223

AXIOMS FOR INDEFINITE METRICS

by Herbert Busemann and John I(. Beem (Los Angeles, U. S. A.)

1. INTRODUCTION

The theory of G-spaces has shown that metric differential geometry is

largely amenable to an axiomatic, purely geometric treatment, which is particu-

larly effective in non-Riemannian spaces and leads to attractive novel problems

related to the foundations of geometry.

The question, whether indefinite metrics lend themselves to a similar

approach is not only of considerable mathematical interest, but was also raised

by physicists in connection with relativity theory. The latter suggests a second

topic, timelike spaces. Although these will be the subject of another paper, we

give the definition because it is needed here (see [13]).

A space is timelike i f its points are partially ordered (x ~ y) and a function

x y is defined for x ~< y satisfying the conditions x x ~ O, x y > 0 for x < y

and x y - k y z-~< x z i f x < y < z.

The best known example is furnished by the n-dimensional Lorentz space,

i.e. the x ~ (xi, . . . , xn)-space where the function

z . ( x ) = x', - 1>1

defines the partial ordering: x < y if x~ < y~ and ~..(x - - y) > O; as well as

the distance x y ~ ~,~2n ,(x--Y). If x < y < z , then x y - 4 - y z < x z unless y

lies on the straight segment from x to z. Thus the segment from x to z

224 H E R B E R T B U S E M A N N a n d J O I I N K . B E E M

maximizes length among all curves y(t) from x to z satisfying y ( t ' ) ~ y( t")

for t ' ( t ' . In the case n ~ 4 this expresses the so-called clock paradox of

special relativity (t).

Here we are interested not only in the case x ( y but in x y = I) ,n(x--y)[ il~

for arbitrary x, y. This and other simple examples show that axiomatizir, g

indefinite metrics offers considerable difficulties and is not feasible in strict

analogy to G-spaces. Overcoming these difficulties has been the principal goal

of the present paper. Its main contribution is therefore of a critical nature

involving the rejection of numerous seemingly plausible approaches. The result

is a system of axioms (T and A~_s in section 2, MI_3 in section 4 and M4.5 in

section 5) which will, most probably, surprise the reader.

Thus the axioms require comments. We make these here, giving our moti-

vation while describing the content of the axioms loosely and indicating why

they are adequate.

As definite metrics go back ultimately to the euclidean plane, indefinite

(or better, not necessarily definite) metrics will be an extension of a distance

x y ~ [ L ( x - y)[l/~ in the (xi, x2) plane where

2 2 L(x) ~ s,x, Jr- ~2x2, Ei = • 1, O.

The significantly different cases are: ~t ~ %-~ 1; ~t ~ 1, % ~ O; ~1 ~-1,

% ~ - - 1 ; r ~ % ~ 0 .

In the first case one has the euclidean plane in which the (ordinary) seg-

ments are the shortest connection of their endpoints.

For E~----- 1, % ~ 0 we have the "neutral" metric x y : [x~--y~l . If x ~ ( y i ,

then all curves z(t) from x to y with z~( t ' )~ z~(t") for t ' ~ t" have length

For s t ~ 1 , % ~ - - 1 we have L ( x ) : ) . : z I x l and obtain, apart from the

pairs x -~ y with L(x ~ y) ~ O, two timelike spaces, where x ~ y is defined,

respectively, by x l < y ~ , L ( x - - y ) > 0 and by x 2 ( y 2 , L (x - - y) .< 0. A seg-

ment whose length is not zero, i.e. whose slope differs from -t-1, maximizes

length in its respective timelike space among all curves z ( l ' ) < z It") for t ' ~ t".

(l) A good elementary introduction to Lorentz space is found in NGII [8]. A detailed n

discussion of the "euclidean" spaces corresponding to the form .~ ~t x~ with ~i ~ 0, ::k 1 as t = l

well as the "noneuclidean" spaces in a similar general sense is found in Rozenfel'd [10]

which provides a great wealth of information.

AXIOMS F O R I N D E F | N I T E M E T R I C S 225

The case ~ ~ ~ = 0 is trivial but cannot be ruled out because such

"null planes" occur in interesting higher dimensional spaces.

In the three-dimensional Lorentz space with x y = ]~,3(x--y)[I/~ consider

the segment cr from ai ~ (0, 0, - - 1 ) to ae ~ (0, 0, 1) and a variable plane P

through a. If P intersects the cone C: ) ,3(x)= 0 only at the origin, then the

metric in P is euclidean. If P intersects C along a generator then it is isometric

to the plane with the "neut ra l " metric I x 1 - yll above. If P intersects C in

two lines it is isometric to the Lorentz plane with distance [~2(x--y)[~/~ and

carries two timelike spaces. Here, null planes do not exist.

Thus, c~ minimizes length (in P ) i n the first case, is " n e u t r a l " in the

second case and in the third, maximizes length among the previously indicated

curves.

Because of the above and since we do not intend to assume the existence

of planes or of geodesic manifolds, we abandoned the idea of defining the

geodesics in terms of the distance (~). Instead we introduce rather mild topolo-

gical conditions (T) and then postulate (A~_s) the existence of a set A of arcs

playing the role of sufficiently small arcs of the extremals (in the Lorentz space

we could take all segments). We show that the space is locally homcgeneous

in the sense of Montgomery [6] and that - - in analogy to a metric theorem of

A. D. Alexandrow [1]-- uniqueness and existence of the connection by arcs

in A implies prolongability, if the space is a topological manifold.

Next we require (JV/'1,2,3) the existence of a continuous function x y with

x x = O, x y = y x ~ 0 defined on a set containing the endpoints of the arcs

in A which either makes a given arc in A isometric to an interval of the real

axis or vanishes for all x, y on the arc.

The main idea is to express the non-trivial metric properties through the

local behavior of the functions p x where p is a fixed point and x traverses

an arc cr of A isometric to an interval (see M4,5).

The basic condition is this : I f p is sufficiently close to a given interior point

o f ~ then either p x > O and [px - - py[ < x y for x ~ y, or p x has exactly

one zero and [px - - P Y l = x y for x, y on the same side of the zero, or p x has

two zeros and [px - - p y [ > x y for distinct x, y on the same side of both zeros.

As to the adequacy of the axioms: the definite case x y ~ 0 for x ~ y

should and does lead to G-spaces. If geodesically complete then the space is

(~) Introducing a function xy ~ taking negative values would not remedy the situaticn. Also, the sign does not seem significant in the basic s~age. 15 - Rend . Circ. Matem. P a l e r m o - S e r i e II - T o m o X V - A n n o 1966

226 HERBERT EUSEMANm and JOHN ~'. BEEM

essentially a G-space (secticn 5) and with a preper definition of the arc system

A, a G-space will satisfy all the axioms.

The axioms comrrise all spaces with indefinite Riemannian metrics, in

particular, the projective metrics ([12]) ar.d the other - - in contrast to the definite

case- - very numerous spaces with a high degree of mobility ([14]). In each

case our distance x y is a suitable function of the number ~(x, y) (for example

x y = [~(x, y)[ or x y = [;~(x, y)l L/e) us~.d for describing the metric in the geometry

under consideration. (The usage is not uniform).

Finally, i f two-dimensional, our spaces are manifolds and we always obtain

the disjunction into qualitative analogues to the example of the (xi, x~)--plane with

the distance I L ( x - y,[~/e (section 6). We either have x y =--0 or x y > 0 for

x ~ y or the surface is locally isometric to the neutral plane with the distance

[x~--Y,I or it is locally covered by two simple families of arcs in A of length 0

and the given distance induces two timelike metrics, to that x y ~ 0 only if

x < y or y < x in one of these metrics.

Thus our axioms accomplish the purpose of delineating an interesting class

of spaces. The theory will be further developed. One of our first concerns will

be the precise analytic character cf the differentiable case, where R is a manifold

of a certain class and the arcs in A as well as the distance x y are differentiable.

2. ARCS AND PATHS

Concerning the space /? we assume throughout (at least) the following

topological properties.

T. t? is a locally compact, connected, Hausdorff space with a countable base.

Since T and dim /? = 0 imply that R consists of a single point we consider

only spaces of positive dimension.

It is known, see [9, p. 74] or [11], that R can be metrized such that it

becomes finitely compact, i.e. so that bounded infinite sets have accumulation

points. Although this metric is not unique it proves most useful to select one

such distance, ~(x, y), once and for all as an auxiliary device. The open ball

tx It(x, p) < ,~1 is denoted by S(p , p).

A subset N of R X / ? is symmetric if (x, y) E~l. implies (y, x) E ~ . The

diagonal & consists of the pairs (x, x) and we put ~ U & ~ - ~ .

We now give the axioms for the arc system A mentioned in the introduction.

(A point is not an arc. Conditions M45 influence the extent of A).

AXIOMS FOR ]ND~FIIqITE M]~TRI~t?.~ 2 ~ 7

There exists a system A of Jordan arcs ~ satisfying the conditions AI_~:

AI I f ~ fi A, then every subarc o f ~ lies in A.

A~ Each ~ E A lies in the relative interior of some o~" E A.

As I f o~, ~" lie in A and have an end point and a further point in common

then either ~ Q ~" or ~" Q ~.

A 3 implies that at most one arc in A exists joining two distinct points x, y.

We denote the symmetric set of all pairs of endpoints of arcs in A by B and

by ~(x, y) the arc in A from x to y if (x, y) fiB.

A 4 I f (x~, y,,) E $I, x~ -~ x, y~ -~ y and (x, y) E $to, then ~ (x~, y~) ..~ a (x, y)

i f x ~ y and a(x~, y~)-~ x i f x = y.

Here the limit is Hausdorff's closed limit.

A5 Bo is a non empty open subset of R X R.

For brevity we denote the arcs in A by Arcs.

It follows from A s that each point p has a neighborhood U such that any

two distinct points in U can be connected by an Arc. In terms of the auxiliary

metric ~(x, y) there is a 9 > 0 such that Sip, P ) X S(p, ~)C Bo. Let B(p) be

the least upper bound of these ~. Then, see [2, p. 33], S(p, ~(p)) X S(p, B(p))C~lo

and either ~(p) ~ co, i.e. Bo = R X R, or

I~ (P) -- ~ (q) l ~< ~ (P, q).

The set formed by the union of all Arcs with endpoints in S(p, ~) shrinks

to p when ~..~ 0 as a consequence of A4. Therefore, R is locally arcwise

connected, and being connected, R is arcwise connected.

We introduce a topology for the Arcs by defining a neighborhood U,, of

(x, y) to consist of all Arcs a(x ' , y') for which x', y ' are, respectively, in

certain neighborhoods Ux of x and U v of y. It follows from A s that Ux X UyC ~lo

for sufficiently small U,, Uy. We deduce from A4 that a,,-~ a if and only if

each U,, contains all but a finite number of a,,. Thus the limit induced by the

U~ for the Arcs coincides with the Hausdorff closed limit.

M. Morse [7] showed, using an idea of Whitney, that it is possible to

derive from the metric ~(x, y) a parametrization of all oriented arcs such that

some of the essential features of arclength as parameter are preserved.

Precisely, the paper of Morse (applied to our special case) yields the fol-

lowing: The Arcs a(x, y) can be parametrized simultaneously as z(o) ( 0 ~ o ~ , ~ )

228 HERBERT BUSEMAI~I~' and JOHN K. BEEM

such that

1) a-~z(a) is a topological map of [0, r162 on ~(x, y) with z ( O ) = x and z(a~) = y.

2) (1/2) diameter ~-~< a,~ ~ diameter ~.

3) If ~ ' = ~(x, u ) C a(x, y) and u ~ y then the parametrization z '(a) of ~"

satisfies ~,~, ~ a,~ and z (a) = z ' (a) for 0 ~ a ~ a,~..

4) If q(cz, a) ( 0 - ~ < a ~ a , , , ) is the point z(a) then q(a, a) depends conti-

nuously on ~ and a (jointly).

The topology for the Arcs is that d iscussed in the beginning. In Morse ' s

paper it is given by Frdchet distance, but for Arcs the two topologies are

equivalent.

In contrast to arclength the parameter a is not additive, i . e . , if y lies in

the relative interior of a(x, z) then in general

c~=(x,y) -Jr- ~=(y,z) :~ O=(x,z).

A first application is the f•llowing: For a given ~r y) form the union T

of all =(x, z) D ~(x, y). Introducing the parameter a on all ~(x, z) we see from

A2 and 3) that T receives a representat ion z(a) for 0 ~ a ~ z, where "~ may be

finite or infinite. If "~ = oo then ~(x, z(r ~ c~ for "~-~ cr For, assume there

is a sequence z(av) with a.,-~ c~ and z(a~)-~ u, then ~(z(a~), z(a,))-~ u by A 4

so that T would be bounded contradicting 2).

If z is finite then a similar argument shows that z(a) converges for a - ~

to a point u. Finally, if x, y E S ( p , B(p)) and s > 0 is given then a ( ~ ) <

exists such that ~(p, z(o)) > ~(p) - - ~ for "~ ~ o ~ o(~). Otherwise, z(o)..~ u

for a--~'r would give that ~(p, u)~< ~ ( p ) - e. Then ~(x, u) would exis t and

not contain =(x, y) or T, hence o~(x, z (a) )~ a(x, u) for a.-~ ~ contradicting A , .

This implies

(1) I f x, y are distinct points in S(p, ~(p) - ~) where 0 < ~ < ~(p) then (x, y) C ~ (x', y') with ~ (p, x') ~- ~ (p, y') = ~ (p) - - e.

We divide the Arcs into classes, called paths, by requiring that ~ and ='

belong to the same path if and only if a finite number of Arcs =~ = - ~ , ~ ,

%, . . . , ~,, = ~' exist such that ~; ~ =;+, for i = 1, . . . , n - l is an Arc.

A representation of a path P is a map x(t) of a connected open set lp of

the real axis into ,P with the fol lowing two proper t ies :

a) For each Arc ~ in P numbers tx ( t~ exist such that x(t)[[t~, t.~] is a

topological map of ~ on [t~, t.2].

AXIOMS FOR I N D E F I N I T E METRICS 229

b) Given t o E lp then tt < to < t2 in Ip exist such that x(t) l[ti, t.,] is a

topological map of [t~, t~] on an Arc in P.

Each point on an Arc r of P is called a point of P. A line elem, ent ), at a

point q is a maximal set of Arcs containing q in their interiors such that any

two Arcs in ;~ have an Arc in common.

If one Arc of ~. lies on the path P, then they all do. We denote this by

) , C P and call ~, a line element of P at q. The multiplicity of P at q is the

cardinal nIJmber of distinct line elements of P at q. A point of multiplicity one

is a simple point of P, all others are multiple points. If all points of P are

simple, then P is called simple.

We list some facts on paths without proofs, because the latter can be

obtained by simple modifications of the arguments for the corresponding pro-

perties of geodesics in G-spaces, see [2, sections 7, 8, 9]. In some cases, for

example in the proof of (5), the parameter a of Morse is very useful.

(2) Each path a possesses a representation x(t).

(3) The multiplicity of a path at any of its points is finite or countable.

(4) A path possesses at most a countable number of multiple points.

(5) If two line elements exist at one pt, int of R titen dim R :>~ 2.

(6) I f dim R : 1, then R consists of one simple path.

A path P represented by x(t), (tElp) is called a line if x(t)l[t~, t~] is an

Arc (and hence in P) for all t t < t~ in Ip.

(7) A line is a simple path and any two distinct lines have at most one

point in common.

(8) I f ~ o : R X R then all paths are lines; the converse is not, in general, true.

This is less obvious and expresses, in fact, a difference in the behavior of

paths from that of geodesics in G-spaces where the converse is true.

Let 2 1 o = R X R and assume that x(t) represents a path but x(t)[[tl, t2] is

not an Arc. Denote by ~ the supremum of all t' :> tl for which x(t)[[tt, t ~] is

an Arc. Then t~ < z ~ t ~ and x(T)~x( t t ) by A4. By hypothesis an Arc ~(x(tl), x(-c))

exists, but a(x(ti), x(t,)) does not tend to a(x(t~), x('c)) for t , ~ ' q (t~ <'~). This

proves the first part.

We establish the second part with an important example. For reference

the space is denoted by R ~ and the system of Arcs by A h, where h indicates

a close relation to the Poincar6 model of hyperbolic geometry.

230 HERBERT BUSEIH&NN and JOHN X. BEEM

R ~ is the half plane x 2 > 0 of the (x~, x~)-plane. The paths are the parts

in xz > 0 of the following curves

a) b) --(x,--~)2--l-x22=y, r > 0 . c) ( x t - - ~ ) ~ - - x 2 ~ = Y , Y ~ 0 , including x i = c o n s t .

In the case b) a path is a branch of a hyperbola, in c) we have one-half

of each of the two branches of a hyperbola. These two pieces count, of course,

as different paths. The Arcs are the subarcs of the paths, so that the paths

are lines.

A h and the system of paths are invariant under the quasi-hyperbola group

(see [2, p. 371 and (46) on p. 407]):

(9) x~ = ~ xl W L 3, x~ = a x~, ~ > 0, ~ arbitrary, which is simply transitive

on x~ > O.

That ~ o - ~ R h X R h and hence the converse in (8) is not true is readily

seen by considering the paths through (0, 1). Their points form the set

[x l x ~ > O , x ~ > - - x x - - l , x ~ > x ~ - - l l

and not R '~.

The path system exhibits another important phenomenon: there are simple

families of paths covering R h (i.e. each point lies on exactly one element of

the Iamilyl which are not topologically equivalent to a family of parallel lines

in the euclidean plane. An example is provided by the paths

x~ = ++_ x, - - Z, "r >~ O, - - x~, + x~ = Y, z > O .

Also, the lines L(q, vn) passing through q = (0, 1) and vn = (2 -1 , 2-t-[ - n -1)

do not tend to L (q, v) where v = (2 -1, 2 -1), although, of course, a (q, vn) --~ a (q, v).

However, lim inf L(q, v~)D L(q, v). This ts a general property.

(10) I f all paths are lines and L(x, y) is the line through x and y # x

(if it exists), then x~ -~ x and y~ -~ y imply lim inf L (x,, y~) D L (x, y).

For the proof we introduce a notation which will be used frequently in the

case of general R satisfying T and A,_~: we say that r lies between q and s

and that (qrs) holds, when (q, s) E $t and r is an interior point of a(q, s). The

betweenness relation is continuous:

(11) If (x~y~z,), x~-~ x, y ~ y, z , - ~ z and x, y, z are distinct then (x y z) holds.

This follows from lira y,~ E lira :~(x,,, z~).

A X I O ~ I S F O R I N D E F I N I T E M E T R I C S 231

In (10) we know that a (x , , y . ) ~ a(x, y~. We orient L(x, y) so that y

follows x. If (10) were not true then lira infL(xn, y,,) being closed, a last point z

following x (or a first point preceeding y, we assume the first case) would

exist such that ~(x, z ) C l iminfL(x~, yn). Then z = l imz , with z~ E L(x~, y~). Choose t~ with (x~t~z,) and 0 < ~ ( t , ~ , z ~ ) ~ 2-1~(z~) and u~ with (t~z~u~),

~(z~, u,) -- min(2-~(z, , ) , 1). Then oc(t~, u~)C L(x~, y~) and ~(z~) -~(z ) > O.

We conclude easily that z cannot be the last point with a(x, z ) C lira inf L(x~, y~).

(12) I f ~to : R X R and x , -~ x, y~-~ y ~ x then L (x~, y,,)-~ L (x, y).

Because of (I1) it suffices to prove that lim supL(x~, y,,)C L(x, y). Let {m I

be a subsequence of ~nl such that u,~ E L(Xm, y,,) and um "~ u. We must show

that u E L(x, y) and may assume (x,, y~ u,,) so u ~ ~(x, y). By hypothesis a(x, u)

exists and ~(x,,, u,,)-~ a(x, u). Then (11) yields (xyu) and hence u E L(x, y).

(14) I f "s X R and x(t) (tE Ip) represents a path then ~(x(0), x( t ) )~co

when either t-~ "q = lim inf t or t-~. % = lim sup t. t ~ [p t ~ lp

Assume there is a sequence t~-~z~ with x(t~)l~y then there is an ~(x(0), y),

and hence a point z with (x(0)yz) would exist which is impossible.

3. PROLONGABILITY AND LOCAL HOMOGENEITY

We conclude the purely topological part of the theory by proving two less

elementary theorems. The first generalizes to our systems A of arcs a metric

theorem of A. D. Alexandrow [I], see also [19, pp. 288, 289] and states roughly

that, if T is strengthened, prolongability can be deduced from the uniqueness

of the connection. Precisely:

(1) Theorem. Let R be a topological manifold in which a system A of Jordan

arcs is defined for which Al, A3, A4, A5 hold. Then A also satisfies A s .

It suffices to prove: for a given ~(p, q) there is a ~(p, wo) D ~ ( p , q)

with Wo -~ q.

Choose ~ > 0 such that S(p , ~ ) X S ( q , ~ ) C ~ . If d i m R = n let B" be

the unit ball in E" with origin 0 as center and S"- ' as boundary of B". Let

be a topological map of B" into S(q, ~) with ~ ( 0 ) = q. Put B : gP(B") and

B ' : R - - B . Then ~(q, B ' ) : 3 0 > 0 .

Let z(a) (0 ~ a .~< a~) be the Morse parametrizaticn of ~(p, q) with z(0) = q.

Properties 2) and 3) of a show that z (a)EB for a ~ 3 0 / 2 . Therefore, u-----z(0)

is an interior point of B.

232 HERBERT BU.gEMANN and JOHN K. BEEM

Choose a ball x in B" with center ~- ' (u) and radius less than t~- ' (u)] /2

such that, in addition, cb(~r C S(u, 0/2). The projection, from the origin of E",

of ~r on S ~-1 lies in a hemisphere of S~-k

For 0 < E ~.. 1 let W(~) be the image of ~S"- ' (the sphere of radius

about 0) under r Because of A4 the set carrying the union of the Arcs a(w, p),

w E W(E) tends to r p) for E-~ 0. Let z(o, w) be the Morse parametrization

of ~(w, pl, where wE W(E) with z(0, w ) = w. We can choose r such

that z(O, w) E ~(~) for all w E W(E), and z(Ot, w) C B for 0-~< t-~< I.

For y E S "-~ and t E [0, 1] set

h(y, t) = [z(0t ,

if defined, i.e. if q~-~[z(Ot, r O. Assuming the map is defined for all

y, t, then property 4) of the Morse parameter guarantees that h(y, t) is con-

tinuous on S ~ - ' X [0, l].

But h(y, O ) = y and h(y, 1) lies in the projection from 0 of ~r on S ~-~, so

that h (y , t) would yield a deformation (homotopy) of S ~-' on itself into a proper subset.

Therefore, ~ - ' (z (0 t,

for wo = ~ (~ Yo) E W (~).

that a(Wo, ff contains q

Our second theorem

homogeneous in the sense dimensional, has many

will be mentioned later.

�9 (~y))) = 0 for some Yo, to which means z(Oto, Wo)=q Here t o > 0 because z(0, w o ) = w o ~ q . This means as an interior point and proves (1).

states that a space R satisfying T and A,_~ is locally of Montgomery [6]. This entails (1. c.) that R, if finite

properties of topological manifolds, a few of which

Let M be a non-empty set in R and let h(x, t) be defined and continuous

on M X [0, 1]. Then h(x, t) is an E-family of homeomorphisms on M if

a) h(x, O)-----x for x E M .

b) h(x, t) is a homeomorphism for fixed t.

c) B(x, h(x, t ) ) < ~ for x E M and 0 ~ t . ~ < l .

A space /~ satisfying T and with a metric ~(x, y) is locally homogeneous

if every point has a neighborhood U with the following property: given any

E > 0 there is a [3> 0 such that for a E / J and ~(a, b ) < ~ (b need not be

in U) an E-family of homeomorphisms on O with h(a, 1) = b exist.

(2) Theorem. If R satisfies T and A,_ 5 then it is locally homogeneous.

Let p E R and ~ > 0 be given.

A X I O M S F O R I N D E F I N I T E M E T R I C S 233

For (x, y ) ~ 0 put r x ) = 0 and r y)=cr,~ for x ~ y where % is

the number using the Morse parametrization zQ) (0 ~< r ~ ~) of ~(x, y) with

z (o ) = x .

(Then r y ) = or(y, x) follows from the definition of % because ~(p, q ) =

= 8 ( q , p)). Put B~(p)= rain(l, ~(p)) and choose ~ ~ 0 such that for U = S ( p , 5) the Arcs ~(p, x) lie in S(p, 8~(p)/2) for all x E ( J . Put ~ ' = sups(x , y) for

x, y E U. We may assume 7' > e.

Property 4) of ~ implies that for a E 0 a neighborhood W(a)C S(p, 2 ~ (p)/3) exists with [=(a, x) - - ~(y, x)l < e when x E U and y E W(a).

Then the union of W(a) such that any two x, y in /J

supremum of the ~.' is the choose a neighborhood V(a)

for all a E /ff covers U and there are positive X'

with 5(x, y ) ~ ~" lie in some single W(a). The

Lebesgue number ~. of the covering. For a E U

with

~r x) C S(a, ~.e/16max(rl ' , 5)) for all x E V(a).

Finally, let ~ be the Lebesgue number of the covering of /.J by the V(a). We must show for each a E 0 and b E R with ~(a, b ) < [3 an e-family of

homeomorphisms h(x, t) on 0 with h(a, 1 ) = b exists. Extend ~(a, b) beyond b to ~(a, r), where r is the first point with ~(a,r)=X/2.

For x E U let z(o, x) (0 ~ o ~ o(5 x)) be the Morse parametrization of ~(x, r)

with z(0, x ) = x . With "(-----o(a, b)/o(a, r) define for x E U , 0 ~ t ~ l h ( x , t ) = = z('co(x, r)t, x). Then h(x, t) is continuous in x and t, moreover It(a, 1)----- b.

We also have

~(x, h(x, t ) ) ~ 2a(x, h(x, t ) ) = 2~'~(x, r)t~<

2[(~,e/16 max (~', 5))(~,/4)-'] (r~' + e) <

[ , ] < 2 4max( '~ ' , 5) (2 7 ' )~<e .

Finally, to show that h(x, to) is a homeomorphism it suffices to see that

it is injective. This is clear since

z(y~(xi , r) to, x l ) = z('(~(x~, r) t0, x2)implies x~ = x e.

We mention a few of the many implications of local homogeneity given in [6].

(3) If dim R = n then a closed n-dimensional set has interior points. Any

open set V ~ ~ contains an open set W ~ 0 such that any (n -- 1)-cycle in W

bounds in V.

234 HERBERT BUSEMANN and JOHN K. EEEM

The question whether every space R satisfying T and AI_~ has a finite

dimension is open (even for G-spaces).

(4) I f dim 1? < co then any two points possess homeomorphic neighborhoods.

This is probably true without the hypothesis dim1? < eo because this is

so for G-spaces, see [2, p. 49].

(5) I f dim R = 2 then t? is a topological manifold.

If, in addition, L~ o = 17 X R, then it follows from (2.8, 12) that R is

homeomorphic to the plane. Because of (2.13) the paths which are lines satisfy

the hypothesis of (11.2) in [2, p. 56] and consequently have the topological

properties derived in [2, pp. 56-60].

A non-empty set C in a space 17 (satisfying T, A,_s) is convex if C X C C ~ t o

and a(x, y) C C for any distinct x, y in C. If ~to = 17 X R then 17 is convex.

If C is open and convex we define A c as the subset of those Arcs a(x, y)

in A for which x, y lie in C, then the axioms T, A,_ 5 are satisfied for C

with A c. Also, c X C is the union of the diagonal of C X C and the set of

pairs of endpoints of the Arcs in A c. For 17 = C we have the previously

discussed case ~o = 17 X 17 so that (2. p. 12, 13) hold.

(6) I f dim 17 = 2 and C is an open convex set in 17, then each compact

subset of C lies in a compact convex subset of C (see [2, p. 58]).

(7) I f dim R = 2 then every point p has a convex neighborhood.

This follows from (6) but can also be seen directly: take points a, b, c, d

sufficiently close to p such that (apdl, (bdc) and a(a, d ) n a(b, c ) = d. Then

~(a, b) U ~(b, c) U ~(c, a) bounds on R an open convex set containing p.

Also, in contrast to R h, a simple family of lines covering the convex

plane 17 is topologically equivalent to a family of parallel lines in the ordinary

plane.

4. SPACELIKE, NEUTRAL, AND TIMELIKE PAIRS

The axioms T, Al-s being assumed we come to the metric axioms Ma-s.

The importance of ,'144 and 1145 is best understood if the rather trivial implications

of MI_ s are discussed first.

M~ A continuous function x y is defined on ~to which satisfies x x---= O,

x y = y x ~ 0 and is bounded on any subset of "Bo with compact closure in 17 X 17.

M~ If x o y o = O for one pair of distinct points of an Arc o~ in A then

x y ~ - 0 for all x, y in ~.

AXIOMS FOR I N D E V l N I r E M g r R I c s 235

M 3 I f x y > 0 for one pair (hence all pairs) of distinct points on ~ E A,

then x y defines on r a metric with which o~ is isometric to an interval of the

real axis.

If x y-----0 for x, y on ~ we call ~ a null arc, otherwise a segment.

M3 means that a s~gment ~(x, y) can be parametrized in the form z(s) with

(1) z(s3z(s ) = Is, - - u s u + x y .

Henceforth a representation of a segment will mean a parametrization of this

type. The remaining representations of a segment can be obtained from (1) as

(2) y (s) = x(E s -[- ~.), e = ___ 1, ~. arbitrary.

If one Arc in a path or a line element is a null arc, then all are. In that

case we call the path a null path or the line element a null element.

A path all (or one) of whose Arcs are segments is called a geodesic.

Among the representations of a geodesic G some have the form z(s), sE lo where

for each ~(x, y) EG a number u exists such that z(s)[[u, u - -kxy] represents ~.

The term representation of a geodesic will henceforth be reserved for this

form. If lo = ( - -oo , c~) for one representation of G then this for holds for all

representations.

The following definition will prove important later; a space satisfying

axioms T, Ai-5, M~-3 is geodesically complete if for each geodesic G the set

lo is the entire real axis.

If a segment or a geodesic is oriented we stipulate for a representation z(s)

that increasing s corresponds to traversal in the sense of the given orientation.

The remaining representations are then given by y(s) ~ z(s -q- ~.).

(3) The points which lie on null arcs (or at which null elements exisO form

a closed set.

For, if points p," lie on the null arcs ~," and p,'-~p, then x~ with ~(p, x ~ ) =

= ~i(Pvl/2 and p x~ = 0 exist. A subsequence of Ix.,} will converge to a point

x with ~(p, x ) = ~(p) /2 and p x - - - O .

We now come to some very important definitions:

Let q be a point and e a segment with (q, x)E ~0 for all x E e. The pair

(q, e) is called:

Spacelike, i f q x > 0 for x E ~ and [qx - - qYl < x y for distinct x, y on ~.

Neutral, i f I q x - q Y l - - - x y for x, y on % unless z with (xzy ) and

q z ----- 0 exists.

236 HERBERT BUSEMANN and 3OHr~ K. ]~E~M

Timelike, i f q x has at least one zero in the interior o f ~ and I q x - qYl ~ x y

or I q x" - - q y" l > x" y" whenever q z = 0 and x, y, z, x ' , y" lie in this order on ~.

We discuss some simple propert ies implied by these definitions.

(4) I f ~ is a segment and q E ~', ~" D ~, then (q, ~) is neutral.

All points other than q are unders tcod to lie on a in (5) to (9).

(5) I f (q, ~) is neutral and q z = O then q x = z x . Also, q y = q x J r - x y

for (z x y) or (y x z) so that q x has no other zeros than z.

The first s tatement is contained in the definition. The second fol lows from

q y > q x for x close to z.

(6) I f (q, ~) is neutral and q x > O for x E a, then ~ can be so oriented

that q x ~ x y = q y i f x preceeds y.

(7) I f (q, ~) is timelike, then q x has at most two zeros.

Assume there are three with (zlz~z3). If (z~xzz! , ( z z y z s ) and x, y are

close to z2 we would have l q x - - qz~l < z~x and IqY - - qz3l < z3Y.

For reference we state the fol lowing corol lary:

(8) I f (q, ~) is spaceIike, neutral, or timelike then q x has at most two zeros.

(9) Let (q, ~) be timelike and let q z = O, where z is an interior point o f a.

I f z decomposes ~ into the segments ~ , ~.,_, then

the inequality q x ~ x y < q y holds for (z x y)

It does not hold on ai i f qz" = 0 for z" E

either on ~t or on a s (or on both)

or z = x and y ~ z.

~ i - - Z.

For the first part assume [ q x ' - - q Y ' l ~ x ' y " for some x', y ' on a s with

( z x ' y ' ) . Then the definition yields for x, y on a t with ( y x z ) that l q x - q y l ~ x y .

Since q y > q x for x close to z, we have in this case q y > q x q - x y , and

therefore a lways if ( y x z ) . Let ( y u z ) and ( u x z ) hold then q y > qu + u y

~>~ q x ~ xy , and q y ~ z y = q z --[- z y fol lows.

That the inequality does not a lways hold on % if z 'E a , . - - z with q z ' = 0

exists is seen as in (7).

The case where both ~i and % satisfy the inequali ty occurs, for example ,

in Lorentz spaces and is responsible for our definition of a timelike pair (q, a).

(10) Let a, b, c be points in Zt and not on one Arc. I f a b q - b c < ac and

bc ~ 0 and ~ is a segment connecting b, c then (a, ~) is neither spacelike nor neutral.

Because a c - - a b ~ b c the pair (a, a) is not spacelike. Assume (a, ~) was

neutral. If a x > O for (bxc) then l a b - - a c I = b c . If az-----O for (bzc) then

a b = bz , ac ~--- c z hence we have ab Jr- bc ----- b z -~- c z = ac.

AXIOM~ FOR INDEFINI T E METRICS 237

5. THE DECISIVE METRIC AXIOMS

The remaining two axioms require that locally all pairs (q, ~) are spacelike,

timelike or neutral.

M4 Let U (p) be a neighborhood of p for which U (p) X U (p) C ~to. Then

a neighborhood V (p) C U (p) exists a such that for any point q E V (p) and any

segment ~ which intersects V(p), lies in ffI(p) and has its endpoints on U ( p ) - U(p)

the pair (q, ~) is spacelike, timelike or neutral.

Ms If p is an interior point of the segment ~ then neighborhoods U~ of

and W(p) of p exist such that for q~ W(p) and ~" E U~ the pair (q, ~') is either

spacelike or is neutral with one zero o f q x in the interior of ~" or is timelike

with two zeros of q x in the interior of ~'.

According to (4.5) and (4.7) the function q x (x ~ ~') in /145 has exactly ene

zero if (q, ~') is neutral and exactly two if (q, ~') is timelike.

The following will be used frequently.

(1) Let U(p), V(p) satisfy M 4. I f q E V(p) and ~ is an Arc in ~J(p)

intersecting V (p) for which q x (x ~ ~) has more than two zeros then ~ is a null arc.

For, ~ can be extended to an Arc ~' in U(p) with endpoints on U(p)--U(p).

If ~ were a segment then (q, ~') would be spacelike, timelike or neutral,

contradicting (4.8).

Before preceeding it is convenient to make the following agreement: we

put U p = S ( p , 3~1(p)/4), remembering ~(p)----- min (~(p), 1) and then choose p ~ 0

such that for distinct x, y in lip = S(p, F) the arc ~(x, y) lies in Up and lip

has the property of V(p) in 344 for U ( p ) = Up.

Moreover ~(x, y) with x, y in Vp can be extended to a standard Arc ctxj,

in Ut, with endpoints on Up-- Up. We call ~y a standard extension r co(x, y).

It is not necessarily unique, therefore, x~-~x, y~-~y will in general not imply

OCxvyv"~xy. However, if we choose ~ y as the minimal standard extension,

then Ux~y~ can be chosen such that it tends to ~t~y. In the following we will

use ~xvy~'~ cr assuming that such a choice has been made.

We now prove two important facts concerning the distribution of nullarcs.

(2) I f p x = 0 for all x in a non-empty open subset W of lip then x y = 0

for all (x, y) ~ No.

If y, z are distinct points in W then p x has infinitely many zeros on %,~,

hence %~ is a null arc by (1) so that y z = O.

238 HERBERT BUSEMANN and JOHN K. BEEM

The set M of all points which have a neighborhood such that all Arcs

containing a point of the neighborhood are nullarcs is open and ncn-empty.

But M is also closed. For if a~ E M and a~-~ a, then all Arcs containing a are

nullarcs. Therefore, we conclude as before in (1), that all Arcs containing a

point b E V are null arcs, so a E M.

The proof of the next assertion is more involved.

(3) Theorem. The set of points at which null elements exist is either empty or the entire space.

The set is closed by (4.3). It suffices to show: if a null element exists

at p then there is also one at any q E V p . There are points a, b in Vp with

(apb) and a b = O . Because of (2) we may assume that a , - ~ a , b , - ~ b with

a,b , ~ 0 exist. Choose ~a~b~ and ~ab with ~v ~ ~ b v ~ ~b .

Suppose no null element exists at q. Then q ~ ~b and M4 give that (q, ~) is spacelike or neutral with no zero of qx on ~ . Hence ] q a ~ - qb~ I ~ avb~ and qa = qb.

Now let r be a point with (qrb) close to q. Then r E Vp. We choose ~qa

and ~r~ such that ~ra is so close to ~q~ that M5 is applicable to q and ~r~.

Because no null element exists at q the pair (q, ~,a) is spacelike, so that

qa < qr ~ ra. But now a null element must exist at r, otherwise we would

obtain, as above for q, that r a = r b , hence qa < q r q - r a = q r q - r b ~ q b .

This would imply that a null element exists at q, see (4.3).

(4) I f (q, ~y) is not timelike for any q, x, y (x ~ y) in Vp then a b q- b c =>~ a c for a, b, c in lip.

We may assume a c ~ 0 and that a, b, c are not collinear. Then ~c is a

segment and a b ~ - b c ~ -0 is impossible since this would make (b, ~a~) timelike.

If bc ~ 0 then ab q- bc < ac and (4.10) would imply that (a, ~b~) is timelike.

If a b e 0 and b c = 0 then (c, ~z~b) is neutral and a c = a b = a b q - b c .

(5) I f x y ~ 0 for (x, y) E ~ , then x y defines in Vp (for each p) a metric, moreover a b -~ b c = a c for distinct a, b, c in Vt, only i f (a b c).

The first part follows from (4).

Suppose non-collinear points a, b, c with a b q - b c ~ a c exist. Choose y

with (ayb) in Vp. Then b y - ] - b c ~ y c , a y - ~ - y e ~ a e , a b q - b e ~ - a c imply

ay ~ yc ~ ac. Let (uac), u E Vp and choose z with (zyc) such that z ~ u for

y - ~ a . Then (a, ~(z, c)) is not spacelike because ay = a c - - y c By M5 the

function a x (x E ~(z, c)) would have a zero for y close to a, contradicting the

hypothesis.

.AXIOMs FOR INDEFINITE METBICS 239

We remember that a metric space or its distance is called intrinsic if the

distance of any two points equals the greatest lower bound of the lengths (in

terms of the given distance) of all curves connecting the points.

(6) Theorem. Let Xoy > 0 for some fixed xo and all y with (xo, y)C Zt. Then the space R can be provided with an intrinsic metric ~(x, y) such that:

a) Each point has a neighborhood W(p) in which p(x, y) = x y.

b) The arcs of A are geodesic curves for ~ (x, y) and the length of o~ (x, y) in terms of p(x, y) is x y.

c) I f R is geodesically complete, then R is, with the distance p (x, y) a G-space.

Conversely, if R is a G-space with distance ~ (x, y) then a system A of arcs

can be defined in R such that all our axioms hold and x y is the restriction of

(x, y) to ~lo.

It follows from (3) that x y > 0 for all (x, y) s ~t.

Let z(t), O~ ~< t~< O~ be a curve in R. For a partition ~: 0~ = to < t~ < - . .

�9 . . < t,, = 0~ of [0~, 0~] we put []a[[ = max (b - - ti-l). For small []~[[ we have

~o for all i. Using (4) we find in the usual way (compare (z (t,), z (t,_,)) e [2, p. 20])that

m

~,(z)= lim ~_~z(t,_,)z(t,) II A II - ~ 0 i = 1

exists if we admit ~ as limit. Consequently this length is additive. Clearly

(7) ~ (z) = x y if z (t) represents ~ (x, y).

We define ~(x, y ) = inf ;~(Cxy), where Cxy is a curve from x to y. Then (7)

implies that p(x, y) is always finite.

If Ixlpx < 8" < ool c vp let W(p) = {xIPx < ~'/21. We know, see (5),

that x y defines a metric in Vp. Therefore any curve joining two points x,

y E W(p) which does not lie entirely in V(p) has at least length ~'.

On the other hand, ;~(C):>~ x y for any curve in Vp and ).(~r y ) ) = xy~<

~< x p -}- p y < ~', see (2). This proves that ~(x, y) is a shortest curve from x

to y, hence

(8) ~(x, y ) = x y for x, y in W(p).

Further, ~(x, y ) > 0 for x ~ y . Since ~(x, y) = ~(y, x) and p(x, y)-Jr-

-t-- ~(Y, z) ~ ~(x, z) are obvious, ~(x, y) is a metric. It follows that the length

).(C) coincides with the length of C in terms of p(x, y). The definition of ~(x, y)

entails that ~(x, y) is intrinsic.

2 4 0 HERBERT BtlSEMANN and JOHN K. BEEM

The isometric map s-~ x(s) of a half open interval [0, ~) ('~ < oo) into a

metric space R is called a finite ray in R. Clearly, either no sequence {x(s~)}

with s~-~'c has an accumulation point, or lim x ( s )= q exists. In the latter case

defining x(-c)-----q makes x(s) isometric to [0, .c] and we say that the finite ray

can be completed. A necessary condition for the finite compactness of R is that

every finite ray can be completed. The Hopf-Rinow Theorem states that this

condition is under certain circumstances also sufficient:

(9) A locally compact metric space with an intrinsic distance is finitely

compact if and only if each finite ray can be completed.

This form of the theorem is due to Cohn-Vossen [3]; it can also be

extracted from [19, chapter IV]. Hopf and Rinow [5] as well as most other

authors deal only with Riemann spaces.

Obviously, finite rays can be completed in geodesically complete spaces.

So R is finitely compact (with metric ~(x, y)) if geodesically complete.

In a finitely compact space with an intrinsic metric any two points x, y

can be connected by a metric segment T(x, y), i.e. a curve of length p(x, y).

Locally the metric segments coincide with the Arcs. As and A4 guarantee the

local existence and local uniqueness of prolongation of metric segments. The

latter implies that prolongation is unique in the large [2, pp. 36, 37].

Conversely, if R is a G-space with distance p(x, y), then each point has a

neighborhood U (see [2, pp. 38, 39]) such that for any two points x, y in U

the segment T(x, y) exists, is unique and lies in the interior of a segment

T(x', y') with endpoints in U. Every subarc of a metric segment is one too,

and T(x~, y~)~ T(x, y) if x~-~x, y~-~y and T(x, y) is unique (T(x~, y ~ ) ~ x if limx~ = lim y,, = x). Therefore, putting re(x, y ) = T(x, y) for distinct x, y

in U we obtain an arc system A satisfying A1_5. The pair (q, T) is spacelike

or neutral for any q and any metric segment. Therefore R satisfies M1_5 and is,

of course, geodesically complete.

6. TWO-DIMENSIONAL SPACES

In this section we will show that there are four types of two-dimensional

spaces satisfying our axioms. One type is x y = 0 for (x, y)E ~1. and is unin-

teresting, a second is x y > 0 for (x, y)E ~ and has been disposed of in

theorem (5.6). The other two types will be discussed in greater detail.

AXIOMS F O R I N D E F I N I T E ME ' IRICS 241

(1) Theorem. Let dim R = 2. Then there is either no null element, or exactly

one at each point, or there are exactly two at each point, or all line elements

are null elements.

We know from (5.3) that there are no null elements, if there is one point

at which no null element exists. We therefore assume that there is at least one

nutl element at each point, and show first:

(a) If three null elements exist at the point p, then all line elements are null elements.

Let 0<~ ~(p, q ) < ~i(P)/4. Then p E Uq. Assume q does not lie on an

Arc in one of the given null elements. In each of these elements take standard

null Arcs % ( i = 1, 2, 3) through p. A standard Arc ~ through q which passes

close by, but not through p intersects r %, %. By 3'/4 (or Ms) ~ is a null Arc.

The union of these ~ contains an open set M homeomorphic to the interior of

a triangle and q E AI. Therefore q satisfies the hypothesis of (5.2) and (a) follows.

(b) The set of points where exactly one null element exists is open.

We assume that a sequence of points x~ tending to p exists with two null elements at x~ and show that this leads to a contradiction.

Let ~0 be a standard Arc in the null element at p. We show first that {x~}

cannot contain a subsequence [xml on %. Denote by ~zm a standard null Arc

through xm not in the line element containing %. Choose a standard segment

through p. For large m the Arc ~m intersects ~ in a point gm (otherwise %~-~

and ~ would be a null arc). If y E ~ with (gmPY) then (Xm, ~) is timelike (for

large m) hence xmy ~ p y. On the other hand (p, C%ny ) is neutral which gives

xmy = py .

olo

16 - R e n d . Circ. M a t e m . P a l e r m o - S e r i e II - T o m o X V - A n n o 1 9 ~

242 HERBERT BUSEMANN and JOHN K. EEEM

Assume now x, ~ a and let a~, a~' be two standard null Arcs through x,

with ~r N a~' =: x, . Then a~ -~ ao, a: ~ a0 moreover, because of the first part

of this proof, a~, ~' do not intersect a0 in a certain neighborhood of p. Choose

a standard segment a through p not containing any x~. Then a~, a: intersect

in points g : , g : on one side of p and we may assume that this is the same

side of p on a for all v. Choose y on a with (g'~py). Then a(x~, y) intersects

ao in a point w, and (p, ax~) is neutral (for large v). This yields p x , ~ x.,w,,,

p y : w , ~ y , hence x~p--kpy-- - - -x~y. On the other hand (x,~, a) is timelike so

that x,, y ) x,~ p -4- P Y.

(c) The set of points at which two null elements exist is open.

The idea of the proof is the same as in (b) and we choose the notation

so that this becomes clear.

Assume there are two null elements and corresponding standard Arcs ~', ~"

(with respect to Ux) at x and that a sequence p ~ x exists with only one null

element at p~. The Arcs ~', ~" divide the plane locally into 4 closed angular

regions and we assume that p~ is always in the same one of these. We choose

a standard segment ~ through x such that one of the subsegments of ~ bounded

by x lies in the same region as p~, and on this subsegment a point y - ~ x

in Vx. Let (y x u), u E V~, then choose u~ with (yp, ,u~) and u~-~ u. Then

(always for large v) ~r u~) intersects ~' and ~" in points g~ and g~'. We I air ts choose the notation ~ , such that (g~ g,, y). The pair (x, ~(y, u,~)) is timelike

by M 5, hence x y > p~ x + p~ y. The pair (p~, ~) is neutral. If w,, E ~ and p w,~ ~ 0

then either w~E~(x, y) and we see as under (b) that x y ~ x p ~ + p , ~ y or

(u w, x) then p~ y - - p , x ~ x y.

Clearly (a), (b) and (c) prove (1).

To study the cases with one or two null elements at a point in greater

detail we introduce open convex sets C whose existence we proved at the end

of section 3, but reserve the term line for paths in C consisting of segments,

and denote the other paths in C as null lines. We remember C X C C ~o .

By (1) either all paths are null lines or all are lines, or there is exactly one

null line through a given point or exactly two. The interest of our results will

partly lie in that they show how repeated application of M5 yields statements

on distances for any two points of C.

Basically there is only one type of C with one null element at each point.

To formulate this precisely we need the concept of a topological isometry.

AXIOMS FOR INDEFINITE METRICS 243

Let M~ be a subset of the topological space /?i and let a function ~i(x, y) be

defined on MeXM,. with ~,.(x, x ) = O , ~i(x, y ) = ~ ( y , x )>10 . The map qb of

M~ on M2 is a topological isometry, if it is topological and ~.~(q~(x), ~ ( y ) ) =

g~(x, y) for all x, y in M~. Obviously the requirement that �9 be topological

cannot be omitted, as in the case where one ~l (and hence also the other)

satisfies the conditions of a metric space.

(2) Theorem. I f there is exactly one null line through each point o f an open

convex set C in a two dimensional space, then C is topologically isometric to a

subset s E IN, Ill < co of the (s, t)-plane with the distance ]si - - s~l , where IN is

a connected open set o f the s-axis.

The null lines cover C simply and are topologically equivalent to a family

of parallel lines in the plane (see the end of section 3). Let N , , N3 be two

distinct null lines. We show that x~ x3, x~ E N~ is constant. Let a, E N~ then the

set of points x, E N; for which x~ a a = a i a a is non-empty and closed. To see

that it is also open let Yi E N, and y, aa = aiaa.

Choose y~ with (y~ Yl a3). Applying M 5 we find a neighborhood U of Yl such

that (y~, ~ (x~, a3) ) is neutral for x~ E U. This proves that x~ a 3 -= ai a3 for xj E Nl.

Reversing the roles of N, , Na we find x t x 3 = a,a~ for x,. E N~.

We put d(Nt , N ~ ) = a~ a 3. If N. 2 lies between N, and N3 it will intersect

(a,, a3) in a point a 2 and a i a 2 + a.~a 3 ~ a iaa yields d(AT,, N,2) + d(N2, Na)

= d(N, , N3). Since the null lines form a set I homeomorphic to a line, we see

that I with d(m~, N.2) as distance is isometric to a non-empty open connected

set IN of the s-axis. If s(N) is the image of N under such an isometry then

[s(Nt) - - s(N.~)] = x,x~ for x~E Nl

whence (2) readily follows.

The following is a corollary of (2) and its proof.

(3) I f Z i o = R X R, direR-----2 and there is exactly one null element at

each point, then R is topologically isometric to a subset s E I~, It] < oo of the

(s, t)-plane. I f a line L exists with a representation x(s), Is] < ~ , then Ia = (--0% oo)

and all null lines intersect L.

We turn to the last case and show first:

(4) I f C is an open convex subset o f a two-dimensional space in which

exactly two null elements exist at each point, then the null lines in C can be

divided into two disjoint families ~'~v, ~'Q such that the lines in each family

cover C simply.

244 HERBERT EUSEMA~N and JOHN K. EEEM

A moment's reflection shows that (4) is equivalent to:

(5) Under the assumptions of (4) there is no triple p, q, r of non collinear

points in C with p q = q r = r p = 0 .

Assume such a triple exists. The null line Nx through x E ~(p, r) not

containing ~(p, r) depends continuously on x because of its uniqueness. If

(p x I-) then Nx intersects ~(p, q) t3 ~(q, r) - - p - - q - - r in a point x' depending

continuously on x. Therefore either always (px 'q ) or always (qx'r). Take the

second case. The map x -~ x' is injective, moreover x ' -~ q for x ~-p. Also

x ' - ~ r for x - ~ r because x ' - ~ x is the corresponding map of ~(q, I " ) - - q - - r

in o~(q, p) U o:(p, r) - - p - - q - - r.

The analogous map of z E **(p, q ) - p - - q in ~(p, r), ~(r, q ) - - p - - q - r

would produce three null lines through the image of z.

Since the families ~N and fQ of (4) form sets homeomorphic to a line

they each have two natural orderings or orientations.

(6) Under the hypothesis of (4) let g~v and g~ be orientations of ~N and

gQ and denote the elements of gN and gQ through x by N~ and Qx. Define

x < y to mean N, < Ny and Qx < Qy. Then C with the partial ordering x ~ y

and the distance x y is a timelike space. Moreover x y + y z = x z for x < y < z

only i f (x y z).

It is clear that x y > 0 for x < y and x y + y z - ~ x z for (xyz) . Assume

that x < y < z and not (xyz). To establish x y + y z < xz it suffices to show

that f ( v ) - - - - - v z - - v y increases on ~(y, x) (meaning as v tranverses ~(y, x)

from y towards x) because f ( y ) = yz .

Let (wyz) and choose a neighborhood W of w and u with (yux) such

that M5 is applicable to the pair (v, ~(w', z)) for v E oc(y, u) and w'E W. We

can choose u such that in addition for v~ E ~(y, u) and w(v,)~ W with (w(v~)vlz)

the lines Nv~, Qv~ intersect ~ (w (vl), vl) in interior points if (v 2 v, y) and v2 E ~ (y, u).

Then (v2, ~(w(vi), z)) is timelike and v~z > v~v~ + v i z or

f(v~) = v~z - - v2y ----- v,~z - - v2v~ - - v~y > v~z - - v~y = f(v~)

the set of points u on re(y, u) such that f (v) increases on ~(y, u) is closed in

(y, x ) - x and contains, as we just saw, a neighborhood of y. But we can

repeat this argument for any u with (yux) and find a u, with (uu, x) such that

u"z ~ u 'u" + u 'z if u" follows u' on ~t(u, u~). Then f (u") > f(u ') follows as

before. This contains that f (v) increases on ~(y, x).

AXIOMS FOR |NDEF/NITE METRICS 2 4 5

We combine our results.

(7) Theorem. Let every point of an open convex set C in a two-dimensional

space R lie on exactly two null elements. Then the null lines form two disjoint

families ~N and gQ which cover C simply. Denote the lines in ~,v and gQ through

x by Nx and Qx let I;r ~ be orientations of IN, IQ. Define two partial orderings

x <, y of C as follows: x <t y means Nx < Ny and Qx < Qy and x ( ~ y

means Nx < N~ and Q~ < Q,. For each x ~ i Y the given distance x y defines

in C a timelike space, i.e. x y + y z ~ x z if x <, y ( i z and equality holds

only if (x y z).

Note that any two points x, y satisfy exactly one of the relations

x y = 0, x < t Y, y < t x, x ~ y, y ~ x.

The present results yield in conjunction with well known topological facts:

(8) If a compact surface R satisfies A~-5 and M~-5 and if at some point of R there exist either precisely one or precisely two null elements, then R is a

torus or a one-sided torus.

In case no null element exists, the argument in the proof of (6) yields

that f(v) decreases on a(y, x) if x, y, z are distinct and not collinear. Moreover,

the position of the zeros of v2x on a(w(v), z) does not enter. So we have:

(9) If C is a convex subset of (a not necessarily two-dimensional) space R

in which no null elements exist, then x y + y z >~ x z for any distinct points

x, y, z in C with equality only for (x y z).

The two-dimensional Lorentz space (see introduction) is the simplest example

for (7). A less trivial example is provided by an analogue to the Poincar~ Model

of hyperbolic geometry leading to the system R h of Section 2.

Consider dSh = [dx2, - dx~['~*x-; ' in x 2 > 0. The curves a) of R h are evidently

the null paths. The extremals for the variational problem with (dx~--dx~)'~x~ '

as integrand are the curves in b) and those for (dx2,--dx~)'l~x; ~ are the curves

in c). The distance x y is defined for (x, y)E ~ as the length of the Arc a(x, y)

in terms of dSh.

Define x < , y by x ~ < y ~ and y ~ - - x ~ > [x2--y2[ provided (x, y) E ~ .

Similarly define x <2Y by x 2 < Y2 and Y2~X2 > [x~--y~l; in this case always

(x, y) E 2t. Then x y + y z ~ x z for x < , y , ( , z with equality only for (xyz).

The cases x ( ~ y or x <~ y correspond to x, y lying, respectively, on a path

of b) or c).

246 H E R B E R T HUSEMANN and J O H N g . B E E n

d Sh and hence x y is invariant under the quasihyperbolic group (2.9). Its

elements are therefore motions of R ~, i.e. topological isometries of R h onto itself.

We notice that if x lies on a line L (i.e. a path in b) or c)) and y traverses

one of the rays into which x divides L then x y - ~ oo only if L lies in c) and

Y2"~ 0. Thus all rays, except for the latter type, are tinite rays in the sense of

the Hopf-Rinow Theorem, the space is not geodesically complete and finite rays

cannot always be completed.

This is in sharp contrast to the analogous situation for metric spaces: if a

locally compact metric space possesses a transitive group of motions then it is

complete, and hence finite rays can be completed.

It is true that R h can be extended to a two-dimensional geodesically com-

plete space R, such that the elements of the quasihyperbolic group acting on R h

are restrictions to R h of motions of R. But then we have another phenomenon

which does not occur in G-spaces, namely a group of motions which is not

transitive although it has a non-empty open orbit, see table in [12, p. 63].

Los Angeles (California), May 1966.

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Riemann-Tagung Forschungsinst. fur Math., Heft 1 (1957), 33-84. [2] H. Busemann, The Geometry of Geodesics, New York, 1955.

[3] S. [4] f t .

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Cohn-Vossen, Existenz kiirzester Wege, DokI. Akad. Nauk SSSR 3 f1935), 339-342.

Freudenthal, Das Helmhollz-Liesehe Raumproblem tel indefiniter Metrik, Math. Ann.

156 (1964), 263-312. [5] H. Hopf und W. Rinow, []ber den Begriff der vollsl?indigen differenlial-geometrischen Fldche,

Comment. Math. Helv. 3 (1932}, 209-225. [6] D. Montgomery, Locally homogeneous spaces, Ann. of Math. 52 (1950), 261-327. [7] M. Morse, A special parametrization of curves, Bull. Amer. Math. Soc. 42 (1936), 915-922. [8] W. Noll, Euclidean and Minkowskian geometry, Amer. Math. Monthly 71 (1964), 129-144. [9] W. Rinow, Die innere Geometrie der metrisc~.en Rdume, Berlin-CSttingen-Heidelberg 1961.

[10] B. A. Rozenfel'd, Non-euclidean geometries (Russian), Moscow 1955. i l l I H. E. Vaughan, On locally compact metrizable spaces, Bull. Amer. Math. Soc. 43 (1937),

532-535. [12] 1. M. Yaglom, B. A. Rozenfel'd and E. U. Yazinskaya, Projective metrics (Russian), Uspehi

Mat. Nauk 19 (1964), 51-113. English translation in Russ. Math. Survey 19 (1964), 49-107. [13] H. Busemann, Timelike spaces, Dissertationes Math. 53 (1967).