Geodesics of two-dimensional Finsler spaces

IUathl. Comput. Modelling Vol. 20, No. 4/S, pp. l-23,

1994 Copyright@1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 08957177/94 $7.00 + 0.00

08957177(94)00118-9

Geodesics of Two-Dimensional M.

Finsler Spaces

MATSUMOTO*

15 Zenbu-cho, Sakyo-ku,

Shimogamo

Kyoto,

Japan

Abstract-A concise description of 2-dimensional Finsler spaces is presented from the viewpoint of their geodesic curves (i.e., extremals of a variational problem). Berwald’s classification and geodesics of l-form metric spaces are studied. Darboux’s solution to the problem of determination of the variational functional (homogeneous Lagrangian) from given geodesic equations is presented. INTRODUCTION It has given the author great joy to have found several recent papers on applications of Finsler to certain fields of science. We feel strongly the necessity for further investigation and diffusion of a geometry which has been unpopular even with mathematicians, though we have had several books published on Finsler geometry in the last decade, after a quarter of a century of having only one book, that of H. Rund [6-lo]. The purpose of the present paper is to exhibit several equations of geodesics, extremals of the variational calculus, for those who are in need of Finsler metrics for further development of their scientific theories. After six preliminary sections, the whole of Section 7 consists of original results, which are stated in two theorems. The last section is an explanatory description of the author’s paper [ll] which shows how to find the fundametal function having a given family of geometry

curves as geodesics. The full proofs are sometimes omitted owing to space; please refer to [6,10].

1. EQUATIONS

OF

GEODESICS

Let TMn be the bundle of all non-zero tangent vectors of an n-dimensional smooth manifold M” and (x,y) = (x1,. . . ,P, y’, . . . , y”) the canonical coordinate system of TM” which is induced from a local coordinate system (x) of M”. Suppose that we have been given a function L(x, y) used to define an integral

s b

J(C)

=

Jw% qt)) 4

jr(t) =

W)

--g-T

a

along an oriented curve C : x = x(t) from P = x(a) to Q = x(b), a 5 t 5 b. It is a matter course that L(x, y) is assumed differentiable in all that follows. For our geometrical purpose, shall suppose three conditions as follows:

of we

*The author would like to express his thanks to P. L. Antonelli, University of Alberta, Edmonton, Canada, for his interesting paper [l], which gave him a new hope of future development of Finsler geometry [2]. His metric given there belongs to the class B’(1). He also express his thanks to V. Dryuma, The Academy of Sciences, Kishinev, Moldova, who showed the author some excellent applications of Finsler spaces belonging to the class B2(r, s) [3]. See also [4,5] for further reading. Typeset 1

by A+@-W

M. MATSUMOTO

2

CONDITION (Ll).

L(x, y) is a scalar function.

That is, the value of L(x, y) is independent of any choice of the coordinate system. CONDITION (L2). J(C) is independent of any change of parameter

t preserving its orientation.

CONDITION (L3). In each tangent space M,” we have a positively conical region R, L(x, y) is positive-valued.

in which

R, of M,” - (0) is called positively conical if y E R, implies py E R, for all positive numbers p [7]. On account of the well-known Caratheodory theorem, (L2) states that L(x, y) is positively homogeneous of degree one in y : L(x,py) = pL(x, y) for all positive numbers p. Then, we have the following well-known theorem in variational calculus. If C provides a weak relative minimum to J, it is necessary that

L(x, y) satisfies the Euler

equations, dL(i) &(C) = --LLi=O, &

(1.1)

along C in each coordinate neighborhood in which C enters, where Lli) = @ and Li = s. If C satisfies the differential equations (l.l), then it is called the extremal or geodesic of M” with respect to L(x, y). The Euler equations may be written as Ei(C) = L(i)(j) kJ + L(i)j k’ - Li = 0.

(1.1’)

On account of the Euler theorem on homogeneous functions, (L2) implies (i)

L(i) yi = L,

(ii)

(iii)

L(,)(j) yi = 0,

L(qj Yi = Lj;

(1.2)

(ii) shows det (L(i)(j)) = 0, identically. Consequently, we can not rewrite Ei(C) = 0 in the normal form gi = fi(x, jC). Furthermore, from (iii) we get Ei(C)ki = 0, hence n equations Ei(C) = 0 are not independent. Now, we are concerned with the function F(x, y) = (L2(x,y))/2

and the normalized para-

meter s of C, given by s = s,” L (x, g) dt. Then, we get Ei(C) = w - Fi/L = 0 and L (x, 2) = 1 along C. Therefore, we obtain the equations of the geodesic C in the form

+ 2Gi(x,y)

dx Y=-&’

= 0,

(1.3)

where the functions Gi(x, y), positively homogeneous of degree two in y, are defined by 2Gi(x, Y) = F(i)j Yj - Fi.

(1.3a)

In view of (1.3), we shall impose one more condition on L(x, y): (L4)

det (F(i)(j))

# 0.

This is called the regularity condition. As a consequence, Equation (1.3) can be written in the normal form. If a smooth manifold Mn is equipped with such a function L(x, y), satisfying the four conditions above, then Mn is called the n-dimensional Finsler space with the fundamental jknction L(x, y) and denoted by Fn = (Mn, L(x, y)). The value of the integral J(C) defines the length of C and the normalized parameter s is the arc-length of C. The value of L(x, y) is called the absolute length of the tangent vector y at the point x. We have used the tensor field g, having the covariant components gij(x, Y) = F(i)(j) = LL(i)(j)

+ L(i) L(j),

Two-DimensionalFinder Spaces

3

called the jundamental tensor of F “. Its contravariant components g”j(x, y) are elements of the inverse matrix of (gij), and Equation (1.3) can be written in the form

(1.4) where we put Gi = gij Gj, that is, 2Gi(x, y) = gij (F(j)k Yk - 4)

*

(1.5)

Further, we have two important tensor fields 1 and h, with the components

(i) called the normal&d

(ii)

L(i),

li =

supporting element

hij = LL(i)(j),

and the angular metric

(1.6) tensor,

respectively. It is easy

to show that the contravariant components 1” = g”j lj of 1 are equal to yi/L(x, obtain gij = hij + li lj 7 which shows that the rank of the matrix (hij) is equal to n - 1. It is noted here that, if we refer to an arbitrary admissible parameter t ($ tion (1.4) is written in the form

y).

Thus, we (i-7)

> O), the equa

(1.4’)

where L = L (x, $)

= $f and L’ = $.

If we are concerned with a Ftiemann space, with a Riemann metric L(x, y) = dm, the covariant components of the fundamental tensor g are nothing but gij(x). In this view, we can construct the “Christ&e1 symbols” {jik} (x, y) from gij(X, y) in a Finsler space Fn:

{jik} (%Y) = i {P where aj = &.

+ akgj, -

(ajgkr

hgjk)}

,

It follows from the homogeneity that (ajgk,)

y"

=

aj (F(k)(r)

yk)= ajF(,),

(ad?jk)

YjY" = 2F,.

Consequently, we have 2@(x,y)

=

{jik}

(%Y)YjYk-

On the other hand, we can construct the “Christoffel symbols” cjik(x,

where dj = &.

(1.5’) y) with respect to y:

From gij = F(i)(j), we get

cj’k

=

f

(P$gkr)= i

(PF(r)(j)(k))*

(14

It is obvious that Cjikyk = 0 and the condition Cjik = 0 is a characterization of a Remann metric. Cj”k are components of a tensor field C of type (1,2), called the C-tensor. Its components Cijk

= 9jr Cirk

me

equal

to (F(i)(j)(k))/2*

M. MATSUMOTO

2. EQUATIONS OF GEODESICS IN THE CASE OF TWO DIMENSIONS We consider a two-dimensional Finsler space F2 = (M2, L(x, y)) . On account of the homogeneity of L(x, y) in y we have (1.2.ii): Y1 +

L(l)(l)

Y2 =

L(l)(Z)

L(2)(1)

Y1 +

-WY’

Y2?

L(2)(2)

Y2 =

0.

Thus, we have the function W(x, y) satisfying L(l)(l)

=

w

(Y2)2,

L(l)(Z)

=

L(2)(2)

=

w

(Y’)2,

(2.1)

which is called the Weierstrass invariant [ll]. Since Lci)(j)(x, y) are positively homogeneous of degree (-1) in y, IV is (-3)phomogeneous in y. (Throughout the remainder, we shall use these abbreviations of the homogeneity.) From (l.l’),

(1.2.iii), we have El(C) = -y2 W(C),

I&(C) = y1 W(C), where we have put dxi yi = dt.

W(C) = Ll(2) - L?,(l) + (yl g - y2 $) w, Consequently, we obtain W(C)

(2.2)

= 0 as the single equation of a geodesic C, said to be of the

Weierstrass form. Next, in F2 we shall use the symbols (z, y) and (k,Y), instead of (x1,x2) and (yl, Y2), respectively. Then the differential of the arc-length s of a curve C is given by ds = L(x, y, i, g) dt,

(&ti) = ($$). kdt.

Supposing k > 0, we have ds = L

Hence, using the usual symbol y’

2 , we

obtain ds = A(x, y, y’) dx,

(2.3)

Ax, Y, Y’) = L(x, Y, 1, Y’).

(2.4)

where we have put Conversely, L is obtained from A as L(x,y,

i,$)

= A

(2.5)

The function A(x, y, y’) of three arguments is called the associated fundamental function of F2 (see [12]). Putting A, = g, A, = $$ and A’ = 3, we have, from (2.5), Lti = A’,

Li. =A-A’y’,

Consequently, we get W = A”/i3,

Lk,=A,-A:,y’,

L+=A;,

and W(C) written as

A(C) = A” y” + A& y’ + A; - A,.

(2.6)

Therefore, we obtain another form A(C) = 0 of the equation of a geodesic C, which is sometime said to be of the Rashevsky form [3]. We shall return to the equations (1.4’) of a geodesic: ji =

ji=-2G’+,

-2G2

+

41

*

L y1

from which we have ji k - Z Y = 2 (G1 Y - G2 5) and the equation of a geodesic in the form y,,=

2(G’Y-G2k) k3

where Gi = Gi(x, y, k,Y). in (i,Y).

’

(2.7)

It is noted that the right-hand side of (2.7) is (0)phomogeneous

Two-Dimensional

3. FINSLER

Spaces

Finsler

5

CONNECTIONS

We shall sketch the Finsler connection theory for later use. In Riemann geometry, we have the Levi-Civita connection RI? = ( {jik} (x)), w hic h is d et ermined uniquely from the given Riemann metric L(x, y) = dm

by the two axioms:

(1) metrical: gij;k = 0, (2) without tOrSiOIl: {jik} = {kij},

where (;) in (i) stands for the covariant differentiation in RI?. We can, however, consider a linear connection I? = before a Riemann metric is given, to define the covariant differentiation (rjik(x)),

Vi;j = 8jV” +V’ rrij for a tangent vector field Vi(x), for instance. On account of the well-known transformation law

xi

rjzk(x)

xt =

rbac(q

x: + ax; axe’ 7

xi

b

_ ax’ dzb’

it is confirmed that Vi;j constitute a tensor field of type (1,l). We have the similar situation in Finsler geometry. In a Finsler space, we can have the concept of pm-Finsler connection Fr = (Fjik, Nij, Vjik) [6,10], before a Finsler metric L(x, y) is given. In order to define two kinds of covariant differentiation (i)

Vilj = Sj Vi + VT Frij,

(ii)

Vilj = ajVi + V’VTij,

(3.1)

for a given Finsler vector field Vi(x, y), for instance, where 6-differentation in (i) is defined as Sj = aj - N’j &;

(3.2)

(i) The connection coefficients Fjik(X,y) must satisfy a transformation law of the same form as it has been shown above; (ii) Nij(x, y) must obey the transformation law which are satisfied by *Nij = yr Fvij; and (iii) Vjik(x, y) are components of a tensor field of type (1,2). On account of (ii) and (3.2) the derivatives SiS of a scalar field S(x, y) constitute a covariant vector field, and together with Vilj of (3.1.i) constitute a tensor field of type (l,l), which is called the h-wvariant derivative of Vi. On the other hand, since 8jVi already constitute a tensor field, Vi 1j of (3.l.ii) are components of a tensor field of type (1 ,l) on account of (iii), which is called the v-covariant derivative of Vi. Now we shall exhibit two kinds of Finsler connections which are determined from the Finsler metric L(x, y). The Berwald connection IX’ = (Gjik, Gij, 0) is determined uniquely by the system of axioms [lo] as follows: (i) (iv)

Lli = 0, . &G’j

= Gk2j,

(ii)

Gj”k = Gk2j,

(V)

Vjik = 0.

(iii)

Gij = yT Grij, (3.3)

The first four axioms determine Gij as Gij = &jGi, where Gi are to be given by (1.5). From the homogeneity, we have Gij(x,y) yj = 2Gi, hence, together with (3.3.iii), it is seen that the equations (1.4) of a geodesic are written in the form

The last axiom shows that Vi 1j in Br are 8j Vi. The h- and v-covariant differentiations in Br will be denoted by (; ) and (.), respectively. Consequently, we obtain (i)

Vi;j = bjVi + VPGrij,

(ii)

Vi.j = 6jVi.

Sj = aj - G’j &,

(3.5)

6

M. MATSUMOTO

By the direct calculation of Vi;j.k - Vi.k;j, we obtain one of the communication formulae, called the Ricci identities: Vi;j.k - Vi.k;j = V’ Grijk,

(3.6)

where Ghijk is a kind of curvature tensor, called the hv-curvature tensor, given by Ghijk = & Gjik = bh Pj & Gi.

(3.7)

Second, the Car-tan connection CI’ = (I’;ik, Gij, Cjik) is determined uniquely by the system of axioms as follows: (i) (iv)

9ijlk = 07 9ijlk= 0,

(ii)

l?T,r, = l?;'j,

(V)

Cjik = Ckijs

(iii)

Gij = yrI’Tij, (3.8)

The first three axioms determine Gij as the same as given by the Bl?, and l?;ik as the “Christoffel symbols” with respect to Si: rj*% = i {gir (bj9kr +

bkgj,

-

bgjk)}

The axiom (iii) as well as (3.3.iii) are nothing but the condition yilj = 0. The last two sxoims, similar to those of the Levi-Civita connection RF, determine Vjik as the “Christoffel symbols” with respect to y, that is, they are components of the C-tensor (1.8). The h- and v-covariant differentiations in CT will be denoted by (1) and (I) respectively. Consequently, we have (i)

Vilj = SjV” + V’l?:ij,

Vilj = 8jVi + VrCtij,

(ii)

(3.9)

where Sj is the very same as those of (3.5). We shall prove the useful relation [lo]: Gj”k = r;‘k + Cj’klO, where the subscript 0 stands for the transvection by y. r$ + Djik. Then gijlk = 0 in Cr implies

(3.10)

To prove the above, we put Gj”k =

Dijk = 9jrDirk;

&'ij;k = -Dijk - Djik,

Equation (3.6) is applied to gij, and we have 9ij;k.l

-

&j.l;k

=

-9rj Girkl - SirGjTIcl.

The left-hand side is equal to -(Dijk+Djik).l_2Cijl;k. Then, focusing on Dijo = 0 and Gircl = 0, the transvection by yk leads US to Dijl + Djil = 2Cijl;o. From the symmetry of Dijl in i and 1, we get Dijl = Cijl;o. Since Cijl;e = Cijrle is obvious, we have (3.10) and gij;k

4. MAIN

=

-2Cijk;O

SCALAR

=

(3.11)

-2cijklOs

AND

CURVATURE

We shall consider a two-dimensional Finsler space F2 and recall (1.7). As we have seen, the components hij of the angular metric tensor constitute the symmetric matrix (hij), of rank one for F’, hence it is easy to show that we get the sign E = fl and the vector field m = (mi) satisfying hij = emimj; (4.1)

Two-Dimensional Finsler Spaces then equation

7

(1.7) is written as gij = lilj +&m~77lj.

(4.2)

Since lj = yj/L, equations (1.2.ii) and (1.6.ii) imply hij Zj = 0, and (4.1), (4.2) show

?7ljlj = 0,

mjmj = E.

(4.3)

Therefore, we obtain the local field (1, m) of orthogonal frames, called the Berwald frame [8,10]. The sign E is called the indicator of the sign&m of F2. Since equation (4.2) implies g = det(gij) = E (21rn2 - Z2mi)2;

(4.4)

e is the sign of g. We consider the C-tensor (Cijk) of F2. Since it is completely symmetric and satisfies Cijklk =O, we get its expression, Lcijl, = Imimjmk,

(4.5)

in the Berwald frame. The scalar function I(z, y) in (4.5), (O)phomogeneous, is called the main scalar of F2. The derivatives of (Z,m) will be needed in our future discussions. First, from (1.6), (4.1), and li = yi/L, we have Lajli=Emimj, L8jli=&mimj. (4.6) Next, mi li = 0 and gij mi mj = E give (L 8j mi) li = -mj tively. Similarly, we obtain L8jmi

= -(I? +EImi)mj,

L8jmi

and (L 8j mi) mi = -I mj, respec-

-&Imi)mj.

= -(li

(4.7)

We consider the covariant derivatives of (Z,m). From gi.jlk = 0 and Yilj = 0 in Cl?, it follows that Zilj = lilj = 0, 7Ttilj= ?7Zilj = 0. (4.8) Next, let S(z, y) be a scalar field which is (0)phomogeneous in y. It is obvious from S,i yi = 0 that S.i has no component in the direction li, hence, we can put L S.i = S2 mi.

S;i = Sli = SJ li + S,2 mi,

(4.9)

Then, from (4.5), we have L Cijklh

which impk!S to

Cijk$

=

(I,1

lh + I,2 mh)

= Cijklo = I,1 T7Limj mk.

&;j = 0,

mi

mj

(4.10)

mk,

Consequently, equations (3.10) and (4.8) lead us

WLi,j= --E I,1 rni mj.

(4.11)

Now we consider the expression of the hw-curvature tensor Ghijk in (I, m). Applying (3.6) to (1, m), we have s!i;j.]E - li.k;j = -1, Girjkr mi;j.k - mi.k;j = -??I, Girjk. On account of (4.6), (4.7), and (4.11) it is easy to show that these equations are written as 1, Girjk = -s

(I,imimjmk)

,

‘W Gi’jk

= i

({I,2

i-

(1,112)

mi

mj

mk) .

Therefore, we obtain [lo], LGihjk = [-21,i

lh + {I,2 + (1,1)2} mh] mi mj

?nk.

(4.12)

8

M. MATSUMOTO

Now we consider the commutation formula S;i;j - S;j;i in Br, for scalar field S(z, y). By direct computation it is easy to show one of the Ricci identities: S;i;j - S;j;i = -S,, where Rhij, called the (v)h-torsion

(4.13)

RTij,

tensor [6,10], is given by Rhij = Sj Ghi - 6i Ghj.

(4.13a)

Applying (4.13) to L(z,y), we have 1, R’ij = 0 from (3.3.i). Further, Rh, in i, j. Consequently, Rhij is expressed in (1, m) as

is skew-symmetric

Rhij = EL R mh (li mj - lj mi).

(4.14)

The scalar field R(x, y), (0)phomogeneous in y, is called the curvature of F2. The main scalar I and the curvature R are two essential scalar fields for the theory of twodimensional Finsler spaces. Applying (4.13) to I, we get (4.15)

I;i;j - I;j;i = -12 R (li mj - lj mi).

REMARK. The tensor field of a Finsler space which corresponds to the curvature tensor of a Riemann space is called the h-curvature tensor [6,10]. It is written as Hihjk and Rihjk in BI? and CT, respectively. These satisfy Hohjk = bhjk = Rhjk given by (4.13a). In the two-dimensional case, we have the expression

Rihjk= E R (Eimh

- lh mi) (lj mk - lk mj),

but Hihjk has not such a simple expression in (1, m) [lo], though it is given by the simple equation Hihjk = Rhjk,i,

5. TWO-DIMENSIONAL

BERWALD

SPACES

As it is obvious from (3.7), the condition Ghijk = 0 is such that Gjik are functions of position x alone. DEFINITION. A Finsler space F” is called a Berwald space, if the hv-curvature tensor Gihjk of BIT vanishes identically, that is, Gjik are functions of position alone.

Consequently, it follows from (3.7) that 2Gi(x, y) of a Berwald space are quadratic forms in yi : 2Gi = Gjik(X) yj yk and these Gjik(X) are nothing but the connection coefficients of Br. As a consequence, equation (3.4) is written in the form d2xi p f Gjik(X) (dxj) x

($)

=O,

similarly to the geodesics of a Riemann space. Let us restrict our consideration to the two-dimensional case. Then equation (2.7) can be written as y” = Y3 (y’)3 + Y2 (y’)2 + Yl y’ + Yo, (5.1) where Y’s are functions of (x, y) for a Berwald space as follows: Y2 = 2 Gil2 - G222,

Y3 = G2l2r

Yi = Gili

- 2GiZ2,

Y. = -Gi2i.

(5.2)

Two-Dimensional

PROPOSITION 1. The equation of a geodesic

Finder

Spacea

in a two-dimensiona

9

Berwald space can be written

in the form (5.1), where the right-hand side is a polynomial of degree at most three in y’ with the coefficients

given by (5.2).

Next, for a Berwald space F2, equation (4.12) implies IJ = 1,~ = 0, that is, I;i = 0 and equation (4.10) leads us to the well-known characterization Cijklh = 0 in CT, for a Finsler space of general dimension to be a Berwald space [SJO]. In the twedimensional implies I2 R = 0. Therefore, we have the following theorem.

case equation (4.15)

THEOREM 1 (BERWALD) . A tw&dimensiond Finsler space is a Berwald space, if and only if the main scalar I satisfies I;i = 0. Ail the two-dimensional BerwaJd spaces are divided into three classes by the properties of I and the curvature R as follows: (i) I is constant a.nd R # 0; (ii) &I # 0 and R = 0; and (iii) I is constant and R = 0. A Berwald space with R = 0 is a locally Minkowsici space [6,10]. A Finsler space of general dimension is locally Minkowski, if there exists a covering by the coordinate neighborhoods of adapted coordinate systems, in each of which L is independent of x, hence equation (1.5) shows that all Gi vanish. The adapted coordinate systems are connected with each other by linear transformations [lo]. Therefore, Theorem 1 has the following corollary. COROLLARY 1 (Berwald). A Berwald space of dimension two has the constant is not a locally Minkowski space. Next, we consider I,i = 12 mi/L,

By changing

for

Chij.k

Chijlk

Thijk

called the T-tensor,

=

main scalar, if it

apart from the Berwald spaces. From (4.5), we have by (4.7),

in the above, we find that the symmetric tensor L Chijlk

+ lh C$jk + li Chjk

+ f!j chik

+ lk Chijr

(5.3)

is written as L Thijk = I2 mh ‘?&mj mk.

(5.4)

Consequently, we have the following theorem. THEOREM 2. The main scalar is a function of position iden tically Now our problem is to two having the vanishing F = L2/2 and G = ,/Ej partial differentiations by

(9 (ii) (iii)

GBi

= Lli-

alone, if and only if the T-tensor vanishes

consider the fundamental function L of a Finsler space of dimension T-tensor. First, we deal with the scalar function B = F/G, where (9 = det(gij)). D enoting by the subscripts i, j, k of B and G the yi, yJ, y”, we obtain BGi,

G2 Bij = Ggij - L (li Gj + lj Gi) + B (2Gi Gj - GGij), G3 Bijk = 2G2 Gijk - {Ggij Gk - 2Lli Gj Gk + L (G& - L Gi) Gjk + (i, j, k)}

(5’5)

- B (G2 Gijk + 6Gi Gj Gk), where (i, j, k) in the parenthesis stands for the terms obtained from the preceding terms by cyclic permutation of i, j, k.

10

M. MATSUMOTO

We consider Gi, Gij, and Gijk. If we put Ci = Cijkgk,

then equation (1.8) shows

2g Ci. Thus, we obtain

Gij = G (Ci Cj + Ci.j),

Gi = GCi,

Gijk = G (Ci.j.k + Ci Cj ck) + G {Ci TO consider Ci.j and Ci.j.k, we refer to the T-tensor.

cj.k

+

(i, j, k)} ,

If we put Tij = Tijhk ghk and C, = C, Ci”j,

then we have LCi.j=Tij-liCj_ljCi+LCij, L2Ci.j.k

=

L (Tijlk

+ CT Trijk)

+ {L (Tir

cj’k

+

-

%

L2 (Ci, Cj’k + Cj, Cirk) - hik Cj - hjk Ci Cjk) - Tij lk + 2Ci lj II, + (i, j, k)} .

Substituting those equations in (5.5.iii), we finally obtain 2G Bijk = -L (T ij k + Cr Trijk) - {LTir

cj’k

+

(4

-

1%)

Tjk

+

(6 j,

k)} .

(5.6)

As a consequence, if the T-tensor vanishes, then we have Bijk = 0, which shows that B is a quadratic form in y”. In this case, we have, from (5.5.ii)

G&j =lilj

+Emimj

+

$(limj +ljmi).

Thus, if we put 2B = bij(X) yi yj, bij = bji, and b = det(bij), then the above shows G2 b = (E - 12/4)(llm2 - 12rn~)~ and equation (4.4) leads us to 4b = E (4 - EI=). Consequently, we can discuss our problem by dividing into four classes, as follows: p = pi(z) yi and 7 = qi(z) yi are independent l-forms, and T = pl q2 - pa ql; (i) (ii) (iii) (iv)

E= E= E= E=

1, 1, 1, -1

I2 I2 I2 :

< 4 : = 4 : > 4 : 2B =

2B 2B 2B /3r,

= p2 +y2, 4r2 = 4 - 12; = p2; = Pr, r2 = I2 - 4; and r2 = I2 + 4.

The procedure to find G is carried out by the integration with fixed x, but we have no space to go into the details of the proof. See [6, 3.5.3; 10, Section 281. Finally, we can show the following result.

THEOREM 3 (BERWALD). AI1the twc&mensjonal

Finsler spaces with the main scalar I = I(z) are divided into four classes which have the following fundamental functions L respectively: (i) e=l,

12<4:

(ii) E = 1, I2 = 4 : L2 = p2 exp (iii) E = 1, I2 > 4:

{~arctan(~)},

L2=(p2+y2)exp (

9

L2 = Pr (a)““,

(iv) E = -1 : L2 = p y (a) I”,

>

J=Jm.

. J = dm.

J = dm,

where fl and y are independent l-forms in yi.

COROLLARY 2. All the two-dimensional Finsler spaces with the constant

main scalar I are divided into four classes which have the fundamental function L given in Theorem 3.

11

Two-Dimensional Finsler Spaces

6. GEODESICS

OF TWO-DIMENSIONAL FINSLER WITH l-FORM METRIC

It is obvious from Corollary 2 that mental functions L(a’, a2) constructed

SPACES

ail the two-dimensional Berwald spaces have the funda, by two independent l-forms e1 and u2, provided that Thus, we shall consider a twodimensional Finsler space

the space is not locally Minkowski. F2 = (M2, L(u1,u2)) with the l-form metric L(a’,a2) [6,13], where ua = ar(z)yi, (Y = 1, 2. We consider its MinkowslEi model m(F2) = (V’, L( d, v”)) , which is defined on a two-dimensional vector space V2 with the Minkowski norm ]]v]] = L(v’,v2). Let (I, m) and (L, M) be the Berwaid frames of F2 and m(F2), respectively. Then we have

p=aQ

L

L ’

If we denote by Haa the angular From (4.1), we get

a

=dL

(6-l)

CW’

metric tensor of m(F2), then we have Hap = L Lap (Lap = a). Hap = sj%

MO,

(6.2)

and Ma of (L, M) is given by MaL, Next, if we denote

=O,

M”M,

by (bh) the inverse matrix

of (a:),

li = $

=E.

(6.3)

then we have

li = L,uT,

= L” b;,

(6.4)

Now, if we put (6.5) we have &a? - a? rjik = 0, which shows that uy are covariant constants with respect linear connection I’1 = (rjik(x)), called he l-form linear connection. This has the torsion Tjik(x) We consider

the function

= bh (ak a: - 8,

Gi of the Finsler

space with

Ug)

to the tensor

a

(6.6)

l-form

metric.

Putting

F = L2/2,

from (6.5), we have $jF=

EJjF = ForuFI’o’j, &

8j

F = (Fc,p at r&)

where by the subscripts have, from (1.5),

Fa ~7

(= yj),

a: + F, (uz rjrk) = gjr rOrk + yr

(Y and p we denote

the differentiations

2G”(x, y) =

rjrk,

by uQ and up. Consequently,

roio+ Ifto,

we (6.7)

where we put T.ij, = gi’ Qjs T,.‘k. Let us restrict

our discussion

to the two-dimensional

case. It is obvious

that Tjik is expressed

as Tjik = (Tl li i- T2 mi) (Zj mk - lk mj), where Tl and T2 are scalar functions.

Then we have

!P.:k = (Tl lj -I-T2 mj) (ii mk - lk mi) ,

CO,-,= -L2 Tl mi.

As to Tl, we have Tl = ETjik li li mk = ETjik (La U:) (L” b$) (MY bt) .

(6.6)

M. MATSUMOTO

12

Hence, putting T*pQ, = Tj”k a: bj bt, we have Tl = ET*~~~ L, (L1 M2 - L2 M’),

T*ra2 = (asay - &a;) (b: bi - 6: b.$).

Thus, if we pay attention to the curl

AQ = &a? -&a;, then we have Tl = E (A” L,) (L’ M2 - L2 M’)/d, Therefore, we obtain the following theorem.

(6.9)

d = det(aq).

THEOREM 4. The quantities Gi(z, y) appearing in the equations of geodesic of a twedimensional Finsier space F2 = (M2, L( al, a”)) with l-form metric are given by

2Gi(z, y) = roio + T&, where l?jik(z)

are defined by (6.5), !r&, = -$ mi :

(A=L,) (L’ M2 - L2 M’) mi,

m’=i(M’a;-

m2=i(M2ai-

M2 ai) ,

and (L’-, M”) and AQ are given by (6.1)-(6.3)

d = det (a:), M’ a?) ,

and (6.9), respectively.

We deal, in particular, with (2.7) of a geodesic in a Berwald space F2 = (M2, L(a’, a”)). Then it follows from (6.7) that T.j-,, must be quadratic forms in yi, hence, we may put

T&e - Z$ Ci =

Yi Yj Y”.

Dij&(Z)

(6.10)

Consequently, the Y’s of (5.2) are written as y3 = r2’2

+ 0222,

&

+ r2’1

= rl’2

- I’222 +

D221 + 0212 + D122,

(6.11)

K = rI1r - r122 - rs2r + D1r2 + Drsr + Dsrr, Yo =

7. GEODESICS

-r121+ 0111.

OF TWO-DIMENSIONAL

BERWALD

SPACES

We consider the equations of geodesics of two-dimensional Berwald spaces with the constant main scalar I. Corollary 2 shows that all the spaces above are divided into four classes as follows: (1) P(1)

:

E = 1, I2 < 4, L2 = (/12 + r2) exp {

(2) P(2)

:

E = 1,

(3) P(r,s):

( y ) arctan

(5) } , J = dm.

I2 = 4, L2 = p2 exp (9).

L2=/P+

r+s=l.

Here (iii) and (iv) of Theorem 3 are put together as the class B2(r, s) because of the following observation. We have T = 1 - I/J, s = 1 + I/J, and

.*. ( ) 1-I 111

I

J

>I,

rs
( )

Thus, we have the following subdivision of B2(r, s): (3-l) P(T,S), (3-2) B2(r,s),

TS < 0 : T,S > 0 :

E = 1, 12 > 4. c = -1.

iv

I < 1, J 1-I

r, s > 0.

Tw+DimenaionalFinder Spaces

13

Class 1. First, we deal with a space belonging to the class B2( 1). In the symbols used in Section 6 L2 may be written as L2=2F=B2E2 B2 = (cz’)~ + (a2)2,

7

E=exp{parctan($)},

p=

-&.

(7’1-1)

We shall proceed our calculation following the way shown in Section 6: LL,

= F, = E2b,,

where two l-forms b, = b,i(z) y” are defined by bl=a1+pa2,

b2=u2-pa’.

(7.2-l)

Next, we have Fc,p: B2+2pa2bl), F12 =

2 (p B2 - 2pa’ bl) =

0

;

2 (-pB2

+2pa2b4,

2 (B2 - 2pa’bz), where aQ b, = B2 is remarkable. Then we have Hap:

Hz2 =

0

;

2 (1 +p’)

(c?)~,

which show E = 1 and k2 =

(A&) = k (--us, a’),

0

g

2 (1+p2).

Next, equation (6.3) gives h=&.

(MO) = h (-bz, bi), Then, we have

m’ :

hag b,

,I=__

d

’

m

2 _ ha?b, --. d

Therefore, Theorem 4 leads us to Tl

=

.oo

(A” hx)(47 b) d2(1+p2)

T2 = _ W .oo

’

(7.3-l)

hJ (af b) d2(l+ps)

*

Since b, are l-forms in y’ and other quantities in the right-hand sides above are only functions of xi, it is concluded that G”(z, y) are certainly quadratic forms in yi. Further, equation (6.10) gives (A* ba) B” (7.41) +~(~)y’y~ yk = (1 +P2)d2 *

M. MATSUMOTO

14

Class 2. We consider a space belonging to the class I12(2). L2 may be written as (7.1-2)

L2 = 2F = (a’)2 E, Then, we have LL, where two l-forms co = Cam

= F, = Eco,

yi are defined by

Cl

=

t-2 -

(;p,

c2=

($2.

(7.2-2)

Next, we get Fap

Fll=E{l-s},

Fn=E($-;),

Fz2=2E.

Then, we have H,p: Hzz = E.

HI1 = E Since E is positive, Hap = EM, Mp shows

E =

1 and

(Il&) = k (--(x2,a’),

k2=&

(Ma) = h (-c2, cl),

1 h = -. k(a1)2

Then we get

7ni :

,l=

_ha%c”,

d

m

2 _ ha?c, --.

d

Therefore, we obtain Tl .oo

(7.3-2)

As a consequence, it is observed that Gi are certainly quadratic forms in yi and (7.42) Class 3. We consider a space belonging to the class B2(r, s). L2 is written as L2 = 2F = (a1)2r (a2)23,

r+s=l.

(7.1-3)

Two-DimensionalFinslerSpaces

15

We have easily

From Hap = E M, Mp, the above shows E (r s) < 0 which causes the subdivision of B2(r, s) above, and (M,) = k (-a2,a1),

(-$5) .

(Ma) = ; Then, we have

L2M’ ,I=

& TV = (rA1a2+sA2a’) =--, kL kda1a2 E(saia’ +raia2) kda1a2 ’

’

E(sa:a1+raia2) kda1a2

Therefore, we obtain

*

-‘-{(rA’a2+sA2a’)(sa~a1+ra~a2)}, T.‘,,= -rsd2

(7.3-3)

T&l = &{(rA1a2+sA2a1)(sa~a1+ra:a2)}, Consequently, Gi are certainly quadratic forms in yi and D&T)

yi yj yk = -

(rA1a2+sA2a1)a1a2

(7.43) rsd2 THEOREM 5. AlI the twodimensional Berwald spaces with constantmain scalar I axe divided into four classes B2(1), B2(2), B2(r, s), r s < 0, and B2(r, s), r, s > 0, the fundamental functions L of which me written as (7.1-l), (7.1-2), (7.1-3), respectively. The functions G’ of those spaces are written as (6.7), where Ijik(z) respectively.

are given by (6.5) and T.&, by (7.3-l),

(7.3-2) and (7.33),

Class 4. In particular, we are interested in the coefficient Ys of the equation (5.1), which is given by (6.11). For a space belonging to the class B2(2), we have, from (7.42), Y3 = $ (a; a2 ai - a: & a$) + f

{ (Aa cap) (ai)2}

(7.5-2)

.

Hence, if ai vanishes, then we have Ys = 0. Similarly, for a space belonging to the class B2(r, s), we have from (7.43), Ys = i (a$&a:

-a:&af)

- ---&

(rA1ai+sA2ai)

aiag.

(7.53)

Thus, if ai or a$ vanishes, then we have Ys = 0. Therefore, we have the following result. THEOREM 6. If the equation of a geodesic for the spaces mentioned in Theorem 5 is written in

the form (5.1), then the coefficients Y’s are given by (6.11), where Dijk(Z) (7.42), and (7.43), respectively.

axe given by (7.4-l),

For spaces belonging to the classes B2(2) and B2(r, s), Ys is given by (7.5-2) and (7.53), respectively.

16

M. MATSUMOTO

8. FROM THE GEODESICS TO THE FUNDAMENTAL FUNCTION Part 1. Suppose that a twoparameter

family of curves

q&b) :

f(z; a,b),

Y=

(8-l)

is given in the (CC,y)-plane, or the coordinate domain of a local coordinate system (z, y) for a smooth manifold M2 of dimension two. We shall consider the problem of finding the FinsleT associated fvndamental function A(x, y, y’) having C(a, b) as the geodesics. It seems not to have been noticed that the problem had been solved affirmatively by Darboux near the end of last century [14]. We shall show Darboux’s method here. First, from (8.1), we get

z (= Y') = f&x; a,b), z’ = fz5(x; a, b).

(84 (8.3)

We solve the parameters (a,b) from (8.1), (8.2), to obtain

a = 4~

Y,

.z), b = P(x, Y, 4.

(8.4)

The equation (8.3) is written in the form

We are concerned with the equation of a geodesic for the Rashevsky form (2.6): A(C) = A,, u + A,, z + A,,

- A, = 0.

Our problem is to find the function A(x, y,z) such that A(C) = 0 becomes the identity [A] = 0 in (x, y, z). Differentiating [A] = 0 by z and putting B = A,,,

we get

B,+B,z+B,u+BuZ=O.

(8.6)

We shall recall a theorem on partial differential equations as follows. Let Pi(x, B), i = 1,. . . ,n, and R(x, B) be given functions of xi and B. general solution of a first order quasilinear differential equation,

We hope to find the

To do so, we construct the auxiliary equations dx’ P’=...=pn=R’

dx”

dB

and find n independent solutions fi(x, B) = Q, c’s being constant. Then, the given equation shows the existence of a functional relation among fi, that is, @( fi, . . . , f,,) = 0 gives the general solution for an arbitrary function ip. Now the auxiliary equations of (8.6) are written as dx=dy=dr=__!$ Z

11

t

Two-DimensionalFinder Spaces

17

From the first two equations we get, of course, the solutions (8.1), (8.2), that is, (8.4). Next, dx = -#$, that is, T = -uz dx yields the third solution, B = c/U(x; a, b), where c is the third constant and U(z; a,b) = exp

Uz(z,f,fi)dx

(J

1

.

(8.7)

Consequently, equation (8.6) shows the existence of a functional relation among a = CX,b = ,B, and c = B V, where V(x, y, z) is defined by

Therefore, the functional relation may be written in the form

H(a, P) w, Y, 2)’

B(x, Y, z) =

(8.9)

where H is an arbitrary function. Then, if we construct A*@, Y, z) =

B(z, Y, z) (W2,

(8.10)

+ ~D(z,Y)

(8.11)

I..

then we obtain A(x, y, z) in the form A(GYJ)

= A*(GY,z)

+ C(Z,Y),

where C and D are arbitrarily chosen, but A must satisfy [A] = 0, which is written as D,-C,=A;-A;,-A;,z--A;+.

(8.12)

It is easy to verify that the right-hand side of (8.12) is independent of z. Therefore, we have the following theorem. THEOREM 7 ( DARBOUX). The associated fundamental function A(x, y, z), whose geodesics are given by (8.1) is obtained as equation (8.11), where A* is given by (8.10) and C(x, y) and D(x, y) are arbitrary functions, such that equation (8.12) is satisfied. B(x, y, z) in (8.10) is found as follows: (i) We find cy, p of (8.4) fkom (8.1) and (8.2), and define u by (8.5); (ii) U is given by (8.7), V (8.8), and B by (8.9), where H is arbitrarily chosen. EXAMPLE 1. We consider the family of straight lines: y = ax + b. Then we easily obtain cY=z,

p=y-zx,

u=o,

U=l,

V=l.

Thus, we have B = H(z, y - z x). According to the general formula

- t) f(t) JJf(u) b-W2= o”(u J

&

we obtain A* = If we put HZ = -1,

then

A; = AtZ =

‘(z-t)H(t,y-tx)dt.

J ’

J0 10

*(z-t)

H2dt,

A;,

= s’ 0

Hz 4

A;, = H,

H2(-t)dt,

(8.13)

18

M. MATSIJMOTO

hence, D, - C, = 0, which implies the existence of a function E(x, y) satisfying C = E, D = E,. Thus, we have A(x,y,z)

=

and

*(z-t)H(t,y-tx)dt+rE,+E,. J0

Therefore, equation (2.5) leads us to the fundamental function

qx, y,i, Q) =

z=Y :. X

=(z-t)H(t,y-tx)dt+E,k+E&

i

(8.14)

A Finsler space F” is said to be projectively flat or with rectilinear extremals, if it is covered by coordinate neighborhoods of local coordinate systems in each of which any geodesic is represented by n linear equations zi = x6 +t ai of a parameter t, or n- 1 linearly independent linear equations a? (xi - x6) = 0, CX= 1,. . . , n - 1 [6]. Such a coordinate system, (xi) is called rectilinear [15]. To find all the projectively flat Finsler spaces, closely related to the 4th problem of D. Hilbert, has been attempted by a lot of authors. The two-dimensional case, however, has been solved as above. THEOREM 8 (DARBOUX). AJJ the fundamental functions of projectively Aat Finsler spaces of dimension two are given by (8.14), where H and E(x, y) are arbitrary functions. REMARK 1. There are some essential mistakes in FtapcsB’s paper [16] on the projectively Finsler spaces. Therefore, his result is false.

flat

REMARK 2. The additional terms E, k + EV ti of (8.14) is the derived form z, hence, it gives rise to the well-known Randers change [17], which preserves geodesics as a point set. Generally, the arbitrariness of H and E may cause a projective change of the Finsler metric [6]. EXAMPLE 2. We consider the family of spirals C(a, b) : r = exp(a0 + b) in the polar coordinate system (T, 0) of a Euclidean plane. Since we have logr = a 0 + b, we put x = 8, y = log r, and equation (8.14) immediately gives

qe,

7-, 4, f) = b

J

*(z0

t)H(t,logr-

t8)dt + E,6+

ET+,

.Z=-,

i

where H and E(r, 0) are arbitrarily chosen (181. The figure shows three spirals Co(-), and Cz(- - -) of the set C,, = C(a,, b,), n = 0, fl, 3~2,. . . , from P(?, 6,) to Q(T2, b), r1 = exp(a,

4 + h),

(8.15)

7-B

~2 = exp{a,

Figure1.

(02 + 2nr) + bn}.

cl(-), where

Two-Dimensional Finsler Spaces Part

19

2.

There are various interesting curves which cannot be written in the form (8.1), but can be written in the parametric form C(a, b) : x = qi(t; a, b), y = $(t; a, b). Darboux’s method cannot be applied in this case. We now consider how to find the fundamental function for such a family of curves. Throughout, we use the notation 4 = 2, 8 = $$. In order to deal with homogeneous functions, we introduce the auxiliary parameter c such that

x = 4(d; a,b), Y = Il(& a,b),

(8.16)

where c must be assumed to be positive, because we must preserve the orientation of the curves. Since equation (8.16) gives p = 2

= $12, q = $

; = f&t; a, b),

= dc, we get two equations z = $(ct; a, b).

(8.17)

From three equations among the four in (8.16) and (8.17), we solve a, b, and ct as functions of For instance, we shall now suppose that from

z, y, and one of (:,I).

2 = $(ct; a,b),

y = qqct; a,b),

: = &ct; a,b),

(8.18)

we can solve a, b, and ct as a =

a* (x,Y,;),

ct=~*(x,~,f).

b=P*(x,y,F),

(8.19)

Then the remaining equation is written in the form (8.20) from which c is solved as (8.21)

c = $GY,P,9). Consequently, equation (8.19) gives three functions (Y,p, and r as follows: a = (II*

(x,Y,E>= Y

4x,

Q),

Y,P,

(8.22) t+

{?

(x42)) =T(x,Y,P,Q).

We have to pay attention to the homogeneity of CY, ,L3,y, and r in (p,q). number k we have from (8.20)

For a positive

$$=v(z,Y,$), and, from (8.21), kc = -y(z, y, kp, kg), hence 7 is (1)phomogeneous in (p, q). Then equation (8.22) respectively. shows that cr,p, and r are (0), (0), and (-l)phomogeneous, Next, from P = djc2, Q = Gc2, we get the functions P, Q:

P=

gr; a, P) Y2= P(x, Y,P,d,

4=

4(YT%P)-Y’= Qh

It is obvious that P and Q are (2)phomogeneous

Y,P,

in (p, q).

9).

(8.23)

20

M. MATSUMOTO

Now our problem is to find L(z, y,p,q) Weierstrass form

such that the equation (2.2) of a geodesic in the

W(C)=L,,-L,,+RW=O,

R=pQ-qP,

becomes the identity [W] = 0 in (z, y,p, q). W e remark R is (3)phomogeneous in (p, q), while W is (-3)phomogeneous. Let us differentiate [W] = 0 by p and q. For instance, differentiating by p, we have [WI, = &Xl - L,,+R,W+RW,. From L, = W q2 and L, = -W,, we have L,, - L,,, account of the homogeneity of P, Q, R, and W, we have Ppp+Pqq=2P,

= -(W,

p + W, q) q. Further, on

QpP+Qqq=2Q,

R,p+R,q=3R,

W,p+W,q=-3W.

By means of these relations it is shown that [WI,, and [WI, can be written as [WI, = -q[W], and [WI, = p [WI,, where we put (8.24)

[W]o=W,p+W,q+W,P+W,Q+(P,+Q,)W.

Therefore, instead of [WI, = [W], = 0, we have the single quasilinear differential equation [W]s = 0. Now we apply the first theorem in Part 1 to [W]O = 0. Its auxiliary equations are written as dx -=_ p

dy

dp

dW

dq

q =P=G=-(P,+Q,)W’

If we put the above to be equal to dt, then dt =

d”c

&

=

!!!

&

=

q’

P’

dp

&

P’

=

3

Q’

give the four solutions. Introducing the fourth integration constant to, those solutions are written as x=qqc(t+to); a,b), y=$(c(t+to); @b), (8.25) :=&c(t+to);a,b),

%=?l(c(t+to);a,b).

The remaining equation dt = -dW/[(Pp dW = -{Pp(4, W

+ Q4)W] is written as $7 c$, ~8) + Qq(4, $7 ~$7 ~1))) dt.

Thus, owing to the homogeneity of P and Q, if we put ~(x, Y, P, q) = Pp + Qq, U(c(t + to); a, b) = exp

WC (t + to); a, b) = 44,1cI, $7 ?I>,

{J

l-I(U;u,b)du

1

,

u=c(t+t,,),

(8.26) (8.27)

then we get W(c (t + to); a, b) = $,

(8.28)

with the fifth integration constant d. From (8.25) and (8.28), we solve for a, b, c, to, and d, and obtain a = CY,b = p, c = y, to = r - t from our discussion above, and d = WV, where V is defined as (8.29) V(x,y,p,q) = V-YT Q,P). We remark that V is (0)phomogeneous

in (p,q).

Two-Dimensional Finsler Spaces

21

Consequently, the differential equation [W]c = 0 shows the existence of a functional relation among o, 0, y, r-t, and W V. It must be, however, remarked that in our problem the following two conditions of W should be requested for the relation: (i) W does not contain t explicitly, hence the relation may be written as W V = H*((Y, /?, y). (ii) W is (-3)phomogeneous in (p, q). Since V, (Y, p, y are (0), (0), (0), (l)phomogeneous,

respectively, for a positive number Ic, we

have

=

WX’Y’Pq)

v(x

y

If we take k = l/y

p

q)

“’

k3

’

and put H*((Y, p, 1) = H(o, /3), then we obtain finally W@,y,p,q)

H(a, P) = y3,

(8.30)

where H is, of course, arbitrarily chosen. Now, from L, = W q2 and L,, = Wp2, we may put L; = q2

II

w (dp)2,

L; = p2

JJw(&I29

where the integrations must be done such that L;, i = 1,2, be (l)phomogeneous, JJ W (d~)~ and JJ W (dq)2 be (-1)phomogeneous in (p, q). REMARK.

(8.31) that is,

If we take 3x2 + 2xy, for instance, then we get

s

(3x2 + 2x y) dx = x3 + x2 y + c(y),

where the function c(y) can be arbitrarily chosen, so that the above is not necessarily homogeneous. We will show how to obtain the homogeneous function of degree r + 1 by the integration of a homogeneous fin&ion of degree T as follows. Suppose that f(x, y) is a given function homogeneous of degree

T

in (x, y). From

f(x,~)=y”f(i,l),wehave

g(x, Y) =

I

f(x, Y) dx = Y’

J

W,

1) Y) dzz,

z=:. Y

If we put h(z) = J f(z, 1) dz, then h( z ) is homogeneous of degree 0, so g(x, y) = y’+r h obviously homogeneous of degree T + 1. Now, for instance, we obtain L in the form

L= L; +Pc*b,Y,q)

= LfhYlPlq)

E

is

+D*(GY,!d.

Since C* and D* must be (0) and (1)phomogeneous and D* = qD(z, y). Therefore, we obtain L(xlY,Plq)

0

+Pc(x,Y)

in q, respectively, we may put C’ = C(x, y)

+qDhY),

i = 1,2.

(8.32)

Finally, substituting the above in [W] = 0, we get the condition for C and D as C, -D,

= (L;),,

- (L&,

+ RW.

(8.33)

M. MATSUMOTO

22

We remark that the right-hand side of the above does not depend on (p, q), as easily verified by making use of (L;),, = -pq W,, (L;),, = q2 W,, etc. THEOREM

9. The fundamental function L(x, y, p, q) whose geodesics are given by the parametric

form x = r#~(t;a,b), y = $(t; a,b), is obtained in the form (8.32), where Lf, i = 1,2, are given by (8.31) and C(s, y) and D(x, y) are arbitrarily chosen such that (8.33) is satisfied. W(x, y,P, q) in (8.31) is found as follows: (i) (ii) (iii) (iv)

a*, P*, and T* of (8.19) are given from (8.18), for instance, and 7 of (8.21), from (8.20). cr, p, and 7 are defined by (8.22), and P and Q by (8.23). II is given by (8.26), U by (8.27), and V by (8.29). W is given by (8.30), where H is its arbitrary function.

EXAMPLE 3. We consider the well-known Bmchistochmne problem, that is, to determine for a heavy particle the cume of steepest descent between two given points in a vertical plane. This leads us to the Riemann metric

Lb,

Y>P, Q)=

Ji-? d

y > 0.

'

The curves are the geodesics, which are cycloids: x=a(t-sint)+b,

y=a(l-cost).

We apply Theorem 9 to this family of cycloids. In this case, we should start from three equations x = &ct; a, b) = a {ct - sin(d)}

+ b,

y = $(ct; a, b) = a (1 - cos(ct)}, because f = a(1 - cos(ct)}

z = 7j(ct; a, b) = a sin(d),

is only equal to y; b=p”=x-a!*T*+;,

c~=a*=&{(~)~+y~}, ct = r* = arctan

2Y (q/c) {(!?lc)2 - Y2) I .

Sincefrom$=ywegetc=y=E,wehave Ly = Y (P2 + q2) 2p2 ’

p=x7+7

7 = y arctan P We have P = y

and Q = (q2 - p2)/2y, so that we have

r=-

2q Y

Consequently, we obtain W = H(cY, fi) {(p2 + q2)2 y3/4p7}. Since this W is written as W = H(a, p) ((r2 y/p”), replacing H(a, p) CY~by H(a,P) from the arbitrariness of H, we obtain finally the simpler expression of W: W=H(cr,/3)

$ (

. >

Two-Dimensional

On the other hand, H(a,/3)

is chosen

as

from the original

Riemann

Finsler Spaces

metric

23

L, we get W = (p2 + q2)-3/2/&7,

hence

H = (~cY)-~/~.

EXAMPLE 4. At the end of the author’s y > 0, is considered. Applying Theorem

paper [ll], the family of semicircles (CT- a)2 + y2 = b2, 9 to this family, we obtain

a=z+~, p=;(p2+qy2, and

W = H(Q)@) y3 (p2 + q2)‘i2/p4.

Since this

W is equal

to

f-f(a)P>P (y2/p3), repl=iw

H(cr, p)p by H(cr, p), we finally obtain W=H(o,@

f (

. >

It is well-known in Riemann geometry that the family of semicircles above are geodesics of the Raemann metric ds 2 = (dx2 + dy2)/y2 of constant curvature -1. In this case, we have L(z) y,p, q) = dm/y and W = (p2 + q2)-3/2/y. Thus H(cr, p) is choosen as H = pe3.

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