of π −1 (Fi ) in M nunder the (Z2 )n -action is a rank one subgroup of (Z2 )n generated by a nonzero
element, say gFi ∈ (Z2 )n . Then we obtain a map λM n : F(P n ) → (Z2 )n where
λM n (Fi ) = gFi , 1 ≤ i ≤ r.

4

SHINTARÔ KUROKI† , MIKIYA MASUDA‡ , AND LI YU*

We call λM n the characteristic function associated to M n . It is shown in [11]that up to equivariant homeomorphism, M n can be recovered from (P n , λM n ) ina similar way as the construction of RZP n in (2). Moreover, λM n determines agroup homomorphism λM n : (Z2 )r → (Z2 )n where
λM n (ei ) = λM n (Fi ) = gFi , 1 ≤ i ≤ r.
It is shown in [11] that the fundamental group π1 (M n ) of M n is isomorphic to
the kernel of the composition λM n ◦ Ab, that is, there is an exact sequence
λ

n ◦Ab

1 −→ π1 (M n ) −→ WP n M−→ (Z2 )n −→ 1.

(4)

Then it follows from (3) and (4) that we have an exact sequence
1 −→ π1 (RZP n ) −→ π1 (M n ) −→ (Z2 )r−n −→ 1.

(5)

In fact, it is easy to see that RZP n is a regular (Z2 )r−n -covering of any small
cover over P n .
Proposition 2.1. For an n-dimensional simple convex polytope P n , the following
are equivalent:(i) P n is 2-neighborly (i.e. any two facets of P n are adjacent).
(ii) The real moment-angle manifold RZP n over P n is simply connected.
Moreover, if there exists a small cover M n over P n , then the above statements
are also equivalent to the following:(iii) The fundamental group of M n is isomorphic to (Z2 )r−n , where r is thenumber of facets of P n .
Proof. By deﬁnition, P n is 2-neighborly if and only if WP n is isomorphic to (Z2 )rwhere r is the number of facets of P n . Then the equivalence between (i) and (ii)follows from (3), and the equivalence between (ii) and (iii) follows from (5). ¤All the relations between the generators of WP n can be formally represented
by a matrix m = (mst ), called Coxeter matrix.
 1, if s = t;2, if s is adjacent to t; (s, t denote arbitrary facets of P n ).
mst :=
 ∞, otherwise.
The triple (WP n , F(P n ), m) is called a Coxeter system of WP n .
For a general Coxeter system (W, S, m), its Coxeter graph is a graph with avertex set S, and with two vertices s 6= t joined by an edge whenever mst ≥ 3.
If mst ≥ 4, the corresponding edge is labeled by mst . We say that (W, S, m) is
irreducible if its Coxeter graph is connected.

5

A Coxeter group W is called rigid if, given any two systems (W, S, m) and(W, S 0 , m0 ) for W ; there is an automorphism ρ : W → W such that ρ(S, m) =(S 0 , m0 ), i.e. the Coxeter graphs of (W, S, m) and (W, S, m0 ) are isomorphic.
Theorem 2.2 (Radcliﬀe [24]). If (W, S, m) is a Coxeter system with mst ∈{2, ∞} for all s 6= t ∈ S, then W is rigid. In other words, any right-angled
Coxeter group is rigid.
Associated to any Coxeter system (W, S, m), there is a symmetric bilinear form
( , ) on a real vector space V with a basis {αs | s ∈ S} in one-to-one correspondencewith the elements of S. The bilinear form ( , ) is deﬁned by:π,(6)(αs , αt ) := − cos
mstwhere the value on the right-hand side is interpreted to be −1 when mst = ∞.
It is well-known that a Coxeter group W is ﬁnite if and only if the bilinear
form of a Coxeter system (W, S, m) is positive deﬁnite. All ﬁnite Coxeter groups
have been classiﬁed by H. S. M. Coxeter in 1930s (see [10] and [16]). It is easy
to see that if W is a ﬁnite right-angled Coxeter group, W must be isomorphic to(Z2 )k for some k ≥ 0. By Theorem 2.2, any Coxeter graph of W ∼= (Z2 )k should
be k disjoint vertices.
An irreducible Coxeter group W is called aﬃne if there is a Coxeter system(W, S, m) so that the bilinear form of the system (see (6)) is positive semi-deﬁnite
but not positive deﬁnite. More generally, a Coxeter group is called aﬃne if its
irreducible components are either ﬁnite or aﬃne, and at least one component isaﬃne. Equivalently, a Coxeter group is aﬃne if it is an inﬁnite group and has arepresentation as a discrete, properly acting reﬂection group in Rn . The reader isreferred to [18] for more information on aﬃne Coxeter groups. A Coxeter groupis called non-aﬃne if it is not aﬃne.
Theorem 2.3 (Qi [23]). The center of any ﬁnite index subgroup of an inﬁnite,
irreducible, non-aﬃne Coxeter group is trivial.
A group is called virtually nilpotent (abelian, solvable) if it has a nilpotent(abelian, solvable) subgroup of ﬁnite index.
Lemma 2.4. If the right-angled Coxeter group WP of a simple convex polytope
e1 )l for some k, l ≥ 0, where k + 2l
P is virtually nilpotent, then WP ∼= (Z2 )k × (Ae1 = ha, b | a2 = 1, b2 = 1i.
equals the number of facets of P and A
Proof. Suppose WP is inﬁnite and let N be a ﬁnite index nilpotent subgroup of
WP . If WP is not aﬃne, then WP has at least one irreducible component which
is neither ﬁnite nor aﬃne, say W1 . Then N1 = N ∩ W1 is a ﬁnite index nilpotent
subgroup of W1 . Since W1 is an inﬁnite group, N1 is also inﬁnite hence nontrivial.

6

SHINTARÔ KUROKI† , MIKIYA MASUDA‡ , AND LI YU*

Now since W1 is inﬁnite, irreducible and non-aﬃne, by Theorem 2.3, the centerof N1 must be trivial. But the center of any nontrivial nilpotent group is nevertrivial. This implies that the Coxeter group WP must be either ﬁnite or aﬃne.
Since WP is right-angled, so is each of its irreducible components. Then by
the classiﬁcation of irreducible aﬃne Coxeter group (see [18]), each connectedcomponent of the Coxeter graph of WP is either a single vertex or a 1-simplex
labeled by ∞. The Coxeter group corresponding to a single vertex is Z2 , ande1 . Suppose
the Coxeter group corresponding to a 1-simplex labeled by ∞ is A
the Coxeter graph of WP consists of k isolated vertices and l isolated 1-simplicese1 )l and k + 2l equals the number of facets
labeled by ∞. Then WP ∼= (Z2 )k × (Aof P .
¤
Corollary 2.5. If π1 (RZP ) is virtually nilpotent, then π1 (RZP ) ∼= Zl for somel ≤ r/2 where r is the number of facets of P . In particular, if π1 (RZP ) is ﬁnite,
RZP must be simply connected and so P is 2-neighborly.
Proof. If π1 (RZP ) is virtually nilpotent, so is the Coxeter group WP . Then by
e1 )l for some k, l ≥ 0 with k + 2l = r. So π1 (RZP ) =Lemma 2.4, WP ∼= (Z2 )k × (Ae1 , Ae1 ] ∼[WP , WP ] ∼= Zl since [A= Z. If π1 (RZP ) is ﬁnite, then l = 0, i.e. π1 (RZP )is trivial. And so P is 2-neighborly by Proposition 2.1.
¤
Corollary 2.6. If a small cover M n over an n-dimensional simple convex polytope P n has ﬁnite fundamental group, then P n is 2-neighborly and π1 (M n ) is
isomorphic to (Z2 )r−n , where r is the number of facets of P n .
Proof. Because of (5), π1 (M n ) is ﬁnite if and only if π1 (RZP n ) is ﬁnite. So theclaim follows from Corollary 2.5 and Proposition 2.1.
¤
Example 1. Let ∆j denote a j-simplex. If P n is a product of simplices ∆n1 ×· · · × ∆nm , then the number of facets of P n is n + m and P n is 2-neighborly if
and only if ni ≥ 2 for all 1 ≤ i ≤ m. The real moment-angle manifold RZP n is a
product of spheres S n1 × · · · × S nm , so π1 (RZP n ) ∼= Zl where l is the number of
i’s with ni = 1.
Small covers over ∆n1 × · · · × ∆nm naturally arise as follows. Recall that ageneralized real Bott manifold is the total space Bm of an iterated ﬁber bundle:
π

πm−1

π

π

m
2
1
Bm −→
Bm−1 −→ · · · −→B1 −→B0 = {a point},(7)

where each Bi (1 ≤ i ≤ m) is the projectivization of the Whitney sum of a ﬁnite
collection (at least two) of real line bundles over Bi−1 . We call the sequencein (7) a generalized real Bott tower. We consider Bm as a closed smooth manifold
whose smooth structure is determined by the bundle structures of πi : Bi → Bi−1 ,
i = 1, · · · , m. Suppose the ﬁber of πi : Bi → Bi−1 is a real projective space of7dimension ni . Then it is easy to show that Bm is a small cover over a productof simplices ∆n1 × · · · × ∆nm . Conversely, [8, Remark 6.5] tells us that any small
cover over a product of simplices is homeomorphic to a generalized real Bottmanifold. When n1 = · · · = nm = 1, we call Bm a real Bott manifold (see [19]),
which is a small cover over an m-cube.
Remark 2.7. The number of (weakly) equivariant homeomorphism types of real
Bott manifolds was counted in [5] and [7]. Moreover, it was shown in [19] and [9]that the diﬀeomorphism types of real Bott manifolds are completely determinedby their cohomology rings with Z2 -coeﬃcient. This is called cohomological rigidity of real Bott manifold. This property relates the diﬀeomorphism classiﬁcationof real Bott manifolds with the classiﬁcation of acyclic digraphs (directed graphswith no directed cycles) up to some equivalence (see [9]). But cohomological rigidity does not hold for generalized real Bott manifolds. Indeed, it is shown in [20]that there exist two generalized real Bott manifolds whose cohomology rings with
Z2 -coeﬃcient are isomorphic, but they are not even homotopy equivalent.
Remark 2.8. There are many 2-neighborly simple convex polytopes which are
not product of simplices. For example, the dual P n of an n-dimensional cyclic
polytope (n ≥ 4) is always 2-neighborly. Recall that a cyclic polytope C(k, n)(k > n) is the convex hull of k distinct points on the curve γ(t) = (t, t2 , · · · , tn )in Rn . Then the number of facets of its dual P n has k facets. But when k ≥ 2n , P nadmits no small cover hence can not be a product of simplices (see p.428 of [11]).
It is an interesting problem to ﬁnd out all the 2-neighborly simple convex polytopes that admit small covers. Note that this problem is intimately related to
the Buchstaber invariant of simple polytopes (see [3, section 7.5] and [13]).

3. Small cover, real moment-angle manifold andinfra-solvmanifold
It is shown in [19] that any n-dimensional real Bott manifold admits a ﬂat
Riemannian metric which is invariant under the (Z2 )n -action. Conversely, anysmall cover of dimension n which admits a ﬂat Riemannian metric invariant
under the canonical (Z2 )n -action must be a real Bott manifold. In fact, this isproved for any real toric manifolds in [19, Theorem 1.2], but the same argument
works for small covers. This suggests us to ask the following question.
Question: If a small cover M n of dimension n admits a ﬂat Riemannian metric(not necessarily invariant under the (Z2 )n -action), must M n be diﬀeomorphic toa real Bott manifold? Or equivalently, must M n be a small cover over an n-cube?

8

SHINTARÔ KUROKI† , MIKIYA MASUDA‡ , AND LI YU*

We will see that the answer to this question is yes (Corollary 3.7). In fact, we
will obtain a much stronger result in Corollary 3.7. But ﬁrst let us introduce awell-known notion in combinatorics.
Deﬁnition 3.1 (Flag Complex). A simplicial complex K is called ﬂag if a subset
J of the vertex set of K spans a simplex in K whenever any two vertices in J are
joined by a 1-simplex in K.
Let P be a simple convex polytope of dimension n. For simplicity, we saythat P is ﬂag if the boundary of P is dual to a ﬂag complex. In other words, acollection of facets of P have a common intersection whenever any two of themintersect. Suppose F1 , · · · , Fr are all the facets of P . Then each Fk itself is an(n − 1)-dimensional simple convex polytope whose facets are {Fki := Fk ∩ Fi 6=∅, 1 ≤ i ≤ r}. Let s(Fk ) denote the generator of WP corresponding to the facet
Fk (1 ≤ k ≤ r) of P . Similarly, let s(Fki ) denote the the generator of WFk
corresponding to a facet Fki of Fk .
Lemma 3.2. Suppose a simple convex polytope P is ﬂag and F1 , · · · , Fr are all
the facets of P . If Fk ∩ Fi 6= ∅ for i = i1 , . . . , iq , then Fki1 ∩ · · · ∩ Fkiq 6= ∅ if andonly if Fi1 ∩ · · · ∩ Fiq 6= ∅. So if P is ﬂag, every facet of P must also be ﬂag.
Proof. The “only if part” is trivial. Suppose Fi1 ∩ · · · ∩ Fiq 6= ∅. Then any two of
Fk , Fi1 , . . . , Fiq intersect, so the whole intersection Fk ∩Fi1 ∩· · ·∩Fiq is non-empty
because P is ﬂag, proving the “if part”.
¤
Proposition 3.3. Let P n be a simple convex polytope of dimension n. If P n is
ﬂag and the fundamental group of the real moment-angle manifold RZP n of P n
is virtually solvable, then P n must be an n-cube and so RZP n is homeomorphic
to the n-dimensional torus.
Proof. When n = 2, P n is a polygon. Then our assumption that P n is ﬂag and
π1 (RZP n ) is virtually solvable will force P n to be a 4-gon (see [3, Example 6.40]).
In the rest, we assume n ≥ 3. By (3), we have an exact sequence
1 −→ π1 (RZP n ) −→ WP n −→ (Z2 )r −→ 1.
Let Fk (k = 1, . . . , r) be the facets of P n . Then similarly, we have an exact
sequence for each Fk
1 −→ π1 (RZFk ) −→ WFk −→ (Z2 )rk −→ 1
where rk is the number of facets of Fk . Notice that WFk is generated by s(Fki )with relations s(Fki )2 = 1 and (s(Fki )s(Fkj ))2 = 1 whenever Fki and Fkj intersect.
But Fki and Fkj intersect if and only if Fi and Fj (having non-empty intersection

9with Fk ) intersect by Lemma 3.2. This implies that the group homomorphism
ϕ : WFk → WP n sending each s(Fki ) to s(Fi ) is injective. Moreover, since
π1 (RZP n ) = [WP n , WP n ], π1 (RZFk ) = [WFk , WFk ],so ϕ maps π1 (RZFk ) injectively into π1 (RZP n ). Then π1 (RZFk ) is virtually solvable since so is π1 (RZP n ) by our assumption. In addition, Fk is also ﬂag byLemma 3.2.
By iterating the above arguments, we can show that for any 2-face f of P n ,f is ﬂag and the fundamental group of the real moment-angle manifold RZf isvirtually solvable. We have shown that such an f must be a 4-gon. So any 2-faceof P n is a 4-gon, which implies that P n is an n-cube (see [28, Problem 0.1]). ¤It is shown in [12, Theorem 2.2.5] that a small cover over a simple convex
polytope P is aspherical if and only if P is ﬂag. Similarly, we can prove thefollowing.
Proposition 3.4. The real moment-angle manifold RZP of a simple convex polytope P is aspherical if and only if P is ﬂag.
Proof. Deﬁne M = WP × P/ ∼, where the equivalence relation is deﬁned by(w, x) ∼ (w0 , x0 ) ⇐⇒ x0 = x and w0 w−1 belongs to the subgroup Gx of WP
generated by {s(F ) ; x ∈ F }. If x lies in the relative interior of a codimesion-k
face of P , then the subgroup Gx of WP is isomorphic to (Z2 )k .
It is shown in [11, Lemma 4.4] that M is simply connected. Let ζ : WP × P →M be the quotient map. There is a natural action of WP on M deﬁned by:
w0 · ζ(w, x) = ζ(w0 w, x), w, w0 ∈ WP , x ∈ P.

(8)

The isotropy group of a point ζ(w, x) ∈ M under this WP -action is exactly Gx .
Claim-1: The commutator subgroup [WP , WP ] of WP acts freely on M.
It amounts to prove that [WP , WP ] ∩ Gx = {1} for any point x ∈ P . In fact, it
is easy to see that the abelianization Ab : WP → WPab maps Gx injectively into
WPab . So Gx ∩ ker(Ab) = {1}, proving Claim-1.
Claim-2: The quotient space M/[WP , WP ] is homeomorphic to RZP .
Suppose F1 , · · · , Fr are all the facets of P . For each 1 ≤ i ≤ r, let s(Fi ) be theimage of s(Fi ) under the abelianization Ab : WP → WPab . Then {s(F1 ), . . . , s(Fr )}is a basis of WPab ∼= (Z2 )r . So the quotient M/[WP , WP ] is homeomorphic to thespace obtained by gluing 2r copies of P according to the characteristic function µ
on P where µ(Fi ) = s(Fi ) ∈ (Z2 )r , 1 ≤ i ≤ r. This coincides with the deﬁnition
of RZP (see (2)). So the Claim-2 is proved.

10

SHINTARÔ KUROKI† , MIKIYA MASUDA‡ , AND LI YU*

Therefore, M is a universal covering of RZP . Then RZP is aspherical if andonly if M is contractible. But it is shown in [12, Theorem 2.2.5] that M iscontractible if and only if P is ﬂag. So the proposition is proved.
¤
The following corollary is an immediate consequence of Proposition 3.3 andProposition 3.4.
Corollary 3.5. Let P n be a simple convex polytope of dimension n. If the real
moment-angle manifold RZP n of P n is aspherical with virtually solvable fundamental group, then P n is an n-cube. Therefore, RZP n is homeomorphic to then-dimensional torus if and only if P n is an n-cube.
The typical examples of aspherical manifolds with virtually solvable fundamental group are infra-solvmanifolds. In fact, any compact aspherical manifold with
virtually solvable fundamental group is homeomorphic to an infra-solvmanifold(see [14, Corollary 2.21]). But in general, we can not replace the “homeomorphic”
by “diﬀeomorphic” in this statement.
Deﬁnition 3.6 (Infrahomogeneous Space). Let G be a connected and simply
connected Lie group, K be a maximal compact subgroup of the group Aut(G) of
automorphisms of G, and Γ be a cocompact, discrete subgroup of E(G) = G o K.
If the action of Γ on G is free and [Γ : G ∩ Γ] < ∞, the orbit space Γ\G is called a(compact) infrahomogeneous space modeled on G. If G is solvable (nilpotent),
Γ\G is called an infra-solvmanifold (infra-nilmanifold ). When G = Rn and
K = O(n, R) (the orthogonal group), Γ\G is a compact ﬂat Riemannian manifold.
In the above deﬁnition, the group law of G o K < G o Aut(G) is deﬁned by:(g1 , τ1 ) · (g2 , τ2 ) = (g1 · τ1 (g2 ), τ1 ◦ τ2 ), g1 , g2 ∈ G, τ1 , τ2 ∈ Aut(G).
The action of G o Aut(G) on G is deﬁned by:(g, τ ) · g 0 = g · τ (g 0 ), g, g 0 ∈ G, τ ∈ Aut(G).
By deﬁnition, we have the following hierarchy of notions:Compact ﬂat Riemannian manifolds ⊂ Infra-nilmanifolds ⊂ Infra-solvmanifolds⊂ Compact aspherical manifolds with virtually solvable fundamental group
Infra-nilmanifolds and infra-solvmanifolds also have some Riemannian geometric interpretations as follows. By a theorem of Ruh [25] which is based on thework of Gromov [15], a compact connected manifold M is an infra-nilmanifold if
and only if it is almost ﬂat, which means that M admits a sequence of Riemannian metric {gn } with uniformly bounded sectional curvature so that the sectional
curvatures of (M, gn ) converge uniformly to 0. Similarly for infra-solvmanifolds,
it is shown in [27, Proposition 3.1] that a compact connected topological manifold
M is homeomorphic to an infra-solvmanifolds if and only if M admits a sequence11of Riemannian metric {gn } with uniformly bounded sectional curvature so that(M, gn ) collapses in the Gromov-Hausdorﬀ sense to a ﬂat orbifold.
In addition, it is shown in [2, Corollary 1.5] that compact infra-solvmanifolds
are smoothly rigid, i.e. any two compact infra-solvmanifolds with isomorphic
fundamental groups are diﬀeomorphic. From this fact and Corollary 3.5, we caneasily prove the following.
Corollary 3.7. Suppose P n is an n-dimensional simple convex polytope.
(i) The real moment-angle manifold RZP n of P n is (or homeomorphic to) an
infra-solvmanifold if and only if RZP n is diﬀeomorphic (or homeomorphic) to the n-dimensional ﬂat torus.
(ii) A small cover M n over P n is (or homeomorphic to) an infra-solvmanifold
if and only if M n is diﬀeomorphic (or homeomorphic) to a real Bott manifold. In particular, if M n admits a ﬂat Riemannian metric, then M n is
diﬀeomorphic to a real Bott manifold.
Proof. The statement (i) follows easily from Corollary 3.5 and the smooth rigidityof infra-solvmanifolds. As for (ii), note that RZP n is a ﬁnite-sheeted covering
space of any small cover (if exists) over P n . So if a small cover over P n is
homeomorphic to an infra-solvmanifold, RZP n must be aspherical with virtuallysolvable fundamental group. Then (ii) also follows from Corollary 3.5 and thesmooth rigidity of infra-solvmanifolds.
¤
Question: Let P and Q be two simple convex polytopes. If RZP and RZQ
are homeomorphic, what can we conclude about the relationship between the
combinatorial properties of P and Q?
By Proposition 3.4, if RZP is homeomorphic to RZQ , then P and Q are either
both ﬂag or they are both non-ﬂag. It is interesting to see more answers to thisquestion.

4. Flag simple polytopes and Real Bott manifolds
In this section, we will get several diﬀerent ways to describe a small cover thatis homeomorphic to a real Bott manifolds. As we know from the deﬁnition, any
real Bott manifold of dimension n is a small cover over an n-cube. It is clear thatan n-cube is a ﬂag simple polytope with 2n facets. Conversely, we can show thefollowing.
Proposition 4.1. Let P n be a ﬂag simple polytope of dimension n. Then P n has
at least 2n facets, and if P n has exactly 2n facets, P must be an n-cube.

SHINTARÔ KUROKI† , MIKIYA MASUDA‡ , AND LI YU*

12

Proof. This is probably known to many people. But since we do not know theliterature, we shall give a proof here for the sake of completeness of the paper.
nSince P n is simple and of dimension n, there are n facets in PT
whose intersection is a vertex. We denote them by F1 , . . . T, Fn and the vertex ni=1 Fi by v. For
each j ∈ [n] := {1, . . . , n}, the intersection i6=j Fi is an edge of P n which has v
nasTan endpoint. Therefore, there is a unique facet
T of P , denoted Gj , such that
( i6=j Fi ) ∩ Gj is the other endpoint of the edge i6=j Fi diﬀerent from v.
We claim that Gj ’s must be mutually distinct. Indeed, if Gp = Gq for somep 6= q, this implies that any two
T of the n + 1 facets F1 , . . . , Fn , Gp = Gq have nonempty intersection because ( i6=j Fi ) ∩ Gj is non-empty for any j. Therefore theintersection of the n+1 facets must be non-empty since P n is ﬂag. However, this is
impossible because P n is simple and of dimension n. Therefore Gj ’s are mutually
distinct and hence P n has at least 2n facets, proving the former statement of the
proposition.
nHereafter we assume that P n has exactly
T Then the facets of P are
Tn 2n facets.
exactly F1 , . . . , Fn , G1 , . . . , Gn . Since j=1 Fj and ( i6=j Fi ) ∩ Gj are both nonempty, any two of the n + 1 facets F1 , . . . , Fn , Gj have non-empty intersection if
Fj ∩ Gj 6= ∅. However, this is impossible by the same reason as above. Therefore
Fj ∩ Gj = ∅ for any j ∈ [n].
We shall prove that\\
( Fi ) ∩ ( Gj ) 6= ∅ for any subset J of [n]
i∈J
/

(9)

(10)

j∈J

T
by induction on the cardinality |J| of J. Since ( i6=j Fi ) ∩ Gj is a vertex of P n
by the choice of Gj , (10) holds when |J| = 1. Suppose that (10) holds for J with
|J| = k − 1. Let J be a subset of [n] with |J| = k. Without loss of generality,
we may assume that J = {1, 2, . . . , k}. By the induction assumption we have
T
Tn
Tk−1
T
n
( ni=k Fi ) ∩ ( k−1
j=1 Gj ) 6= ∅. Since P is simple, ( i=k Fi ) ∩j=1 Gj ) is a vertex
Tk−1
Tn
n
of P , denoted w, and ( i=k+1 Fi ) ∩ ( j=1 Gj ) is an edge of P n which contains
the vertex w. Therefore, there is a unique facet H of P n such that
(

n
\

i=k+1

k−1
\

Fi ) ∩ (Gj ) ∩ H is a vertex of P n diﬀerent from w.

(11)

j=1

We claim that H = Gk . In fact, since the intersection in (11) is a vertex, H mustbe either Fp for 1 ≤ p ≤ k or Gq for k ≤ q ≤ n. However, the intersection in (11)is empty unless H = Fk or Gk by (9). Moreover, H 6= Fk because the intersectionin (11) is diﬀerent from w. Therefore we can conclude H = Gk and this showsthat (10) holds for J with |J| = k, completing the induction step.

13

Let P ∗ be the simplicial polytope dual to P n . Then the facts (9) and (10)show that the boundary complex ∂P ∗ is isomorphic to the boundary complex ofa crosspolytope C of dimension n, which is isomorphic to the n-fold join of S 0 .
Therefore the simplicial polytopes P ∗ and C are isomorphic combinatorially andhence so are their duals P n and C ∗ . Since C ∗ is an n-cube, this proves the latterstatement of the proposition.
¤
Remark 4.2. The above argument shows that if the geometrical realization of aﬂag simplicial complex K is a pseudomanifold of dimension n−1, then the numberof vertices of K is at least 2n; and if it is exactly 2n, then K is isomorphic to theboundary complex of the crosspolytope of dimension n.
Combining all our previous discussions, we get several descriptions of a small
cover that is homeomorphic to a real Bott manifold as follows.
Theorem 4.3. Suppose M n is a small cover over a simple convex polytope P nof dimension n. Let b1 (M n ; Z2 ) denote the ﬁrst Betti number of M with Z2 coeﬃcient. Then the following statements are equivalent.
• M n is homeomorphic to a real Bott manifold.
• M n is aspherical and b1 (M n ; Z2 ) ≤ n.
• P n is ﬂag and the number of facets of P n is ≤ 2n.
• P n is an n-cube.
Proof. Suppose P n has r facets. It is known that r = b1 (M n ; Z2 ) + n (see [11]).
Then b1 (M n ; Z2 ) ≤ n ⇐⇒ r ≤ 2n. In addition, M is aspherical ⇐⇒ P n is ﬂag
by [12, Theorem 2.2.5]. Then this proposition follows from Proposition 4.1. ¤5. Riemannian metrics on small covers and real moment-angle
manifolds
Geometric structures on small covers were ﬁrst discussed in [11, Example
1.21] for 3-dimensional cases. Later, Davis-Januszkiewcz-Scott [12] systematically studied the piecewise Euclidean structure on any small cover. A very niceresult obtained in [12, Proposition 2.2.3] says that the natural piecewise Euclidean
cubical metric on a small cover over a simple polytope P is nonpositively curved
if and only if the boundary of P is dual to a ﬂag complex (this is also equivalent
to saying that the small cover is aspherical).
In this section, we will study Riemannian metrics on small covers and real
moment-angle manifolds in any dimension with certain conditions on the Ricci or
sectional curvature. By our discussion of the fundamental group of small covers
and real moment-angle manifolds in section 2, we will see that these curvature

14

SHINTARÔ KUROKI† , MIKIYA MASUDA‡ , AND LI YU*

conditions put very strong restrictions on the topology of the corresponding small
covers and real moment-angle manifolds and the combinatorics of the underlying
simple polytopes.
A Riemannian manifold is called positively (nonnegatively, negatively) curved
if its sectional curvature is everywhere positive (nonnegative, negative). It isclear that a positively (nonnegatively, negatively) curved Riemannian manifold
has positive (nonnegative, negative) Ricci curvature.
5.1. Positive curvature. By a classical theorem of Bonnet and Myers, any compact Riemannian manifold with positive Ricci curvature must have ﬁnite fundamental group (see Chapter 6 of [21]). Then by Corollary 2.5 and Corollary 2.6,
we have the following corollary.
Corollary 5.1. Let P n be an n-dimensional simple convex polytope. If the real
moment-angle manifold RZP n admits a Riemannian metric with positive Ricci
curvature, then P n must be 2-neighborly and RZP n is simply connected. Similarly,
if a small cover M n over P n admits a Riemannian metric with positive Ricci
curvature, then P n must be 2-neighborly and the fundamental group of M n isisomorphic to (Z2 )r−n where r is the number of facets of P n .
The following is a well-known fact on positively curved Riemannian manifolds(see Chapter 6 of [21]).
Theorem 5.2 (Synge 1936). Let M be a compact Riemannian manifold withpositive sectional curvature.
(i) If M is even-dimensional and orientable, then M is simply connected.
(ii) If M is odd-dimensional, then M is orientable.
Notice that a small cover is never simply connected (this is an easy consequenceof (4)). So by Synge’s theorem, we can conclude the following.
Corollary 5.3. If an even (odd) dimensional small cover admits a positively
curved Riemannian metric, then it must be non-orientable (orientable).
It is well known that a simply connected closed smooth n-manifold which admits a Riemannian metric with positive constant sectional curvature is diﬀeomorphic to the standard sphere S n in Rn+1 .
Theorem 5.4. Let P n be an n-dimensional simple convex polytope. If the real
moment-angle manifold RZP n (or a small cover M n ) of P n admits a Riemannian
metric with positive constant sectional curvature, then RZP n (or M n ) is diﬀeomorphic to the standard sphere S n (or the real projective space RPn ).

15

Proof. If RZP n admits a Riemannian metric with positive constant sectional curvature, then it is simply connected by Corollary 5.1. So it is diﬀeomorphic to S n .
If a small cover M n over P n admits a Riemannian metric with positive constantsectional curvature, so does RZP n . Then RZP n is diﬀeomorphic to S n and RZP n
is a universal covering space of M n . In addition, Corollary 5.1 tells us that thefundamental group of M n is isomorphic to (Z2 )r−n where r is the number of facetsof P n . So M n is the quotient space of RZP n ∼= S n by a free (Z2 )r−n -action. But
by the classical Smith’s theory (see [26]), we must have r − n ≤ 1. So P n must
be an n-dimensional simplex and then M n is the n-dimensional real projective
space RPn .
¤The following geometric problem for small covers should be interesting to study.
Problem-1: ﬁnd out all the small covers (or real moment-angle manifolds) which
admit Riemannian metrics with positive sectional or Ricci curvature in each dimension.
It is well known that the only 2-neighborly simple polytopes in dimension 2 and3 are the 2-simplex and 3-simplex. So by Corollary 5.1, the only small covers indimension 2 and 3 that admit Riemannian metrics with positive Ricci curvature
are RP2 and RP3 . But in dimension ≥ 4, the answer to Problem-1 is not so clear.
In particular, it is interesting to see if there exists any small cover which admits apositively curved Riemannian metric but not homeomorphic to a real projective
space RPn .
5.2. Nonnegative curvature. By Cheeger-Gromoll splitting theorem (see [6]),the fundamental group of any compact Riemannian manifold with nonnegative
Ricci curvature is virtually abelian. This fact leads to the following description of
the fundamental group of the real moment-angle manifold or a small cover that
admits a Riemannian metric with nonnegative Ricci curvature.
Proposition 5.5. Let P n be an n-dimensional simple convex polytope with r
facets. If the real moment-angle manifold RZP n of P n admits a Riemannian
metric with nonnegative Ricci curvature, then π1 (RZP n ) is isomorphic to Zl forsome l ≤ r/2. Similarly, if a small cover M n over P n admits a Riemannian
metric with nonnegative Ricci curvature, then there is an exact sequence
1 −→ Zl −→ π1 (M n ) −→ (Z2 )r−n −→ 1
where l ≤ r/2.
Proof. If RZP n admits a Riemannian metric with nonnegative Ricci curvature,then π1 (RZP n ) is virtually abelian as remarked above. Therefore, the former
statement in the proposition follows from Corollary 2.5. If a small cover M n overP n admits a Riemannian metric with nonnegative Ricci curvature, then so does

16

SHINTARÔ KUROKI† , MIKIYA MASUDA‡ , AND LI YU*

RZP n because RZP n is a ﬁnite cover of M n . Therefore, the latter statement in
the proposition follows from (5) and the former statement.
¤
Example 2. The real moment-angle manifold over a product of simplices ∆n1 ×· · ·×∆nm is a product of standard spheres S n1 ×· · ·×S nm =: S. In the product, leteach sphere S ni ⊂ Rni +1 be equipped with the standard Riemannian metric whose
isometry group is the orthogonal group O(ni + 1, R). Then S is a nonnegatively
curved Riemannian manifold with respect to the product metric.
A generalized real Bott manifold Bm discussed in Example 1 is a small cover
over ∆n1 × · · · × ∆nm . It is known that Bm is the quotient of S by a free (Z2 )m action on it (see [8, Proposition 6.2]) and one can easily see that the (Z2 )m -action
preserves the product metric on S. So the quotient space of this free (Z2 )m -action,that is Bm , inherits a nonnegatively curved Riemannian metric from S.
Remark 5.6. It is possible that Bm is the total space of several diﬀerent generalized real Bott towers. Therefore, we will get several diﬀerent nonnegatively
curved Riemannian metrics on Bm which may not be isometric.
Problem-2: ﬁnd out all the small covers which admit Riemannian metrics with
nonnegative sectional or Ricci curvature in each dimension.
In dimension 2 and dimension 3, the small covers which admit Riemannian
metric with nonnegative Ricci curvature are exactly the generalized real Bottmanifolds (this follows from the classiﬁcation of 3-dimensional compact Riemannian manifolds with nonnegative Ricci curvature in [17]). So by Example 2, we
can conclude that all the nonnegatively curved small covers in dimension 2 anddimension 3 are exactly generalized real Bott manifolds. In dimension ≥ 4, theanswer to Problem-2 is not so easy. In particular, it is interesting to see if there
exists any small cover which admits a nonnegatively curved Riemannian metric
but not homeomorphic to any generalized real Bott manifold.
5.3. Negative curvature. A theorem due to Preissmann [22] says that any
abelian subgroup of the fundamental group of a negatively curved compact Riemannian manifold is inﬁnite cyclic (also see [4, section 9.3]). Using this fact, we
can easily show the following.
Proposition 5.7. If the real moment-angle manifold RZP (or a small cover) of asimple polytope P admits a negatively curved Riemannian metric, then no 2-faceof P can be a 3-gon or a 4-gon.
Proof. If RZP admits a negatively curved Riemannian metric, then it follows from
Cartan-Hadamard theorem that RZP is aspherical. So the simple polytope P is
ﬂag by Proposition 3.4. Then by Lemma 3.2, any 2-face f of P must be ﬂag. So f

17

can not be a 3-gon. In addition, since P is ﬂag, the proof of Proposition 3.3 implies
that there is an injective group homomorphism from π1 (RZf ) into π1 (RZP ). Thisimplies that f can not be a 4-gon, because otherwise π1 (RZf ) ∼= Z ⊕ Z and so
π1 (RZP ) contains an abelian subgroup Z ⊕ Z which contradicts the Preissmann’s
theorem mentioned above. Finally, if a small cover over P admits a negatively
curved Riemannian metric, then so does RZP .
¤Remark 5.8. It was shown in [11, Example 1.21] that a small cover over a
3-dimensional simple convex polytope P admits a hyperbolic structure (i.e. Riemannian metric with negative constant sectional curvature) if and only if P has
no 3-gon or 4-gon facets. Its proof uses the Andreev’s theorem [1] on when a3-dimensional simple polytope can be realized as a right-angled polytope in thehyperbolic 3-space. But since there is no analogue of Andreev’s theorem in higher
dimensions, it is not clear how to judge the existence of hyperbolic structures on
small covers (or real moment-angle manifolds) in general.

Acknowledgements
The authors want to thank Y. Kamishima and J. B. Lee for helpful comments
on infra-solvmanifolds.

References[1] E. M. Andreev, On convex polyhedra in Lobačevskiı̆ spaces (English Translation),
Math. USSR Sbornik, 10 (1970), 413–440.
[2] O. Baues, Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups,
Topology 43 (2004), no. 4, 903–924.
[3] V. M. Buchstaber and T.E. Panov, Torus actions and their applications in topology andcombinatorics, University Lecture Series, 24. American Mathematical Society, Providence,
RI, 2002.
[4] D. Burago, Y. Burago and S. Ivanov, A course in metric geometry, Graduate Studies inMathematics, 33. American Mathematical Society, Providence, RI, 2001.
[5] M. Cai, X. Chen and Z. Lü, Small covers over prisms, Topology Appl. 154 (2007), no. 11,
2228–2234.
[6] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Diﬀerential Geom. 6 (1971/72), 119–128.
[7] S. Choi, The number of small covers over cubes, Algebr. Geom. Topol. 8 (2008), no. 4,
2391–2399.
[8] S. Choi, M. Masuda and D. Y. Suh, Quasitoric manifolds over a product of simplices,
Osaka J. Math. vol. 47 (2010), 109–129.
[9] S. Choi, M. Masuda and S. Oum, Classiﬁcation of real Bott manifolds and acyclic digraphs,
preprint, arXiv:1006.4658 (2010).

18

SHINTARÔ KUROKI† , MIKIYA MASUDA‡ , AND LI YU*[10] H. S. M. Coxeter, The complete enumeration of ﬁnite groups of the form ri2 = (ri rj )kij = 1,
J. London Math. Soc. 10 (1935), 21–25[11] M. W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions,
Duke Math. J. 62 (1991), no.2, 417–451.
[12] M. Davis, T. Januszkiewicz and R. Scott, Nonpositive curvature of blow-ups, Selecta
Math. 4 (1998), no. 4, 491–547.
[13] N. Yu. Erokhovets, Buchstaber invariant of simple polytopes, Uspekhi Mat. Nauk, 63(2008), No. 5(383), 187–188; translation in Russian Math. Surveys 63 (2008), no. 5, 962–
964[14] F. T. Farrell and L. E. Jones, Classical aspherical manifolds, CBMS Regional Conference
Series in Mathematics, 75, American Mathematical Society, Providence, RI, 1990.
[15] M. Gromov, Almost ﬂat manifolds, J. Diﬀerential Geom. 13 (1978), no. 2, 231–241.
[16] L. C. Grove and C. T. Benson, Finite Reﬂection Groups, Graduate texts in mathematics,
vol. 99, Springer, (1985).
[17] R. S. Hamilton, Four-manifolds with positive curvature operator, J. Diﬀerential Geom. 24(1986), no. 2, 153–179.
[18] J. E. Humphreys, Reﬂection groups and Coxeter groups, Cambridge Studies in Advanced
Mathematics, 29. Cambridge University Press, Cambridge, 1990.
[19] Y. Kamishima and M. Masuda, Cohomological rigidity of real Bott manifolds, Algebr.
Geom. Topol. 9 (2009), 2479–2502.
[20] M. Masuda, Cohomological non-rigidity of generalized real Bott manifolds of height 2,
Proceedings of the Steklov Institute of Mathematics, 2010, Vol. 268, 242–247.
[21] P. Petersen, Riemannian geometry, Second edition. Graduate Texts in Mathematics, 171.
Springer, New York, 2006.
[22] A. Preissmann, Quelques propriétés globales des espaces de Riemann, Comment. Math. Helvetici 15 (1943), 175–216.
[23] D. Qi, On irreducible, inﬁnite, nonaﬃne Coxeter groups, Fund. Math. 193 (2007), no. 1,
79–93.
[24] D. Radcliﬀe, Rigidity of graph products of groups, Algebr. Geom. Topol. 3 (2003), 1079–
1088.
[25] E. A. Ruh, Almost ﬂat manifolds, J. Diﬀerential Geom. 17 (1982) no. 1, 1–14.
[26] P. A. Smith, Permutable periodic transformations, Proc. Nat. Acad. Sci. 30 (1944), 105–108.
[27] W. Tuschmann, Collapsing, solvmanifolds and infrahomogeneous spaces, Diﬀer. Geom.
Appl. 7 (1997), 251–264.
[28] G. M. Ziegler, Lectures on polytopes. Graduate Texts in Mathematics, 152 (revised ﬁrst
edition), Springer-Verlag, New York, 1998.

19†

Osaka City University Advanced Mathematical Institute (OCAMI), Sugimoto, Sumiyoshi-Ku, Osaka, 558-8585, Japan
E-mail address: kuroki@kaist.ac.kr‡

Department of Mathematics, Osaka City University, Sugimoto, Sumiyoshi-Ku,
Osaka, 558-8585, Japan
E-mail address: masuda@sci.osaka-cu.ac.jp*Department of Mathematics and IMS, Nanjing University, Nanjing, 210093,
P.R.China
andDepartment of Mathematics, Osaka City University, Sugimoto, Sumiyoshi-Ku,
Osaka, 558-8585, Japan
E-mail address: yuli@nju.edu.cn