O n t h e L a t t i c e S p a c e F o r m s

by JOSEPH A. WOLF1), Princeton (N. J.)

1. Introduction

In a recent paper [1] we gave a global classification for homogeneous pseudo- RIE~L~N~ian manifolds of constant nonzero curvature. We described certain families of such manifolds, and proved [ 1, Theorem 12] that a complete connected homogeneous pseudo-RIEMANNian manifold of constant positive curvature is isometric to an element of one of those families; for negative curvature, one replaces the metric by its negative. The most complicated of those families are the families .~, .Z?a, ~?a, .L~e and .z5~8, the families of so-called lattice space forms. The purpose of this note is to improve their description, examine their interrelations, and correct a minor error in their enumeration.

We first give a criterion (Theorem 3.1) for two manifolds, each isometric to an element of .s or to an element of an ~f?~, to be isometric to each other, i.e., to be equal as elements of .~ or .L~,~. We then decompose ~ into a disjoint union of subsets .z~(-}-, h, n), -L~( - , h, n) and . ~ ( • h, n), and decompose ~/7,, into a disjoint union of subsets A?~ (h, n) ; here (h, n) refers

h to the dimension n and the signature of metric ds 2 = - X dx~ ~- } dx~ of

1 h + l the manifolds under consideration. Our first main result (Theorem 4) is that A?(-~, h, n), - s h, n) and ~A~4(h, n) are each in one to one correspondence in a natural fashion with the quotient of the upper half plane by the modular group, and that .~ ( - - , h, n) is in one to one correspondence with the quotient of the upper half plane by a maximal parabolic (= unipotent) (mod 2) subgroup of the modular group. The other main result (Theorem 5.2) is that .L~3 (h, n), ~6(h, n) and .L~6(h, n) each has just one element.

2. Preliminaries

2. 1. Let R~ +1 be an (n -}- 1)-dimensional real vectorspace with a symmetric bilinear form Q of signature (h - 's, {n - h -k 1) q- 's), and let 0a(n q- 1) denote the orthogonal group of Q. A n orthonormal basis of R~ +1 is an ordered basis {vl , . . . ,v~+l} such that Q(vi, v~)=(~i~ if i :>h,---- - ~ij if i g h . A skew basis of R~ +1 is, ff 2h _~ n -}- 1, an ordered basis

{h . . . . . h ; v~+l . . . . . v~; e~ , . . . , e~} 1) The author was supported b y a National Science F o u n d a t i o n fellowship while preparing

th i s paper .

JOSEPH A . WOLF O n t h e L a t t i c e S p a c e F o r m s 103

where k ~ n - - h ~ - 1, Q ( / ~ , e ~ . ) ~ 2 ~ i ~ - - 2Q(v~,v~) and Q(f~,f~) =- Q(e~, ej) -~ Q(/t , v~) : Q(v~, e~) -= o. A similar definition holds if 2h g n ~ 1. There is a one to one correspondence between or thonormal bases and skew bases which sends an or thonormal basis {v~} to

(Vh_bl - - V I , �9 �9 �9 , Vh-}-k - - V k ; Vk-t- 1 . . . . . V h ; V h + l "-~ V 1 . . . . . Vh+lr - ~ V k }

when 2 h ~ _ n ~ 1. I f A is a linear t ransformat ion of R~ +1 and fl is a basis of R~ +1, then A~

will denote the matrix of A relative to fl. J~ will denote the matrix ( _ 1 ~ ) ,

and, if p is even, J~, will denote the matr ix " . of order p. I will

J2 denote the ident i ty t ransformation, and Iq will denote the iden t i ty mat r ix of order q. Finally, if p is even, R(O)~, will denote the rota t ion mat r ix

cos (2~0) I~ -t- sin (2~0) J~.

2 .2 . Now suppose tha t 2 h > n and / ~ = n - ~ 1 - h ~ 0 (mod 4), and ehoose a skew basis fl of R~ +1. Given an ant i symmetr ie nonsingular k • k

mat r ix d, we have t(d)eOh(n + 1) defined by t(d)~ = Ih_ ~ . Now

, 0 ~

suppose tha t dJ~ + Jkd = 0; this requires /c ~ 0 (mod 4). We identify the

complex number field e with the matrices (aI~ + bJk) by a + ~ lb -+ (aI~ -Jr- bJk), and observe tha t ud is an an t i symmetr ic nonsingular mat r ix for every 0 C u e G . Le t s be one of the signs ~ - o r - - or • Given u E C , u r R, we define ~ (u , s) to be the subgroup of 0h(n -~ 1) generated by t(d) and et(ud).

Now suppose, in addition, t ha t h is even, i.e., t ha t h - /c is even. For every nonzero integer m, we have r(m)eOh(n-~ 1) defined by r(m)~ : = R(1/m)~+l. Given u e G, u ~ R, let L(u) be the addit ive latt ice {1, u}

in C. I f m ~ - 3 , 4 , 6 or 8, and if e x p ( 4 ~ Y ~-~ 1/m) L(u) : L ( u ) , then we define ~ ( u ) to be the subgroup of Oh(n ~ 1) generated by r(m), t(d) and t(ud).

In [1, w167 9 .3 -9 .4 ] , it was s ta ted but not proved tha t ~ (u , s) and ~ ( u ) are well defined, up to conjugacy in Oh(n ~ 1). This will follow from w 2.4.

2 .3 . S~ is the quadric Q(x, x) ~- 1 in R~ +1, together with the induced s t ructure as a pseudo-RIEMA~ian manifold with metric of signature

h _ +

1 h + l

104 JOSEPH A. WOLF

0 h (n -~ 1) is the full group of isometrics of S~, and acts transitively on the points of 82. S~ is thus homogeneous, and it has constant sectional curvature ~- 1. We refer to [1, w 4] for details.

We have proved [1, w167 9.3-9.4] that S~/~(u, e) and S~/~(u) inherit from S~ the structure of homogeneous pseudo-Rx~MAN~ian manifold of constant curvature + 1. The manifolds S~/~ (u, e) form a family .L', where we iden- tify isometric manifolds. The manifolds S~/~m(u) form a family ./?m, again identifying isometric manifolds. ~/? and ~?~ are the families of lattice space forms.

To study the lattice space forms, we must know when elements of the fam- ilies ,L ~ and ,s arc isometric. An element of one family cannot be isometric to an element of another family, so we may study the families separately. As 0n(n + 1) is the full group of isometrics of S~, it is easily seen that a manifold S'~/~(u, e) is isometric to S'~/~(u' , e') if and only if n -- n' , h --~ h' and ~(u, e) is conjugate to ~(u ' , e') in On(n Jr 1), and that S~/~(u) is iso- metric to S~: /~(u ' ) if and only if n --~ n' , h =- h' and ~ ( u ) is conjugate to ~ ( u ' ) in 0n(n ~- 1).

2.4. In order to study conjugacy of groups ~(u, e) and ~ ( u ) , we will need

Lemma 2.4. Let d and d' be antisymmetric nonsingular real ]c • Ic matrices which anticommute with J~. Then there is a real nonsingular k • ]c matrix gl which commutes with J~, and such that gldtg~ -= d'.

Proo]. Let ~ ---- {s 1 . . . . . sk} be an orthonormal basis of V --~ R0 k and let J be given by J r ---- Jk. V carries the structure of a complex vectorspace of

dimension k/2 where J is scalar multiplication by ~ 1; now

r ' = {s~, s~ . . . . . s~_~}

is a C-basis for V. We define a real-linear transformation V of V by ~](s~) =- ( - 1)~+ls~. Now V, d and d' each anticommutes with J ; thus d~ and

d'~ commute with J , i.e., d~ and d'~ are (~-linear. Let Q' be the sym- metric C-bilinear form on V defined by Q' (s~+l, s~.+~) ~- di~.; it is easily checked that Q' (x, d r y ) ~ Q' (d~ x, y) = 0 = Q' (x, d' ~y) ~ Q' (d' ~] x, y) for eve- ry x, y E V; thus (d~?) v, and (d'~)v, are nonsingular antisymmetric complex matrices. I f follows that there is a nonsingular complex matrix h~ of order k/2

such tha t hl(d~)r,~h ~ -~ (d'~7)~,. I f h is defined by h r, = ha, then (~h~-~)~,----h~. I t follows that d' -= hrd~h~. Define g~ -= h r. Q.E.D.

I t is clear that the choice of the original skew basis fl of R~ +~ does not effect the eonjugacy class of 2(u, e) or 2,~(u) in 0a(n ~- 1). The above

On the Lattice Space Forms 105

lemma shows that the choice of d does not effect the conjugacy class. Thus the groups :E(u, e) and ~ ( u ) are well defined up to conjugacy in Oa(n ~- 1).

3. The Equivalence Theorem

3.1. Let II denote the upper half plane, consisting of all complex numbers with positive imaginary part. The modular group is the group F consisting of

�9 a u ~ - b a b all transformations y : u - > y ( u ) = cu-u-~ d of H where [ t is an inte- (:b\/ \c d/ gral matrix of determinant @ 1. We write y = 4- d}. The meal~_2 parabolics.~

modulargroup F ' is the subgroup of F consisting of all , : 4-(7 ; ) s u c h \( ,

t h a t \C ~ (meal 2), i ,e �9 s u c h t h a t c is e v e n b u t a a n d d a r e

odd. F ' is maximal among the subgroups of F with all elements congruent mod 2 to parabolic (= unipotent) transformations of It . F ' has a subgroup of index 2, the "modular group of level two" given by b -- 0 (mod 2), which is well known.

Convention. Observe that ~ ( u , e ) = ~ ( - u ,e) and ~m(u) = ~ m ( - u). As u has nonzero imaginary part, we may replace it by -- u if necessary�9 We will now always assume u EII when we speak of ~ (u , e) or ~m(u).

The Equivalence Theorem is:

Theorem 3.1. Suppose that 2h > n and k = n -k 1 - h = O (mod 4), and let u, u' � 9 Then ~ ( u , e) and ~ ( u ' , e') are conjugate subgroups o/ O h(n + 1), i / a n d only i[ (1) e = e',(2) there is an element y �9 F such that y(u) = u ' , and (3) ~ �9 F ' i[ e = - - . Suppose/urther that h - - 0 (rood 2), that m = 3 ,4 , 6

or 8, and that e x p ( 4 ~ V - ~ / m ) L(u) = L(u) and exp(4~V ~- 1/m) L(u ' ) = = L ( u ' ) where L(v) denotes the additive lattice {1, v} in C. Then ~m(u) and 2,~(u') are conjugate subgroups o] Oh(n ~- 1), i / and only i/ y(u) = u' [or

8ome y e I ' . The rest of w 3 is devoted to the proof of this theorem.

3.2. Let u , u ' � 9 let L(u) = {1,u} and L(u ' ) = {1,u'} be the corre- sponding additive lattices in C, and suppose that ~(u) ---- u' where y =

= 4 - ( ; ; ) � 9 Let o ~ = ( c u ~ - p ) - l , nonzero e l e m e n t o f C. Then k

= ~-- = ~ {cu -~- p, a u Jr b} = ~ L ( u ) . ' c u §

Now a is represented by sIk -~- tJ~ under 1 -->I~ and ] / /~ 1 -+Je , and, as d is an antisymmetric nonsingular k • k real matrix which anticommutes with Jk, the same is true for d r = ~d. By Lemma 2.4, there is a nonsingular

106 Jos~Pg A. WOLF

real mat r ix gl of degree k, which commutes with Jk, such tha t g~dtgl = o~d.

Let g ~ 0h(n + 1), g~ = Ih-~ . Then g t ( w d ) g -1 = t(o~wd) = t ( w ~ d ) 0 tg~l

for every w e C. This proves tha t g ~ ( u , + ) g-1 _~ P~(u', + ), g ~ ( u , :]:) g-1 = - - ~ ( u ' , ~ ) , and, i f h ---- 0 (rood 2), gP~m(u)g -1 = ~m(u ' ) .

Now suppose tha t ? e F ' . As a and p are odd, and c is even, it follows that , given integers s and v, v is odd i / and only i / v p - - s c is odd. Now u ' - - - o ~ ( a u + b ) and 1 - - - - ~ ( c u + p ) ; thus u o ~ - ~ p u ' - - b and oc-~ - - c u ' + a . Given integers s and v, we have

g . t ( d )8 ( - t ( u d ) ) v . g - ' - ~ ( - - 1 )V .g t ( ( s + u v ) d ) g-1 = ( _ 1)".t((8 + u v ) ~ d ) =

= ( - 1) v. t ( ( a s - s c u ' ) d + ( v p u ' - vb )d) = = ( - 1) v. t ( ( a s - vb )d ) �9 t ( ( v p - s c ) u ' d ) = t(d)a*-vb( - t ( u ' d ) ) v~-Sc

because v - - v p -- sc (mod 2). Thus gP~(u, - ) g - l = P~(u', - ) . This proves the sufficiency of our conjugacy conditions.

3.3. Let ~(u) be a subgroup . !~(u,s) or ~m(u) of Oh(n + 1), and let s be another subgroup ~ ( u ' , d ) or s of the same type. Suppose tha t there exists an element g e Oh(n -~ 1) such tha t gP~(u)g -1 ~-- ~ ( u ' ) .

The various possibilities for ~ ( u , e) are character ized as follows. E v e r y eigenvalue of every element of ~ (u , + ) is equal to + 1. ~ (u , + ) contains -- I . ~ (u, --) has elements with eigenvalues equal to - 1, bu t it does not. contain -- I . Thus gP~(u, s) g-1 _~ ~ ( u ' , d) implies s ---- d . Again looking at eigenvalues, we see t ha t g~(u , + ) g-1 = ~(u , , + ) , whether ~(u) is

(u, e) or ~m(u). We will prove t ha t u ' = ~ (u) for some ~ s F , as a consequence of

g~(u, +)g-~ = 2~(u', +).

For g must preserve the nullspace of t(d) -- I . In our skew basis fl of R] +1,

it follows tha t g~ = g4 g5 , where g e 0h(n + 1) implies g6 = ~g~l.

0 g~ Thus w e C gives g t ( w d ) g-~ = t(gl wd~g~), where C is identified with the R-linear combinations of I~ and J~. As u and I~ span C over R, it follows tha t g~J = 4- Jg~ and g~d~g~ = o~d for 0 :/: ~ e C. We m a y replace gl by gl . d i a g { 1 , - 1 ; . . . ; 1 , - 1} i f necessary, and assume g ~ J = Jg~. Thus g . t ( d ) ' t ( u d ) ' g - ~ - ~ t ( a d ) ' t ( ~ u d ) ~ for any integers s and v. I t follows tha t the lattices L ( u ) = {1,u} and L ( u ' ) = ( 1 , u ' } in C are re la ted b y L ( u ~) = o~L(u), i .e . , t h a t {1, u ~} = { ~ , a u } . As u , u ~ H , L ( u ) and

On the Lattice Space F o r m s 107

L(u ' ) carry the same orientation; thus u' = ao~u § boc and 1 = co~u § po~

where (a ~ ) i s an integral matr ix of determinant § 1. Now define /

Y /

y = "A= e F , and observe tha t u' . . . . . 1 cocu§162 c u § ?(U).

Except for the groups s (u, - ) and s (u', - ) , this proves the necessity of our conjugaey conditions.

Suppose tha t g 2 ( u , --) g-1 = s - ) . We must prove 7 r

g . t(d)s( - t(ud))Vg - ' = ( - 1Ft (a(s § vu)d) = = ( - 1) v- t ( (as -- vb)d) �9 t ( (vp -- sc )u 'd )

as at the end o fw 3.2. Thus (-- 1) v --~ (-- 1) ~p-'r for any integers s and v. I t follows tha t p is odd and c is even. Thus l = a p - - b c - - - a (rood 2)

tha t a is odd. Now ( : : ) - ~ ( 0 ~ ) ( m o d 2), proving ~ / ~ ' . This ~ x

proves \ - - ~[-/ \ l

completes the proof of the necessity of the conjugacy conditions. Theorem 3.1 is now proved.

3.4. In [1, w 9.3], I made the error of stating tha t 2 (u , e) is conjugate to s s), if and on]y i f the lattices L(u) and L(u ' ) are equal. F rom the proof of Theorem 3.1, it is clear tha t this condition is too strong. I f s ---- § or e ---- • then the correct condition is t ha t L(u ' ) ~- o~L(u) where 0 ~ a ~ C.

A similar error [1, w 9.4] was made with respect to the conjugaey of ~ ( u ) and s Here the correct condition on the lattices is L ( u ' ) - ~ o~L(u) where 0 # ~ e C .

Remark. The condition of Theorem 3.1 is not surprising, for the condition tha t two lattices L~ and L2 in C are related by L1 ---- ~L2, 0 =fic~ e C, is clearly the same as the condition tha t an integral unimodular fractional linear trans- fol~nation sends the ratio of a pair of generators of Lx to the ratio of a pair of generators of L 2.

4. Parameterization of ~ and ~'4

~;' is a disjoint union ~; ' (§ U ~ ' ( - - ) U .L~(:k) where

~ ( ~ ) = U U . C ( s , h , 4 t + h - 1) h = 4 t t = l

and ~ ( r h, n) consists of all manifolds S~/s (u, e) in ~ . Similarly, .L~ = oO cO

= U U .L~(2s, 4t § 2s - 1) where ~/?~(h, n) consists of all manifolds ~=2t t= l

S~/2~ (u) in -s We will s tudy the structure of .L~(e, h, n) and .L~ (h, n).

108 JOSEPH A. WOLF

We first look at the easy cases. Le t u e H = upper half plane. For appropr ia te h and n, and any e, s (u, e) = Oh (n + l) is defined. The latt ice L(u) is

preserved by mult ipl icat ion by exp (4g 1/------ 1/4) = - 1, so, for appropria te h and n, s c 0 h ( n + 1) is defined.

Theorem 3.1 now yields:

Theorem 4. Let M = H/F, identification space o/the upper half plane under the modular group; let M' = H / F ' , identification space under the mod 2 parabolic modular group. Then each ~L~( + , h, n), each .~( • h, n) and each ~4(h, n) is parameterized by M, and each .~(-- , h, n) is parameterized by M'. More precisely,/or appropriate h and n, the maps

u -~ S ~ / ~ ( u , § u -> S ~ / s • u -~ S ~ / ~ ( u )

and u-->S~/~(u, --) induce one to one maps o/ M onto .~(~-, h, n), M onto . ~ ( • h, n), M onto ~D4(h , n), and M' onto ,~ ( - - , h, n), respectively.

Given u, u ' e l l , we can find an analyt ic arc (such as the straight line segment) in H from u to u'. This results in a real analyt ic deformat ion of ~(~, u) (or ~4(u)) over to ~(e , u') (or ~4(u')) . I t follows tha t any two ele- ments of ~?(e, h, n) (or of -~'4(h, n)) are real-analyt ical ly homeomorphic. This is notable because, by Theorem 4, ~ ( e , h, n) and .L~4(h, n) carry the s t ructure of a real analyt ic V-manifold in the sense of BA~LEu On the other hand, we cannot assert t h a t two dist inct elements of ,~(~, h, n) or of A?4(h, n) be affinely equivalent, for, utilizing irreducibil i ty of the holonomy group and the fact t ha t the manifolds have the same constant curvature, such an affine equivalence would be seen to be an isometry.

5. Uniqueness in ~f?3, d e and ~A~ S

5. 1. We will prove t ha t each K s (h, n) , each A? 6 (h, n), and each ~'s (h, n), has just one element. This will require a lemma on lattices in C. The lemma also gives an explanat ion as to why m is not a rb i t ra ry in the definitions of the groups ~m(u).

Le t L be a discrete addi t ive latt ice in C, and suppose tha t L = x L for some non-real unimodular complex number x. As L is discrete, we m ay choose a e L with the p roper ty tha t ] b l ~ [ a [ > 0 for every nonzero b E L ; we then define L ' to be the sublatt ice of L generated b y a and xa. Suppose

t h a t x -= cos (t) ~- } / ~ 1 sin (t), t real. Then

Ira -~ s x a l ~ = [al * Ir * § 2rs cos(t) § s*l for any r , s r I t .

Suppose t ha t we have b E L, b r L' . b = ra A- s xa for some real r and s

On the Lattice Space Forms 109

because a and x a are linearly independent over R, consequence of x r R. Adding an element of L ' to b, we may assume tha t I rI g �89 [s I < �89 and b r Thus 0 < l b [ 2 ~ [ a ] 21�89189 < ]al ~.

Now ]a[ ~ [a ~- xa] implies cos(t) ~ - �89 As x~L = xn- lL . . . . . L , the same must be true of every power of x. Thus cos(nt) :> ~ �89 for every integer n > 0. I t follows tha t

x ~-- e x p ( 2 ~ Y ' Z 1/3) or x = exp(~= 2 ~ Y "Z- 1/4),

whence cos(t) = - �89 or c o s ( t ) : 0. In either case, we have 0 < ] b [ 2 < [ a [ 2, contradicting b r L ' . We have proved:

Lemma 5. 1. Let L be a discrete additive lattice in C ; suppose that

x ~ C , x ~ R , I x I = I ,

and x L ~ - - L . Then L = {a, xa} , where ] a [ g ]b] /or every nonzero b c L ,

and x - - ~ e x p ( • ]/3) or x : 4 - V ~- 1. I n other words, there is a nonzero complex number a such that either

L : a {1, exp (2u ~ 1/3)} or L = a{1, ~ 1}.

Remark. Lemma 5.1 provides an alternative t rea tment of the last part of [1, w 10.4].

5.2. Theorem 3.1, the remark at the end of w 3.4, and Lemma 5.1, combine to give a proof of:

Theorem 5.2. I f m : 3, 6 or 8, then each ~' , , (h, n) has just one element. I n other words, for appropriate h and n, we have:

1. Every S~/9.z(u) is isometric to 8~/~3 (exp(2~ V "Z 1/3)) .

2. Every S~/s is isometric to S~/s V ' ~ 1/3)) .

3. Every S~/~s(u) is isometric to S~/s (V ' ~ 1) .

5.3. Let M~ and N~ be connected homogeneous pseudo-RIEMA~c~cian manifolds of constant curvature ~- 1 with isomorphic fundamental groups. gl(M~) --- G --~- zl (N~) for some abstract group G. Let Z denote the infinite cyclic group.

I f G is finite, then [1, Theorem 12] M~ is isometric to N~. I f G is an extension of Z by an element of order 4, then [1, Theorem 12] M~ is iso- metric to N~. I f G is an extension of Z x Z by an element of order m # 1, 2 or 4, then ([1, Theorem 12] and Theorem 5.2) m = 3 , 6 o r 8 and M~ is isometric to N~. I f M~ is not isometric to N~, then ([1, Theorem 12] and

110 JOSEPH A. WOL~' On the Lattice Space Forms

Theorems 4 and 5.2) ei ther G is an extension of Z b y an e lement of order 1 or 2, or G is an extension of Z • Z b y an e lement of order 1, 2 or 4. This covers all possible G.

The Ins t i tu t e for Advanced S t u d y

REFERENCE

[1] J.A.WoLF, Homogeneous manifolds o/constant curvature, Comment. Math. Helv. 36 (1961), 112-147.

(Received March 2, 1962)