Metric Methods of Finsler Spaces and in the Foundations of Geometry. (AM-8) 9781400882298

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M e tric M e th o d s in Finsler Spaces and in the Foundations of Geometry BY

H ERBERT BUSEMANN

PRINCETON P R I N C E T O N U N I V E R S I T Y PRESS LONDON:

H U M P H R E Y M IL F ORD

OXF ORD U N I V E R S I T Y PRESS

1942

Copyright 1942 P

r in c e t o n

U

n iv er sity

P re ss

L ithoprinted in U.S.A.

E D W A R D S ANN

B RO T H E R S ,

ARBOR,

MI CHI GA

1942

INC.

PREFACE Among the earliest speculations on the foundations of geometry we find many attempts to introduce the straight lines as geodesics. But no abstract concept of a metric being known, let alone metrics other than the euclidean and perhaps the spherical, these attempts were futile. Today it is not difficult to formulate axioms for a space in which geodesics exist. This hook treats some of the many problems which arise when one attempts to de­ velop geometry with the geodesic as basic concept. The problems studied here fall essentially under four topics which may be listed, roughly as Finsler spaces, parallels, convexity of spheres, and motions. The choice of these topics is, of course, due partly to personal preference but partly also to the desire to impress the reader with the large variety of questions which fall under the scope of the metric methods. It goes without saying that there are many unsolved problems, very different in character and difficulty. A number of these will be formulated in the text. The book is divided into five chapters, each of which is preceded by a rather detailed introduction. A reader who wishes to get information beforehand concern­ ing the whole content is asked to turn to these intro­ ductions. The idea of completing Prechet1s axioms for a metric space, so as to ensure the existence of geodesics, is due to Menger. His results as far as we shall need them, will be proved in the text. Some familiarity with the topolo­ gy of metric spaces is assumed, and theorems on convex bodies are used. Results of Riemannian geometry will be frequently referred to for comparison, but not actually

PREFACE applied (except in Chapter V §3 ). All facts from other tneories, if not proved in the text, will be stated in exact form and reference to literature will be made. Although Menger was the first to study geodesics in metric spaces, both his and his students1 contributions to the foundations of geometry and the calculus of varia­ tions have so different a trend that the existence of geodesics plays hardly any role in their work. For that reason the material presented here is almost entirely different from the theories found in Blumenthal1s Distance Geometries. Because ;the topics of this book are interrelated with several different fields, even a moderately complete list of the pertaining literature was impracticable. To avoid inconsistencies the bibliography contains (with the exception of Finsler!s dissertation) nothing but references for results actually quoted in the text. Herbert Busemam Illinois Institute of Technology

TABLE OP CONTENTS Preface Table of Contents Chapter I. METRIC SPACES WITH GEODESICS ........ 1 §1. Metric Spaces; Notations ......... 1 §2. The Basic Axioms .................... 11 ....................... 17 §3 . Geodesics §4 . Topological Structure of One- and Twodimensional Spaces With Axioms A - D ...... 2k II. METRIC CONDITIONS POR FINSLER SPACES . . . 30 Convex Surfaces and Minkowski Metrics . . . . 31 ...... ko Riemann Spaces and Finsler Spaces Condition a (P) and the Definition of the Local Metric .................... kl §4 . Equivalence of the Local Metric with the Original Metric, and its Convexity . . . . 53 §5. The Minkowskian Character of the Local Metric . 57 §6. The Continuity of the Local Metric ...... 63

Chapter §1. §2. §3.

Chapter III. PROPERTIES OP GENERAL S. L. SPACES . . . 72 (Spaces with a unique geodesic through any two points) ...... 73 §1. Axiom E. Shape of the Geodesics §2. Two Dimensional S.L. Spaces .......... 79 §3. The Inverse Problem for the Euclidean Plane 89 $4 . Asymptotes and Limit Spheres ........... 98 §5. Examples on Asymptotes and Limit Spheres. The Parallel Axioms ................. 105 §6. Desarguesian Spaces ................. 113

TABU3 OF CONTENTS

.

. . . . . . 119 . . . . . . . . . 120 . . . . . . . . . . . . 124 . . . . . . . . . . . . . . . . . 132 . . . . . . . . . . . . . . . . . . . 139

Chapter IV SPACES WITX CONVEX SPHEBE3 $1 The Convexity Condition $2 Characterization of the Higher Dimensional Elliptic Geometry $3 Perpendiculars in Spaces with Spheres of Order 2 $4 Perpendiculars and Baselines in Open S L. Spaces $5 Definition and Properties of Limit Bisectors 56 Characterizations of the Higher Dimensional Minkowskian and Euclidean Geometries $7 Plane Minkowskian Geometries $8 Characterination of Absolute Geometry

. . . . . .

. . . . 154 . . . . . . . . 160

.

.

. . . 168

.

. . . . . . . . . . . . . . . . 175 . . . . . . . . . . . . . . . 176 . . . . . . . . . . . . . . 184

Chapter V MOTIONS $1 Definition of Motions Involutoric Motions i n s * JJ SPCeS $2 Free Movability $3 Example of a Non-homogeneous Riemann Space in which Congruent Pairs of Points Can be Moved into Each Other $4 Translations Along g and the Asymptotes to 8 $5 &uasi-hgperbolicMetrics $6 Translations Along Non-parallel Llnes and inclosedplanes $7. Plane Geometries with a Transitive Group of Mot ions $8 Transitive Abelian Groups of Motions in Higher Dimensional Spaces $9 Some Problems Regarding S L Spaces and Other Spaces

. . . .

. . .

.

Literature

Kind

. 146

.

. . . . . . . . . . . 192 . . . . . . . . . . . . . . . . . . 198 . . . . . . . . . . 208

. . . . . . . . . . . . . . 214

. . . . . . . . . . . . . . . . . 220 . . . . . . . . . 228

. .

. . . . . . . . . . . . . . . 232 . . . . . . . . . . . . . . . . . . . 235

Chapter I. METRIC SPACES WITH GEODESICS Introduction. The geodesics of the Riemann Spaces and Finsler Spaces which one usually considers have the following properties: 1) Any two distinct points P,Q can be connected by a shortest arc. 2) If the point Q is sufficiently close to P, this shortest connection is unique. 3) Any shortest connection between two points is contained in one and only one geodesic. In the present chapter we give a set of axioms for a general metric space which guarantee that properties 1), 2), 3) hold. We first compile the definitions and theorems on metric spaces which we shall need later. In Section 2 we formulate the basic axioms A, B, C, D and show that 1) and 2) hold. In Section 3 we define geo­ desics and prove that they have property 3). Finally (Section 4 ) we discuss the topological structure of the spaces which satisfy axioms A - D and have dimensions 1 or 2. In both cases we shall find that the space is a manifold. The corresponding question for higher dimen­ sional spaces is open.

51 . METRIC SPACES; NOTATIONS Points, unless expressed by their coordinates, will be designated by Latin Capitals. A point set £ is a metric space if a real number XY, the distance from X to Y (or of X and Y), is defined for every pair X, Y of points in X and satisfies the conditions: 1

2

I,

METRIC SPACES WITH GEODESICS

A1 XX « 0 A2 X Y = Y X > 0 for X 4 Y (symmetry) Y (symmetry) A^ XY + YZ XZ (triangle inequality) Any subset a of Kind a metric space E becomes itself a metric space, if we define as distance for points X,Y in cr the distance X Y of these points in £ . Whenever we speak of a subset cr of a metric spaces without defining its metric, we imply that or is metrized in this way. We say the point Y lies between the points X and Z, and write (XYZ), when Y is different from X and Z and X Y + YZ - XZ. The relation (XYZ) has the following ob­ vious but very useful properties: THEOREM 1. If (XYZ) then (ZYX). If (WXY) and (WYZ), then (XYZ) and (WXZ). The use of the words limit p o i n t , closed s e t , open set being uniform we do not re-define these concepts. (Definitions and proofs, which are omitted here, can be found in standard works as HAUSDORFF [2] or KURATOWSKI [1 ].) We call the set a in a metric space s bounded, if a point P and a number cx exist so that PX < cc for X C a A metric space X will be called compact, if every infinite sequence of point5=1 i-n. 2 contains a converging subsequence| finitely compact, if every bounded, infinite sequence contains a converging subsequence. We remind the reader of the following facts: THEOREM 2 . A closed subset of a compact metric space is compact. A bounded closed set in a finitely compact metric space is compact. A closed set in a finitely compact space is finitely compact. THEOREM 3. A finitely compact metric space I! is separable, i.e. there is a se-

S1. METRIC SPACES: NOTATIONS quence of points P1, P2, ••.in E , so that every point of E is limit point of a suitable subsequence of fPv |.

1

Consider two metric spaces E and E1and a mapping of the subset a of E onto the subset cr1 of E1 , i.e. for every point X of a the image X f « F(X) of X in a1 is uniquely determined and F(X) traverses all of cr* when X traverses cr . The mapping X —»F(X) is called a) continuous, if lim Pv = P , Pv , P C cr 9 implies lim F(PV ) = lim F(P). b) topological, if it is one-to-one and continuous both ways. The sets a and a1 are homeomorphlc if a topological mapping of a onto or1 exists. c) a congruence, if FQ - F(P) F(Q) for any points P,Q in cr . A congruence is a topo­ logical mapping. The sets crand a1 are congruent, if cr can be mapped onto cr1 by a congruence. In case E or E1 is the realaxis, special terms are used. Namely, if E is the real axis - oo ) the endpoints of cr , and say that cr con­ nects A and B. This definition does not depend on the choice of the isometric representation of cr because A and B can be characterized by the property that (AXB) for every point X ^ A,B of cr . The notation AB will be used for any segment with A and B as endpoints. It is convenient to put AA = A. Of any three points on a seg­ ment one is between the two others. If £' is the real axis, and X F(X) = X ! maps the subset cr of the metric space 2 onto the set cr1 in 2 1 , then F(X) becomes a real valued function of the point X, with o as domain of definition and cr' as range of F(X). Many known theorems on functions of real variables can be extended to mappings of metric spaces. .We shall need only:

THEOREM k. If X — > X* = F(X) Kind is a continuous mapping of o onto or* , and if o is compact, then F(X) is uniformly con­ tinuous ; i.e. for a given € > o a 6 > o can be found such that F(P) F(Q) < € as soon as FQ < 6 . and THEOREM 5. A continuous real-valued function F(X), defined on a compact subset cr of a metric space reaches its minimum and its maximum. As distance a t (or d( cr , t ) when there is a possibility of taking 4 ] who studies much more general cases than ours.) Let c : P(t), a £ t £ b, be a continuous arc. Take any subdivision A : a = tQ < t1 < t2 < ... < tn - b of the interval (a,b) and form n-i

SI. METRIC SPACES: NOTATIONS 1 The least upper bound of L(A ) as A traverses all sub­ divisions of (a,b) is called the arclength or simply the length L(c) of c ( oo admitted). For every sequence A^ = ( t ••• , t^ ) of subdivisions of (a,b) with .lim max 1_*°° l^n, we have PROPERTY L a)

(t^ - t A ) = o J J~1 L( A±) — » l (c )

For, obviously For a given e > o let division of (a,b) with

L( A±) ^ L(c). A - (tQ, ..., t ) be a sub­

L( A ) > L(c) - -§- (L( A ) y N+^when L(c) =■ oo ) The interval (a,b) being compact it follows from Theorem i* that a 6 > 0 exists such that (3 ) P(t)P(t«) £ 4 for |t-t'| < t issuch Kind a mapping (isometric representation). PROPERTY L c)* For every continuous curve c = P(t), a ^ t ^ b we have (k) P(a)P(b) i L(c) and if the equality sigtf holds the points P(t) form a segment. If the points P(t) form a segment and if P(t1) « P(t2) for t1 < t im­ plies P(t) = P(t1) for tj t ^ tg the equality sign holds in (4 ).

(

PROOF. We have for every subdivision A = •••,t^) • P(a)P(b) i

Z P(t1)P(t1+1) i

L(o).

If P(a)P(b) - L(o) we have P(a)P(b) = Z P(t±)P(tl+1) for every A therefore (5 ) P(t')P(t") + P(t")P(t"') = P(t')P(t" ') for any t,i < t1’< t111. We now map P(t) on the point t => P(a)P(t) of the interval 0 £, x av

and tJ1_1 < b v hence

putting t^ = a v , t^ = b v , t^ = t^_ for 1 ^ L(cv ) i

£

1=0

Pv ( t > v (t’+1 )

-

0,n

I

)

1=0

Now (6) follows from the arbitrariness of € or N. PROPERTY L e)• Existence of a minimum. In the finitely compact metric space 2 let a continuous arc of finite length from A to B exist. Then a continuous arc c from A to B exists whose length is smaller than or equal to the length of any continuous arc from A to B in 2 . PROOF. Call b the greatest lower bound of the lengths of all continuous arcs from A to B. There is a sequence of (not necessarily different) arcs cvfrom A to B, so that bv = L(cv ) —» b. On cv we may intro­ duce the arc length s as parameter because the length is thereby not increased. We may thus get the representation Pv (a), o £ a £ bv , Pv (0) « A,PV (bv ) = B for c v . For every sv with 0 sv £ bv the se­ quence of points Pv (sv ) is bounded on account of L c). Let r1, r 2, ... be the sequence of rational numbers between o and 1. We choose the subsequence [1v | of }v | so that the points P- (r.-b- ) converge, then in 'v ' 'v |1V | a subsequence (2V | so that the points Pg (**2b2 ) converge and so forth. We -form the diagonal sequence v Pv (a). We conclude from the triangle inequality that PVy (s) converges for every s between 0 and b, to a limit point P(s) and that P(s), 0 0, with the following property: For any two pointa A ^ B in this neighborhood and every e y 0 there ia a positive 0; let for in­ stance tQ0Q1 - QqR = m Because of Theorem I 1.1 every point X between R and Q1 is also between Qq and Q1. Hence it follows from the definition of m that |Q0X - tQ0Q1| = |RX + QqR - tQQQ1| - |RX - m| ^ m, or RX ^ 2m. Hence, if X traverses the points between R and Q1, the function RX reaches a positive minimum ^ 2m at some point S. But according to Axiom C there is a point Y with (RYS). We should have (RYQ1) and RY < RS, which contradicts the definition of S. We now return to the originally given points PQ, P1. Let r1,rg,... be the different rational numbers be­ tween 0 and 1. As we just saw, a point Pr with P0Pr = rlPQP1 and (PQPr P1) exists. Assume that ihe points 1 P ,P.,P ) ... ,1P have already been determined so that 1 (O

P^.Pr.. = Irj - Pjl • Pq P,

Li xj To determine Pr we choose among the numbers 0,1,^,..., rn-1 30 pair r,r so that r < rn < P and. the difference r-r becomes as small as possible. According to the first part of this proof we can determine P so that (Pn r P„ r P=) r and n _ r -r -gfi — — p p. . ( r - r)P.P.. p_p . r rnr - r rr 'n 'o i This shows that the points , n « 1,2,.., can be de­ termined in such a way that n (1 ) holds. Prom (1 ) we conclude: if r£ and r£f are any two sequences of ration­ al numbers between 0 and 1 tending to the same limit t, then the sequences P^, and Pp,, converge and have the 1

_U__________I. METRIC SPACES WITH GEODESICS_________ same limit, which we may call P^. Prom the continuity of the metric we see that PtfV- I*’ - ‘"I popl hence P^ —» t.PQP1 Is a congruent mapping of the points 0 ^ ^ ^ 1 0Ilt0 30 interval 0 £ 3 ^ P0P*| of the real axis. Prom this theorem and from Theorem I 1.1 we conclude THEOREM 1!. If (AQB) then a segment from A to B exists which contains Q. If* we want statements regarding the uniqueness of the segment connecting two given points, we must use Axiom D. To formulate such a result we introduce a no­ tation, which will he used often throughout the hook: for any set a we designate by v ( a , p ) the set of all points X for which X a < p . In particular, v (P, p ) consists of all the points X with XP < p . Occasionally we shall call v (P, p ) a spherical neighborhood of P. We show first THEOREM 2. Let the neighborhood v (P, p ) of P satisfy D(P). If two points PQ,Pa can be connected by a segment which lies wholly in v(P, p ), then there is no other segment (in s) connecting PQ and P^ NOTE. The condition of the theorem is satisfied for Pa = P and an arbitrary PQ in v(P, p). PROOF. Assume that there are two different segments cr and ct from P^ o to P., a formed by the points PT and PT , o £ t^ a = PQPa respectively, and. so that Pt1 Pt 2 “ *T1 f r 2 ” IT2 - T1 I for any Tj_ 30 in (0,a). Suppose a lies in v (P, p ). Let PTO PT -O and denote by Xj the first T / T0 ^or which PT = PT . Put P„ = P_ == B. According to D(P) there is a point 1 1

§2. THE BASIC AXIOMS

15

v (P, p ) with (P„PDA), hence (Theorem I 1 .1 ) we O Or ^

A in

also have (PQBA), (PT BA) and. (PT BA) for every t < t 1 . But E,.

^ PT

for T0

and every 6 £ e ( A B B ) and BB^ A and

T

< Ti

ao that for e «= x1 - xQ

two different points B^ o

with the following property: For any two distinct point a A,B in

v (P, p ) the

points X with (AXB) or (XftB) and XA £ 2 p , or (ABX) and BX

2p

form together with A and B a segment

ftEf of length AB + 4p , and this is the only segment connecting A' and B'. It should be noticed that for any two different points of a S.N.

v(P,p ) the following statements hold:

a) Given a number t with o < t < 1 , there is exactly one point C with (ACB) and AC = t AB. b) Given a number s with o < a £ 2 p there is exactly one point D with (ABD) and ED ** s and exactly one point E with (EAB) and AE - s. c) S B is unlquie and contained in a unique segment of length 4p with center A. Clearly a space which satisfies Axioms A,B,C, and in which every point has a standard neighborhood, also sat­ isfies Axiom D. The converse holds too. Therefore we have THEOREM 3 . A finitely compact, convex, metric space satisfies Axiom D, if, and only if , each point has a S.N. To prove the necessity we show

16

I. METRIC SPACES WITH..GEODESICS THEOREM 3 '.

If Axioms A,B,C hold

and the neighborhood fies D(P), then PROOF.

v(P,p ) of P satis­

v(P, -^p ) is a S.N. of P.

Let A,B c v(P, p), o < p < £ - ^ - p

.

For

(AXB) we have (1 >

PX

£

|BP

+ B x}
6 p ).

v(P, 3 p ) and (VZW)

we

Hence a segment W

lies wholly in v (P, p ) and is, because of Theorem 2 , the only segment connecting V and W.

This holds in particu­

lar for any two of the above points X. The set M consisting of A,B and the points X with (ABX) and BX £ 2 p is closed and bounded, therefore com­ pact. B'.

Hence BX reaches on ^ its maximum at some point We have B B 1 = 2 p .

For if BB' = 2 p - e , e > o,

Axiom D would imply the existence of a point Btfwith (AB'B^ ) and o < B'B^

£

B B 1. BB' - 2 p exists.

, but then (AEB,j ) and BB^

=

We see that a point B' with (ABB1) and

We now consider the set |V consisting of A and the points X with (XAB*) and XA 0 auch that

P(t 1 )P(t2 ) - |t2 - t1| for |t± - t|
P(

r) =

±1

and

a is an arbitrary real number and

Tit+a ) the mapping t — ► Q(t) will be another

isometric representation of g. As we shall see later, all isometric representations of g can be obtained in this way. Let P(tQ ) he any point of g. For ■ 0

b ' £ "E < b, and a sequence of positive numbers

would exist such that P("t + riv ) ^ Pft + n v )• We have (P(T - r|v ) P(t)P(t + r^v )) and for sufficiently large

v, (T(T - n v

) T(T) T(T +

n v )}, since T(t)

la a iaometrlc representation of a geodesic.

But then

the point P(T) = P(t') could not have a S.N.

This proof

also yields the LEMMA 2 .

If two aegment a have the

aame center and contain a common aubaegment (not necessarily with thia center) then the longer of the first two aegmenta, contains the shorter one. It is now easy to prove Theorem 1 b. a S.N. of Q.

Let

v(Q, p ) be

If the point P(tQ ) of the geodeaic P(t) ia

contained in v(Q, p ) the pointa P(t) with |t - tQ | £

6

will form a aegment cr for aufficiently amall 6 > 0 and lie in

v(Q, p ).

It then follows from Theorem I 2 . 3

that P(t0 ) is center of a aegment of length Up

which

contains cr. This segment ia contained in P(t) on ac­ count of Lemma 1 . Hence ry (tQ ) ^ 2 p . The main Theorem of this aection ia THEOREM 2 . At leaat one geodeaic passes through any two given points. If P(t) a £ t £ b repreaenta a segment iaometrlcally, exactly one isometric representation "F(t) of

_________

_I.

METRIC SPACES WITH GEODESICS____________

a geodesic g exists for which T(t) - P(t) for a £ t £ b. PROOF.

Since two points can always be connected by

a segment (Theorem I 2.1), the first part is contained in the second.

We prove first that a segment cr is al­

ways contained in a geodesic.

Let P(t), -a £ t £ a, be

an isometric representation of I 2.3 that a segment

cr.

We see from Theorem

cr.J: P^t), a -

^

£ t £ a + o, exists with PJ (t) =» P(t) for max (a , -a) £ t £ a. If o-|' : p j '(t), a £ t ^ a + 2p - p - p If we traverse A !B (or AB1) from B (or A), let A 11(B!1) be the first point with PA!1 « p (PB11 « p ). if A,B c v(p ~ p ), then AB c v(PQ, p), hence ATTBTT ^ 705. Calling t-segment a segment which con­ tains points of v(PQ, Jyp ) and whose end points have distance p from PQ and whose interior points lie in v (PQ, p ), we see: Any two points of v(PQ, p ) are contained in one and only one t -segment. Because of 2 p < L( u ) the segment a contains a t-segment Ta . Call then t' is at (See KURATOWSKI [1],

We have t' • fl' = 0

least one dimensional.

Kind129, Theorem 3)

p.

since t'

0

F(x) is called the distance function of [3 with re­ spect to o.

The surface

p has the equation F(x) = 1.

12. A function which satisfies (2a,b) Kind is the distance function of a convex surface with respect to the origin o if and only if F(x) is convex. Putting (3) we have

m(x,y) = F(y-x)

(3a)

m(x,y) > o for x ^ y

(3b)

m(x,y) + m(y,z) ^ m(x,z)

because F(x) is convex.

m(x,y) will in general not sat­

isfy the symmetry condition m(x,y) = m(y,x) or F(x-y) = F(y-x).

This relation is satisfied for all x,y if and

only if F(x) = F(-x) for all x or |!(x)| = |!(-x)| for all x.

This means geometrically that o is the center of

ft . Since m(x,y) depends only on y-x, we see that m(x,y) is invariant under all translations x* = x + a, y 1 = y +b. ever

More generally we have m(x',y‘) = p m(x,y), when­ Kind x'-y* =. p(x-y), p y 0 or In words:

13. Let o be the center of p. If the euclidean straight lines xjr and x'y1 are parallel, then

•

§1 . CONVEX SURFACES AND MINKOWSKI METRICS

35

If o ia the center of ft, the function m(x,y) sat­ isfies the axioms of a metric apace (aee I 1.A1,2,3 and (3a,b)). ft ia the aphere K(o,1)of the metric m(x,y). Equation (k ) ahowa that 1 k. If ft haa o as center, the apherea K(P, p) of the metric m(x,y) are homothetic convex aurfacea, whose centers are the euclidean centers of the aurfacea. The relation (4 ) also ahowa that the euclidean cen­ ter (x+y)/2 of the two points x,y ia also a center of x and y for the metric m(x,y), so that m( ) satiafiea Axiom C. Axiom B is obviously true.

15 . If ft haa c aa center, the metric Kind m(x,y) satiafiea Axioma A,B,C. Axiom D does not hold, unless ft is strictly convex. For if ft is not strictly convex, l i containa a aegment, aay the aegment connecting ^ = I(x1) and i2 » l(x2). For the point x = + l2 we have F(x-f1 ) - P(|2 ) - 1 - F(|1 ) = F(x-I2 )

and

_ F(x) = 2 F(|) = 2

becauae x/2 = (f1 + |2 )/2 liea on the aegment from / to l 2 and therefore on ft. Consequently (cf. (3 )) m(o,x) - 2 « m(o, ^ ) + m (^ ,x) » m(o, f2 )+m(l2 ,x), which shows that both the points J1 and f2 are cen­ ters of o and x. Because of (2b) the points pf. j and p 12 will be centers of o and p x for every p > 0. Hence no standard neighborhood of o exists (see property a) of a S.N. in I 2), so that Axiom D does not hold. Conversely, if ft ia strictly convex, then D is satisfied. We have the stronger theorem

56

II. METRIC CONDITIONS FOR FINSLER SPACES

16 . m(x,y) satisfies Axiom D if and only if the spheres are strictly convex. (Because of 14 . all spheres are strictly convex if ft is.) In that case there passes exactly one geodesic xj through any two dis­ tinct points; xy is an open straight line and coincides with the euclidean straight line through x and y. (The definition of an open straight line was given in I 1 .) PROOF. It,follows from 13 . that the euclidean straight line through x and y is at the same time an open straight line for the metric m( ). All we have to show is therefore, that (abc) can only hold when b is on the euclidean segment from a to c. Assume that (abc) holds for a point b not on the euclidean segment from a to c. (abc) is equivalent to (5 ) F(e-a) - F(b-a) + F(c-b). Consider the points b- « a + (c-a)

1

and b

F(c-a)

2

= a + (c-b) —

F(c-b)

With the help of (5) we find F(b-a)b + F(c-b)b2 ^1 “ F(c-a) hence the point b 1 is on the euclidean segment from b to bg. On the other hand, we draw from (2b) that F(b-a) - F(br a) - F(bg-a) so that the three points b,b-j ,bg would also lie on the sphere K(a,F(b-a)). This contradicts the strict convex­ ity of the spheres. A metric m(x,y) belonging to a closed strictly con­ vex surface 13 with o as center, or any metric congruent to such a metric, will be called Mlnkowskian. We say the euclidean coordinate system (x) belongs to m(x,y). This

§1. CONVEX SURFACES AND MINKOWSKI METRICS 37 system Is determined up to an affinity. (Others use the te.no Minkowsklan An awider sense, namely also when o is any point in the interior of ft and ft is convex, but not necessarily strictly so.) There are many interesting characterizations of the Minkowsklan geometries, some of which will be discussed later in the book. Two very trivial ones are contained in theorem 17. Let the metric m(x,y) be defined in E*1, topologically equivalent to the euclidean metric |x-y| and convex, i.e. satisfy Axiom C. Then the metric m(x,y) is Minkowsklan and be­ longs to (x) (a) If the distance m(x,y) satisfies D and de­ pends only on x-y. (b) If the spheres K(P, p) are homothetic, strictly convex surfaces whose centers P are the euclidean centers of the surfaces. A proof of (a) is obtained by tracing our steps badk; We see that m(x,y) has the form F(x-y). We have F(x-y) « F(y-x) because m(x,y) is symmetric. For any two points x^x2 and I = (x.j+x2)/2 we have |-x1 = x2- I . There­ fore | is a center of x1 and xg with respect to the metric m(x,y). We conclude herefrom that 13. holds. Hence F(x) satisfies (2a,b). Since Axiom D holds for m( ), we conclude from 16. that F(x) « 1 is strictly convex. To prove (b) we observe the euclidean center of any two points to be also a center of these points for m( ). Therefore the euclidean straight lines are open straight lines for m( ), and on any one line the euclidean dis­ tances are proportional to the distances in terms of m( )* Let the euclidean segments T ? and x fy1 be parallel. Since the. distances on the euclidean line xx! are propor­ tional to the distances in m( ), and K(x,y) and K(xf,y*) are homothetic, we see that 13. holds. Now we proceed

38

II. METRIC CONDITIONS »FOR FINSLER' SPACES

as under (a). Among the Minkowski metrics m( ) belonging to a definite euclidean coordinate system (x) there are euclidean metrics. (We mean, metrics congruent to |x-y|, but the congruence may not be given by m(x,y) = |x-y| .) It is well known and very easy to see that 18. Among the convex surfaces with center o, the ellipsoids and only these yield Minkowski metrics which are 'euclidean. Now consider a definite Minkowski metric m(A,B). The geodesic through A and B will be designated by AB. We show 19. In a Minkowski space any point P^g has exactly one foot F on a given geodesic g. All points of PF have F as foot on g. That P has at least one foot F on g follows from Theorem I 1.5. It is clear that g contains no interior points of the sphere K(P,F). There cannot be another foot F1, since then the segment would lie on K(P,F). If Q Is any point ^ F on PF, the sphere K(Q,F) has also g as supporting line at F because the spheres K(P,F) and K(Q,F) are homothetic and have Q and P as euclidean centers. The uniqueness of the foot F of P on g can also be expressed in the following way: Orient g, fix a point G on g and put s(X) = m(X,G), If X follows G on g and s(X) « -m(X,G) if X precedes G. Then the function g(s) = g(s(X)) ™ PX reaches its minimum at exactly one point, namely F. But a much stronger statement holds: 20. g(s) is a convex function of s. PROOF. Let P « (Pj^), and let a and b be any two

§1. CONVEX SURFACES AND MINKOWSKI METRICS points of g. If X we have for 0 £ 1

39

designates the point ta + (1-t) b,

PX - F(t a + (1-t )(b - p))= F(t(a - p) + (1 - t)(b - p;)£ £ F(t(a-p)) + F ( (1-t)(b-p)) = t F(a-p) + (1-t)F(b-p)= - t PA + (1-t) PB. This proves that PX is a convex function of the parameter t.

But a being a linear function of t, the function

g(s) is also convex. It is worth observing that 1 9 . follows from the con­ vexity of the spheres alone and has really nothing to with the fact that the spheres are homothetic (cf. Theorem IV 3.U).

But for 20. the Minkowsklan character

of the metric is In a certain sense essential (cf. Theorem IV 7 .k) Finally we consider the limit spheres of a Minkowski metric. To define them we take an oriented geodesic g 1, on it a point P and a variable point X which follows P. All spheres K(X,P) have at P the same tangent cone, be­ cause these spheres are homothetic. When X tends on g 1 in the positive direction to infinity, K(P,X) will tend to this common tangent cone.

We put

A ( g ' , P ) - U * K(X,P)

and call it the limit sphere with g' as central ray through P.

21. The limit sphere A(g',P) la the tangent cone of any sphere K(X,P) at P, where X follows P on g 1. Using 6. we get 22.

All limit spheres of a Minkowski metric are planes if and only if the spheres of the Minkowski metric are differentiable

kO

II. METRIC CONDITIONS FORFINSLER SPACES everywhere. §2. RIEMANN SPACES AND FINSLER SPACES

The theory of Finsler spaces developed from the calculus of variations as well as Riemannian geometry. BERWALD [1] gives a concise introduction to this theory. We shall explain how these theories are connected with metric spaces. To understand our later definitions we first discuss the example of a sufficiently smooth surface ct in E^. Let P he any point of a and tt the tangent plane of a at P. We designate "by ^e(X,Y) the distance of two points X,Y in tt and by XY the geodetic distance on a of the two points X,Y of a. If the circular disk 6p : "e(P,X) < p In tt is suf­ ficiently small the perpendicular to tc at a point X of cSp will intersect a at exactly one point X and the map­ ping X —> X of 1 for Xv

» P, Xy + P

— > 1 for Xy-* P, Xj~* PKind Xv P + PYV

< M < oo

______ §2. RIEMANN SPACES AND FINSLER SPACES_______ 4l_ (1a) means the existence of a tangent plane Tt of at P in the weak sense, that tt contains any straight line which is limit of a sequence of lines through P and points X^of cc which tend to P. From this one concludes easily that (ib) follows from (1a). Hence (1b) is In this case not more restrictive than (la). But (1) is really a stronger condition than (1a). It means the existence of a tangent plane In the strong sense that tt contains the limit of any converging sequence of straight lines through points Xv and Yv of cc which tend to P. In case a tangent plane In the weak sense exists everywhere (1) Implies the continuity of the tangent plane at P. There are, of course, many ways of defining a euclidean metric In a certain neighborhood v of P on a , which is equivalent to XY and for which (1 ) holds. With respect to the metric on cc one way presents itself as very natural: We map P on itself and a point X ^ P of e(X,P) .< p on the point X of a with XP ® e(X,P) for which the geodesic arc from P to X has at P the same direction as the segment PX. We. then put e(X,Y) = e(X,Y). The new distance e(X,Y) is defined in the disk XP < p and we have XY = e(X,Y) for pointfe which are on the same geodesic arc through P. This way of introducing a euclidean metric is gen­ erally described as the Introduction of normal coordi­ nates. One obtains such coordinates by introducing in Tt cartesian coordinates with P as origin and ascribing the coordinates of the point X in "e(X,P)