Curvature Tensors on Hypersurfaces of a Finsler Space endowed with TM-connections

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NII-Electronic Library Service MnMoms oF SAGAM:

INsTITvTH oF TucgNoLoGY

Vol.19,No. 1,1985

Curvature

Tensors

on

Hypersurfaces

of a

Finsler

Space

endowed with

TM.connections

Dedicatedto the latePb'qfessorDr. Akitsugu Kdwaguchi

Mamoru

YosHIDA'

Introduction

H. Yasuda

[9]i)

introduced TM-connections on an n-dimensional Finslerspace from

the standpoint of tangent Minkowski spaces and developedthe theory of these connections

([10],

[11],

[12]).

Moreover inhis_paper

_[13]

the theory of hypersurfacesof a Finslerspace endowed with a TM-connection was investigated axiomatically. Especiallyitisirnportant

that the induced TM-connection isa TM-connection on a hypersurface in consideration.

The firstpurpose of the present paper isto derivethe so-called Ricciformulae,Gauss and Codazzi equations fora hypersurfaceof a Finslerspace endowed with a

TM-connec-tion

(S

4). And the second purpose isto investigatethe curvature tensors with _respect _to

the induced TM-connection, in _particular,those with respect to the induced

AMR-connec-tion and IS-connection

(S5

and

g6).

In the firstthree sections we shall eutline TM-connections in a Finslerspace, its

hy-persurfaces and the inducedTM-connection from H. Yasuda's papers stated above. The

terminologies and notations refer to those of the _paper

[13]

unless otherwise stated.

S1.

ATM-conneetion

According to H. Yasuda

([9]rv[13]),

letM; be an n-dimensional Finslerspace with a fundamentalfunctionL(x`,y`)and be endowed with a

TM-connection

TMr=(rjtk, rtk,Cj`k).

This connection isdefinedas follows:

1) The v-connection is

given

by

(1.1}

C,･`k:=gthCink,

where C,･ltk:=(1!2)agjlt/ayk,gjh:=(lf2)02L2fOy"ayltand gtltare _the components of _the inverse

matrix of

(gih)･

2) The non-linear connection is_given by

(1.2)

Fi,=ptV4i,=Gi,+Tt,,

where G`kisthe non-linear connection of Cartan

(or

Berwald)and

(a)

T`kisa _positivelyhomogeneous tensorof degree1in yiand indicatric,i.e.ykTi,= ytTl=O

(y,:

==gijyO･

3) The h-connectionisgiven by

*#KecR _#nt ₁₉₈₄ _tF ₁₀ _n ₁₁ _Hec"

1) Numbers in brackets refer to the references at the end of the paper.

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Shonan Institute of Technology

NII-Electronic Library Service

ShonanInstitute ofTechnology

igeczmetig#reetig19 ig za1 e

a.3)

r",=art,/tQJ7j+OJi,=G",+Tlit,+QJi,, where Gj`kisthe h-connectionof Berwald, 11f`k:==OTI/ayJand

(b)

Qj',

isa positively homogeneous tensor of degreeO in _y`and indicatricwith

re-spect toindicesiand _1',i.e._{y,eji,==yQji,=O.}

The absolute differentialsof vectors yt and X`(x, y) on M;, are definedin the following

way:

(1.4)

llyt:==dyi+ri,dxk,

(1.5)

DXt:=dXi+(4`k+C"hrhk)XYdxte+Cji,JWdxk=Xir,dxk+Xi]kbyk,

where the h-resp. v-covariant derivativesMre resp. X`lkare _givenby

(1.6)

Xil,:=6,Xt+rjt,XV

_{(6,:=e/Oxk-rJ,O/Qyi),}

(1.7)

xvl,:=oxi/ayx+c,i,xv.

Here we require that llyi=giJllyi.

Then,

H. Yasuda

<[13],

Theorem 1.1)proved the followingtheorem.

Theorem 1.1. A TMLconnection 7 on II(.ischaracterized

by

the

following

six axioms

(TMI)

r-

(TM6)

:

(TMI)

r ismetrical, i.e.

L･k==O･

(TM2) The det17ectiontensorvanishes, i.e.yj4ik-r`k=O.

(TM3)

7

is

v-metrical,

i･e.

giilk=O･

(TM4)

The v-torsion tensor vanishes, i.e.Cj`k-Ckij=O.

(TM5)

Ilyi=gtjllyf,or equivalently

(TM5)' 7:heabsolute doferential

of

giyisindicatric,i.e.y`Dg,,=O.

(TM6)

The

Paths

with resPect to r are always _geodesics

of

Ml,.

Consequently,a TM-connection is uniquely determined by the axioms

(TMI)rv(TM6)

if

tensors

T`,

and

Qj`k

satisfying

(a)

and

(b)

respectively are given.

The so-called Ricciformulae fora vector X`(x,y)are given as follows

([10],

(1.9),

(1.14),

(1.M)

:

(1.8)

XVrfk-X`JkFj=X7t]ilinivk-X`i.rJ'k-XVI.R-'jk,

a.9)

Xtfjl,-Xt]ntj--XhAij,-XVf,C:･,-Xt[.,P,rj,

(1.10)

Xiljl,-.Xii,Id=llPLS,tf,, where

a)

R.ijk

:=Ktjk+ Cnt,RSjk , ]i}h'jkgri= :R-nijk, ny .- -v b) KLtjk:=6kl"nij+I'h'jl-'.ik-1'1le,KL'jkg.`=:KAijk, c) T"ic:=I-'jik-1"lk, rj'kg.t=:Tjik,

(1.11)

d) RNid,:=6,I'i,.-ik=y"R'"b`jk=ryki?liJk, ji}'jkg.i=:]ii}iiic _,

e)

At,-k:=:r"hk-Chikdi+CntsA'J,

A'Jkg.i=:iE,tjk,

N -- -- iLJ

f)

atj:=rijr[k-rktJ:=-ektj=pthadik,

Arjg.i=:Il,j,

g) Sndi,･ic:==ChidltD+Ch's-C,iic-jlle=ChSicC,iJ-1'1le _, Slt'ikg.i=:Shi,-k.

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Curvature 7lensorson llyPersteijkecesofa Finsler SPace endowed with TM-connections

Here, -J'Ik means the interchangeof indices_]',k inthe foregoingterms and subtraction,

and "k:=Olayk.

Rh`jk

(or

Rnijk),

Phifa

(or

Phijk)

and Sh`ik

(or

Shijk)are called h-,hv- and

v-cur-vature tensors respectively. Also R`j,

(or

Rific)iscalled the curvature tensor with respect

to the non-linear connection I"ik. Tjik,

Eik

and Cjiicare called h-, hv- and v-torsion tensors

respectively.

In the definitionof a TM-connection, ifthe tensor T`, in

_(1.2)

isdefinedby the

follow-ing weak condition

(c),

then the connection iscalled a WTM-connection:

(c) Tikisa _positivelyhomogeneous tensor of degree 1 in _yi such that T`,y,=O.

Consequently,

a WTM-connection ischaracterized by the fiyeaxiorns

(TMI)."(TM5).

S2.

Hypersurfaces of M.

Let Mhm, be a hypersurfaceof M;,represented _{parametrically} be the equations

(2.1)

xi=xi(ua) ,2}

where we suppose that the variables ua form a coordinate system of MATi. Also,we

as-sume that the matrix with the components

(2.2) Bi.:=ex`leua

isof rank n-1. Then B`.(u)may beregarded as n-1 linearlyindependent vectors tangent to Ml,-iat a _point

<u").

Consequently,

if we denote the components of a tangent vector

yi to a curve contained in

an-i

by ya3)in terms of u"-system, then we have (2.3) yt==Bi.y",

ayi/dy"=B`..

Also,fora general tangent vector X` to MA-i, we have

(2･4)

X`== B`aXtt,

where Xlrare the cornponents of the vector interms of u"-system.

The fundamental function L(u",y") on M;,-iinducedfrom L(x`,y`)of M. is_given by

(2.5)

L("a,_y")=L(x`

(ua),

B`.ya).

From this,for the r-metric tensor _g.p:=(112)02L!/ayaayP, we have

(2.6)

g.p=gtjBi.BJp.

The covariant vector _y. corresponding to _ya isexpressed by

(2.7)

Y.=g.pYP=LOLIdy"=YiB`.･

At each point

(u")

of Mh-i, we can choose a unit normal vector M(u",y") to ML.i such that

(2.8)'

a) _{g,,B`.IVY=:O,} b) _{g,,NilVli==1.}

Since the matrix

(B`.,Ni)

is non-singular, we shall denote by

(B",,ATL)

the inverse

2)3)HereThisandinthe following, Latin indicesrun from 1 to n, while Greek indicesfrem 1 to n-1. notation will not cause any confusion.

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ShonanInstitute of Technology

Nec=*Jlt\reet ag19# za1 e

rnatrix of

_(B`.,

N').

Then

we have the followingrelations:

(2.g)

a) B`aBPi=bPa, b) B`.IV}=O, c) iV'Ba,=O,

d) IVtN,=1, e) B"iB'.+IV}N'=6'i,

(2.10)

a) IVtpt,=O,b) N}y`=O, _c) Ark=g,,M.

Making use of _the _cornpanents _g"P'_of _the inversematrix of

(g.p),

we obtain

(2.11)

Ba,=gaPg,jBdp,

from which we further have

(2.12)

a) gaPBfp=g`fB",, b) g.pB",=g,iBJp, c) _{gaP=gWB",BPd.}

In the sequel, forbrevitywe shall use the followingnotations:

(2.13)

BYp:=BiaBJp, B`"f:=B%gi', BkSter:=Bt.BifiBk,,

Bff-p"kr:=B"iBfpBicr,BaS'%`:==B`.BjpBkrBio.

Also,

ifwe defifie

(2.14)

a) _{pt.p:=Ci,･thB2"'pNk,}b) _{pt"r:=g"Pgepr,} c) _{pt.:=CifkB`.IV'fNic,}

we have

(2.15)

Ba,Jir==2ptariV}

_("r:=O/dyr),

(2.16)

a) IVUtir=-2FtarBi.-prN`,b) IV}rrr=ptrlV}･

S3.

The induced TM-connection on M;,-i

Let y` and X` be vectors related by

(2.3)

and

(2.4)

respectively. Then _thelabsolute differentialsof _them _are _expressed _as

(3.1)

lly`=(Bo`r+ri,Bkr)du'+B`.cly",

(3.2)

lly`== B`.dXZ'+

(Bp`r+

I-'j`kB'iicr+ Cj`hl"hkB'pkr+ CjikBjpBokr)XPd"'+ Cj`icB'fi'krXPaly' ,

where Botr:=BstryP,Bhi,:=OB`plOu'.

Now we definethe absolute differentialsof vectors _y" and Xt'on

ua-t

as follows:

(3.3)

Qya:=Ba,Llyi,

(3.4)

DXZr:=Ba,DXt.

Thus a connection r:=:(rpar,rar,

Ce"r)

on Ml,-iis

defined.

In_terms of thisconnection, Dya

and DXi' are expressible in

(3.5)

Llya=clya+rardur,

(3.6)

-

_{DXlr=dXtv+(rs"r+Cpaer"r)dur+CsarX]'dyr=Xlrfrdur+XZrlrPyr,}

where

-68-NII-Electronic Mbrary Service

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Curvature Z7nsorson Hly'Persutlfacesofa FVnslerSPace endowed with T,l4-connections

(3.7)

XlrLr:=o-rXti+rparXP

(Dr:=OIOu'-rPrelayP),

(3.8)

. Mrlr:=OXt'lay'+Cp",XP.

Substituting

(3.1)

into

C3.3)

and cornparing the result with

(3.5),

we have

(3,9)

r"r=B",(Bo`r+PkBkr).

Next, substituting

(3.2)

into(3.4),comparing the result with

(3.6)

and making use of

(3.9),

(2.9)e),we obtain

(3.10)

7s"r=B"i(Bsir+rj`kBfkr+Cp`kNkHl),

{3.11)

CB"r==C,ikBfSkr,

where Csik:==Cj`teBjp and

(3.12)

H}:=Al}(Be`r+r`kBkr)･

Then

transvecting

(3.10)

by

yP we have

(3.13)

yPrp"r==rar.

Ifwe put

e 4

(3.14)

H),r:=H),r+ltpH},Hbr:=Nl(Bp`r+4ikBikr),

from

(3.10)

and

(2.9)e)

we have

(3.15)

Bbir+rj`kBe'icr+Cp`kNkH}=rp"rBi.+HbrlV`.

We call a cennection

i=(bar,

rar,

CsaT)

on ML-i obtained as above the

induced

TM-eonnection and denoteitby ITMr. Also, Hi and Hbr are called the normal curvature

vector and the second h-fundarnentaltensor respectively, while _"firof

(2.14)a)

the second

v-fundamental tensor.

Differentiating

(3.12)

by yP and using

(2.16)b)

and (1.3),we obtain

p

(3.16)

H}nB=ptpH}+IVL(Bpir+rik"jBk'kr)=HbrrQp"r==Hisr+ypH}rQp"r,

where

Qpptr:

==Qj`klV}B'p'k,.

Transvecting

(3.12)

and

(3.14)

by yr and _yP respectively, we have

(3.17) Hb=IV}(Be`.+2G`),

Let Xijasbe an object definedon MU-i such that itisa tensor

(with

respect to indices

iand J')in Mh and at the same time a tenser (withrespect to indicescr and

fi)

on M;,-i.

Then the relative h-and v-covariant derivatives of X`jafiare definedas follows([13],

(5.18)

(5.19)):

(3.18)

XtJ-afi,r:= o-rXiJ-ap+Xk,asl'kir-Xi,apl',kr+XijeproaT-Xtjafirpar ,

(3.19)

Xij"fiir:==X`j"pMr+XidapCkir-X`k"pCjkr+X'jefiCoar-X`j"aCpbr,

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igecx#Jsc\teet zz19# eg1e

where

rkir:=rktdBir+CkiJIVYH}, CJkr:=Cjk,Bir.

Ifwe apply these derivatiyesto Bi.,we have

(3･20)

a) B`.rB=HltplV`,b) BS.lp=paplVi･

Further, applying the h-covariantdifferentiationto

(2.8)a),

b) and making use of

(3.20)a),

we have

a) gisErB`.NV+Hltr+gijBi.Arelr='O,

(3.21)

b) gtJ1rN`AJY+21V}IVtir==O.

Transvecting

(3.21)a)

by B"h and using

(2.9)e)

and

(3.21)b),

we _get

(3.22)

Nitr=-IZlarBP`+gJkli('ll-NilVY-g`f)BtrNic,

where we used gjte[r=gyicireBhr.

Similarly,

we have

(3.23)

N`lr='ptprBPi.

Next, we notice that

<3･24)

fir=Birtit+IVblMO/dyi.

Also, forthe h- and v-covariant derivativesof _g.pwe have

(3･25)

a) _{gap:r==gwJkBkjpkr,}b) _{gaprr=gi,･lkB2jpkr=O,}

where we used the axiom

(TM3).

Note 3.1. In a TM-connection and itsinduced TM-connection, the non-linear

con-nections r`, and rar are related by

(3.9).

Now ifwe transvect

(3.9)

by Bf.,we have

r"rBJa== B,Sr+rYteB"r-IVYH} ,

which isexpressible in

(3.26)

Bojr-r"rBJ.+rJkBkr=iVYH}.

Then itfotlowsfrom

(3.26)

that H} vanishes ifand only ifthe followingequation

holds:

(3.27)

B.fr-PcrrBj.+rlkBkr=O.

Hence we can state

Proposition3.1.

With

respect to the induced TMLconnection, the normal c"rvature

vector HL,vanishes

if

and _only

if

the _equatien

(3.27)

holds.

According to the papers

([6],

[7],

[1]),

itissaid that the non-Iinear connection r`khas

the

(H)-property

with respect to the non-linear connection rar ifthe equation

(3,27)

holds.

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Cz{rvatureTlrnsorson HZyPersutlfitcesofa FVnsterSPaceendowed with TtlfLconnections

Therefore from Proposition3.1we can state

Proposition3.2. T7ienon-linear connection ri,has the

(H)-PrQPerty

with respect to

Par

_if

and _enly

if

the _normal curvat"re vector HL, vanishes with resPect to the induced

TMLconnection.

Further

frorn

Theorem 8.1and Theorem 8.5in

[13]

_we _can _state

Proposition3.3, The non-linear _connection r`, has _the _(H)-PrQPertywith respect to r"r

_if

and only

if

a

dyPersudece

MA-t

of

MA istotally_geodesic with _resPect to the induced

TMLconnectien.

in

thiscase, the induced TMLconnection isintrinsic.

S4.

The Ricciformulae, the Gauss and Codazzi _equations for

MTt

In thissection, we first

derive

_the _so-called Ricci

formulae

fora vector Xi.. Applying

the relative h-covariantdifferentiationto Xi.･p,we obtain

02X`. Orap OXV.

O?Xia

oxv.

orjtp oxz,

Xzaorrfiir=oupour- our oye -rep ouTepfi+ our rJZp+XVa eur-- our I-'aEfi

-xi, Oerior,`p-rEr(

siio\li,"p--

Or6y",paSil

',a -rnp

oOy2i\iSfr,+ Oaiyi'.a

r,tp+xv. OrayJl'p

- nilil'i_{r.ep-xioOraycr,fifi)+(}ao.I

lpct--rfip

OoiYy:iii!-+xk.r,jp-x],r.Ep)rfr

-( Ooiliie-r6p eoXyie-+xli,rdtp-Xiar,ip

)r.Er-Xi./,PpEr

_.

Hence

we have the firstRicciformula forXi.

(4･1)

XialpLr-Xia[rle=MaK`fir'X`ort.6er-X`.16"pfir-OX`.fOy"'ROpr,

where the curvature tensors K)･`pr,KLeBr,Repr and the h-torsion tensor ffibrare definedas

follows:

a) Kj`pr:=6r4`p+4kprk`r-Plr,

(4.2)

b)

tgcrafir:=6rraSp+l-'.tpl-'Lbr-P[r,

-. R6pr:=arrap-PIr, Reprgab==:Rcrpr, c) d) 7pfir:=:: rfiSr-Plr , T-fiSrg.6= : "L pcrr･

Substituting

6X`./aya==Xi.la-Xi.C"e+Xi,C.eb into

(4.1),

we have another form for

_(4.1)

(4.3)

X`alpLr-X`.lrlp=Xi.Rj`prmX`eR.Jpr-X`.lb7fier-Xi.leR6pr,

where the curvature tensors

Rj`pr,

RN.efirare definedas follows:

a) Rf`pr:=K)`pr+Cj`aRepr,Rj`prgiic=:Rjkpr,

(4'4)

b)

R.6pr:=iZl,Spr+C.a,Repr,

JiliaSprgaE=:RaEpr'

Next, applying the relative v-covariant differentiationto X`.ip,we obtain

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Shonan 工nstitute of _Teohnology

相模工業大学紀要第 19 巻第 1 号 x ’ ・’・

1

・一 ∂

諾

一∂

畿

∂

寮

一_r・・

篇

・＋

黌

厂ノ・＋鴎

肇

一

｛

券

r． e β一Xi，

∂

多

β ＋

（

零

鍛

一rOp

∂

器

一壬一Xkal”k’β一

X

」e1 −’ atfi

）

C」 ‘ r

＋

（

零

1 ≡

≡

…

！

− ₁一δ β

∂

雛　

＋−

Xj

，r“β一Xiδ1 −一畧 δ β

）

（）atT − Xial，CB ’ r

，

while _∠applying 　the　relative　h−covariant 　

differentiation

　to　X ‘

α］r，　we 　have

x・ a …β一

講

・

夥

・ c_’鵬

鴒暢羨

俳・− x ・ ∂

默

一_丁δ P

（

蠶

蕃

＋ ∂

畢

c_ノ・＋x・a ∂

審

L

夥

c

・・ r− Xie ∂

舞

）

・

（

∂

艶

瑞一襯・

）

蝋

黌

部ノ轟

_・− x ・．

1

，・， “ ，．

Hence　we 　have　the　second 　Ricci　formula　for　Xia

（4．5）XialpレーX ‘ alnp ＝X 」α1「iiPr− X ‘ eP． Op アーX ， aloCp δ r− X ，α』君，

where 　the　curvature 　tensors 」P 」

‘ pr，」％

δ

βr　and 　the　torsion　tensor　jt5rsp　are　defined　as　follows：

　　　　　a_）P 」・ iPr ：＝ rfiPIlr一δ_pC 」 ir 十1’」 kpCkir −

C

／r∫’〜β一Cii，1 ▼tPsrr ，　∫〜 ‘ βγσ ＝；ノ〜卵 r ，（4 ．6）

b

） P・ δ βr ：＝「・δ β」1・一δ・C・ δ r＋厂〜β

C

〜

rC

・ ‘

4

δ ・＋C・ δ ・riβli・　　　　　　　＝ ∬ T ． δ PlIr− CaOr」P十Cα δ ，PrSβ，　・PhδPrge_，＝：Pa_，_Pr_，　　　　　c_）

P

_，δ β：＝ rδ_Pllr− r ，δβ．

Finally，　Iet　us　apply 　the　relative 　v・covariant 　differentiation　to　

X

ε_α

1

β．　 Then 　we 　obtain x・・

1

・

1

・一

券

蒜

＋

崇

cノ・＋x ・a ∂

象

L

欝

（）_at_厂 Xte ∂

舞

・＋

（

∂

黔

＋x・。C・’・

− _X ・・

殉

q

・＋

（

∂

寮

榔ノ・−X ‘・

殉

C・・ r− X ・．

1

，・・’・　， from 　which 　we 　have　the　third　Ricci　formula　for　X

（4・7）X， α

1

β「r − _X， α

lrlp

＝ Xd．

S

」 ‘ βr− X‘aSα δ fir，

where 　the　curvature 　tensors　

S

」

‘

pr　and 　S_．e_β_rare 　de且ned 　as　fo110ws：

（4．8）

a） S・ ’ ・・：＝ C・ ‘ ・・r− C・ k ・Ckt・一β

lr

・ S・ ‘ ・・9・k＝ _・S ・・… 　　　　　　 b）_Saδ βr：＝

C

α δ PIIr−

C

α erC ‘ δ β一βレ，　Sα δ βr9 δ、＝：Sa、β7 ．

　　Next，　we 　shall　find　the　so ・called 　Gauss　and 　Codazzi　equations ．

　　Firstly，　applying （4．3＞to β and 　using （3．20），　we 　have

（4・9_）Biarpir− _Btalr ］P＝ B’ α五〜」 tPr− _」_Bi ，A〜ectPr− 」Ua、2＞ β ε r一μα 、ハμ距・ fir．

Here，　because　of （

3

．20）a）and （3．22），　the　left　hand　side 　of （4．9）is　expressible 　in 一 72 一

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Curvature11ensorson IisPersunyfaces_ofa Finsler SPaceendowed with TMLconnections

BZalp:r-Bt./rlfi=Hl,plrNi+Hl,p[-HlrBE'+gdkli(-il'N`IVli-g")BtrNk]-Plr _.

Consequentlythe expression {4.9)hasthe form

<4.10)

H}p]TNi+Hl,p[-HLrBEi+g,kit(tlVilVV-gt')B`rNk]-Plr

== BjaRjiBr-Bt,R.eprLHL,Ntr"'per-pt.,NiR`pr.

Transvecting

(4.10)

by B"i,we can see

(4.11)

R."pr-Be,R,`prBS.=HLp(Har+gj,[,B`rBejM)-Plr,

where Har:=geEHLr. Lowering indexofiin

(4.11),

_we have

(4･12)

RaBra-RijreB:'p=HZ,r(Hisa+giji icNiB'ika)'riO ･

Also, transvection of

(4.10)

by N, yields

- -. 1

(4.13) Hl,a7p"r+pt.bROpr-IViR,tprB'.= Jl}'Hltpg,k:tBirAPN't-HLptr-filr_･ Secondly,ifwe apply {4.5)to B`.,we have

<4.14)

B`.pprr-B`airlp=Bj.RiinrrB`,P.SfirrHl,,N`Cp'r-pt.,NZPrEp.

Here,

because of

(3.20),

(3.22)and

(3.23),

the lefthand side of

(4.14)

isexpressible in

Bialplr-BZalrlfi=HLrelrNi-H},ppterBet-ptaTlpN'-".r[-HLpBEe+gjkre(-ii-N`AP-giS)B:plVk]. Hence the expression (4.14)has the forrn

(4･15)

HLplrN`-HL,p"ErBtt-".rlpN'Np.r[mHLpBSt+gjk/i(-ll-N`NV-gzj)B`pNk]

=Bi.Pj`fir'Bi,R.tfir'Hl,,N`CpEr'pt.EN`Pr'p _･

Transvecting

(4.15)

by Bni,we can see

(4.16)

Pd"fir-BetPj`prB'.==H}ppar-pt.T(Hbp+gj,L,BipBejAPt).

Lowering index 6 in

_(4.16),

we have

(4･17)

Papre'PijreB2'p=HLrptfie-".s(Hlr+gi,:

icN`B]p' kr) . Also, transvection of

(4.15)

by N} gives

(4.18)

H}sCifr+p.oPrap-AT}PjiprB'a='Hltfilr'(JIIgargJkitBi3N'IVte-parip)･

Finaliy,letus _apply

(4.7)

to Bt.. Then we _get

(4･19)

B`alelr-Bi.lrlp=Bj.SXer-Bi,S.epr.

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reutXmeJk\reee ca19 g ca1 e

Here, the Iefthand side of

(4.19)

can be written as

Bia

lplr-Bi.Irlp

== _pt.p

lrNV-pt.ppt,rBei-Plr

.

Hence

_(4.19)

has the form

(4.20)

g.plrM-pt.p",rB"-Plr==Bj.Si`prmB`.S.`pr･

Transvecting

_(4.20)

by BO,,we haye

(4.21)

S.eprmB6iSiiprBd.=pt.ppteT-Plr.

Lowering

index e in

(4.21),

we have

(4.22)

S,pre'S},reB:'p'=ptar"pe-rl6･

Also,transvection of

(4.20)

by IV}yields

(4･23)

MSi`prB'a=Aaplr-Plr･

(4.11)

(or

(4.12)),

(4.16)

(or

(4.17))

and

(4.21)

(or

(4.22))

are called the Gauss equations

in our spaces. Also,

(4.13),

(4.18)

and

(4.23)

are called the Codazziequations inour spaces.

In the remaining _partof thi$section we shall findthe relations between _the curvature

tensors

Ri`pr

and

R"t.,

Eipr

and ]lifit.,

Sf`pr

and

Sf`ke

respectively.

Let us apply

(4.3)

to a scalar X; then we have

(4･24)

Xipir-XtrJp=-Xl.T"'p"r-Xl.Rapr.

The lefthand side of the above isrewritten as

(-XliT"k-XltR-'fjk)BS}-(-XliC,`k'XliA`j)(B'plr-Pir)IVk

+(Hkr-Plr)XltNt+Xli(flkN`Jr+HisdriV`'Blr),

while the right hand side isexpressible in

-(XiiB`.+XIilV'iHL)7p"r'XliB`.Rutfir.

Thereforesubstituting the above two expressions into

(4.24)

and comparing the components of

Mi

_and Xii,we obtain respectively

(4･25)

Tj`kB'pkr+C"k(Bdpza-fi1r)Nk=7p"rB`.+(Hhr-Plr)N`,

(4.26)

Bt.R"pr+N`Hlfpar=R-ijicB'p'icr+]PkS(B'pHi-Plr)Nte-(HklViLr+HbsrlV`-P[T)･

Now making use of

(4,26),

after directcalculation, Rjdprof

(4.4)

isexpressed as

(4.27)

Rj`pr=R,i,.BkM,+P,iz.(B`fia-PIr)IVin'.

Next, using

(2.15),

(2.16)a),

(3.16)

and

(3.18),

ILiprof

(4.6)

isexpressed as

(4･28)

"P,･ier=E`i.B`pMr+S,･`t.NZBMrHk. Finally,

Svtpr

of

(4.8)

isrewritten as

(4･29)

S"pr=

_Sy

'kt_Bkpir･ NII-Electronic Mbrary 1

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Curvat"re717nsorson Hl}rPersurzfticesofa FinslerSPaceendowed with TM-connections

By virtue of

(4.27)N(4.29),

the Gauss equations

(4.12),

(4.17)

and

(4.22)

are also

ex-pressed respectively as follows:

(4･30) R.prs=RwntB2jpkrS+PifkiBYp(BkrHb-r16)IVi+[ILrr(Hb6+gtjikN`B'iica)Hr[6J,

(4.31)

Aprfi=PA'tdbiBkjpkrS+SijkiBl'p'}IV'tHl+Hlrgpi-tt.b(Hsr+giiikNiB'pkr),

(4.32)

Sa,ero=SijktB:'p'kr6`+(ptarptpAe-r16);

while the Codazzi equations

(4.13),

(4.18)

and

(4.23)

are also expressed respectively as

follows:

(4.33)

Hl,eTA'eSr+p.aROpr"Nt[Rjii.BfoMr+Pjit.(B`pHl･-P[r)IV'"]B'. 1

='i}-HLtpgikitBirNYNk-HL,p,r-Blr,

(4.34)

HheCper+pt.sPr6fi'IV`(Riii.BfoMr+Sjtt.IV`BMrElis)Bj.

=-Hlplr-(-ll-ptargJk,tBipNdNk-garip) _,

(4･35)

NiSjikiB'akSr=ItafiIT-PbJ･

Note 4,1. If we consider the Cartanconnection

(r*"k,

G`k,CJik},the curvature tensors Rniyk,l%iJkand the inducedconnection, then the Gauss and Codazziequations correspond-ing to

(4.30),

(4.31)

and

(4.33),

(4.34)

are expressed as follows:

(4.36)

(4,37)

(4.38)

(4.39)

where b ee Rapr6=Ri,kiB2//"ra`+PLjkiBE"p(BkrHls'7"IO)N`+(Hl,THkfi-r16>

_(cf.

_[5],

_(5.16))

, bc

Rrpr6=:aiktBYpkrS+StfkiB:'p'SNkK+(Hl,ryps-alP)

(cf.

[5],

(5.19))

, ec b c Hl,s:ifr+".aRapr-N`[Rnc.B'sMr+I}ti.(B`pe-Plr)IVM]Bja=-Hlrsir'Plr

(cf.

[5],

c c b e H}fiCfiOr+".eP?fip-Ni(R,'icmBfoMr+SjiimN`BMrHb)Bfa=-Hkplr+ptarlp

(cf.

[5],

(5.17)),

(5.20)),

b c b Hh:=AI}(B.ii+G`icBks), Hlxr::=M(Ba'r+r*j`kB".'icT+C.'kNkH}), c b e b

rpS,:=ptSpH}-P[r,

R6p:=eip+"6THIi

, PVSfi:=Rti,BiJikp,

R,-`k:=Gjik-r*jik

. The expressions

(4.36)

and

(4.37)

will be used later.

Note 4.2. Ifwe consider the Hashiguchi connection

(Gjz',,

G`,,Cjt,)and the induced

connection, then the Gauss and Codazziequations corresponding to

(4.30),

(4.31)

and

(4.33),

(4.34)

are expressed as follows:

h h h b hh

(4･40)

Raprs=RiikiB'.''fi'icr

.!+ajktBia' 'p(BkrH}7rl6)2V`+[Hllr(Hhi-2Prgpfi)-rla] ,

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.75-Shonan Institute of Technology

ShonanInstitute ofTechnology igecz# rt\reaj ee19 gg ag1 e b h bh h (4･41)

Rtpre=efkiB2'pkrS+S,zeBE'pk;NkH}+Hltrptpe-pae(Hhz'2Pnrpr),

hh h b h b (4.42) Hl,bTp"r+p.aR"pr-N'[R,,t.BtpMr+llEliit.(BipH}-Plr)IVb"]B'.=-HlpR-HLplr-Plr, b h h b h (4.43) Hl,bCper+p.eP}ep-IV'i(Riii.B`pMr+SiiimBYIViHk)B'a=-Hltfiir+garRe+"artp, t where h b h b c

Hl,r:=A7}(B.'r+GXkB2kr+C.'kNkHI-),

RliTpe:=Rjk2V'Pkks,

Tper:==pOpHl･-Sr=Tp"r,

b h

lle::=

4jtNilVIBkr

,

a"p

;= pt",Hll.

Nete 4.3. The third

Gauss

and

Codazzi

equations for both

Cartan

and Hashiguchi

connections are

just

the same as (4.32)and (4.35)respectively.

S

5. An AMR-connection and _the induced AMR-connection

A TM-connection issaid to be r-metrical ifLig,,=O, which isequivalent to

([9],

p.7)

{5･1)

Zjk+Zite+Owre+OJite+2<CtJrT'k+Rfk)=O,

where

Zjk

: = 7}'kg.i

,

Qiik

:--

Qi'tg.j

,

Rjic

:= ll'reg.J.

An

r-metrical TM-conneetion iscalled an RTM-connection. In thiscase itisknown

([13],Theorem 2.1)that an RTM-connection r ischaracterized by the followingfive axloms:

(RTMI)

r ish-metricai,i.e.gijik:=O,

(TM2),

(TM3),

(TM4)

and

(TM6).

A TM-connection r issaid to be semi-symmetric ifr satisfiesthe followingaxiom:

(Wl)

The h-torsiontensor rjt, has the forrn

(5.2)

rjtk=6%sk-6`ksf

forsome positivelyhomogeneous covariant vector si of degreeO inyi.

An r-metrical semi-symrnetric WTM-connection iscal]ed a Wagner connection

(cf.

[2],

p.61). Hence thisconnection ischaracterized by the fiveaxiorns

(RTMI),

(TMb,

(TM3),

<TM4)

and

(Wl)

(cf.

I13],

S2).

A TM-connection iscalled an AMR-connection ifTi,and

Qj`,

have the forms

(5.3)

Tt,=.fZhi,,

(5.4)

Oj`k=-Ltflljhik+flkh`y-.f;LrCj`k-Lik

forsome positivelyhomogeneous scalar

f(x,

y) of

degree

O

in

M,

provided

h`re:=ai,-lil,,

l`:=y`IL,lk:==gi,y`.Itfollowsfrom

<5.4)

that the h-connectionhas the form

(s.s)

rji,==r*ji,+;f(ljhi,-lihJ,-LCji,),

where hjic:=gik-ljlk. In thiscase, H.

Yasuda

([13],

Theorem

2.7)proved

Theerem 5.1. An AMR-connection ischaracterized

by

the

five

axioms

(RTMI),

(TM2),

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Curvature Tlensorson H]},Persut:facesofa FinslerSPaceendowed with TndLconnections

(TM3),

(TM4)

and

(AMRI)

71heh･torsiontensor Tji, isgiven

by

(s.6)

T,.i,=f(li6i,-l,6i,･)

for

a

Positively

homogeneo"s scalar

f

of

degreeO iny`.

Consequentlyan AMR-connection isa Wagner connection satisfying Tiikyt=O.

Then we can state

Lemma 5.1. VVith resPect to an AMR-connection the_.following relatiens hold:

a) Sltli=-I-'*,thl'-fait,, b) 6nlj=r*lrnlr-fhpm , c) 6,hi,=-F*.i,h',+O*k',h`.+f(l`hklt+lkh`h),

(5.7)

d) 6hhjk=l-'*j'hh.k+I'*k'nhjr+f<ljh,,ic+lkhjn)m2.f:LCjklt, e) ahCjik=Cjikrh-r*.`ltCjrk+I-'*j'hC.ik+I"*icrhCj`.-f(LCiiki,,+Cjikl,), f) 6,r*ii,:=d,I-T*jik-fLI-T*jikl,h,

where the oblique short line means the Cartan'scovariant difflerentiationwith resPect to xltand dh:=O/Oxn-GihOIOyi.

Proof. Ifwe notice that with respect to a TM-connection we have

lilh=O, lj,,h=O,hik[h=O,

then

(5.7)a,

b>and c) are easily _proved. d)followsfrom that with respect to an

RTM-connec-tion hjksh=Oholds_good. For e), we have from the definitionof the Cartan'scovariant differentiation

(5.8)

Cji,,,,=d.Cji,+I'*.i.Cjr,-I"*jr,C.i,-r*,r,Cji..

Here,taking account of

d,=o/exh-Gr.6/a),r=a/axlt-l-'r,O/Oyr+7"'.OIOyr

=6,+fZLh',e/O",'=a. +fL(6'.-L-il,vr)O/6yr_,

we obtain

6hCjik=

Cjikth-fL(6'h-L'ilny')C.iikEi,-P.ihCj'it+rj'hC.ik+r*k'hC".

.

Further noticing that Cj`kishomogeneous of degree -1 iny`,we have e). Lastly from

the factthat r*j', ishomogeneous of degreeOin _y`we can see f).

Q.E.D.

Let us give another form of the curvature tensor Kl,jkwith respect to an AMR-con-nection by H. Yasuda

([li],

(4.5)).

Then

we can state

Preposition5.1. With respect to an AMR-connection, the curvat"re tensorKl,i,-kis

empressible in

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Kl,wk=KLtjk-]IL(lltwk-Rtik)+f2L2Shijk+[(f-ir,,ji-LChid)6kf+.ftzaktli

(14)

+enc]z*v<\vaetrg 19 g ag1e

1<}ikn j j'

I-'*"klln-felh!=I]titjrg'i+(C.`nRf'k-leIh),

Cj'kiih=-2Cf'kC,th+g`'Cikn][,

(e,g.

[3],

(2.3)(d))

and

(5.6),

then the above equation isreduced to

K}iklt=K}`iclt-.fZLIIthj.g'`+f2L2Si`kn+[flL(-Cj`kln'CrthRi`k+.flnCj`z)

+(fT'g'iTik.-LC"k)fih.f+foriTih.lk-f2hfnh`k-klh]. Further noticing that

C"kih+CrihRi'k-le]h=-Rheh.,g'i

(e.g.

[3],

(2.3)(a)),

R,'khi-hli=

Ilziff'lk,

and replacing the indicessuitably, we have

(5.9).

Further we can state

-f2(h,,h,,+LC,,jl,)-1'lk],

where KAifkisthe curvature tensor

of

Rund.

Proof. Firstsubstituting

(5.5)

into the definition

(1.11)b)

of K)ik,and using

5.1,we have

L`teh=dhrj`k-.rtLrj`k:ln+fihf(lih`k-l`hjk-LC"k)-f2(hjhhik-ljlkhih+lilkhjn)

-fZl:.Cjt,,.+f2L2C",11.+f2LCji,l.+IT*Js,I-T*,i,+f2L2Cit,C,t,-klh

Here ifwe remark that

:=d.r* i,+r* s,r*,t.-klh _,

Lemma

Q.E.D.

Propesition5.2. With resPect to an AMR-connection, the curvature tensor Rhijkis erpressible in

(s.lo)

R.,J,=R.,j,-fL(R,,j,-ll,,,j)+f2L2S,,j,+(tiT,jP,fH-ft,,,ld

Ef2h,,h,,-1'lh).

Proef. Substitutingintothe definition

_(1.11)a)

of R.i,,

(5.9)

and the equation

(cf.

[11],

(1.7))

,lir,,==KV,,+(4hr,-i'lk) _,

i

where K',k:=pt'Kh'pm Fli:=L(fLAis+f21j=f/d)[=-Lbff+f2Lt,, and rernarking Rnijk:= K]z`jk+ CltirKb-jk

,

we have

(5.10).

Q.E.D.

In virtue of

(5.10)

we have

Corollary5.2.1. IVithresPect to an AMR-connection, the curvature tensor

khtJk

is sleew-symmetric in the

]irst

tzvDindices.

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Cttrvature71ensorson I(}TPersut;flices

of

a FinslerSPaceendowed with TM:connections

In

the similar way as the _proof for

(5.7)e)

we have the fellowinglemma.

Lemma 5.2. With respect toan AMR-connection, we have

(s.11)

C,i,.j=C.t,,,j-f<liC,j,+l,Cvi,+l,C,1+lfC.i,)

-.EL(Chikl:j+C.ijChritTClt'jC.ik"Ck'jChir).

Making use of Lemma 5.2we can state

Proposition5.3. With respect to an AMR-connection, the curvature tensor

Phi,･k

is eopressible in

(5.12)

A,j,=:Il,,j.-fLS,w,+<Ai,l.h,i+.iL'ih.,g,s+fl.C,j,-hli)

･

Proof. Substituting

(5.5),

(5.11)

and

(5.4)

(cf.

P,',=-Qk'j) into the definition

(1.11)e)

of Pn`'jitand taking account of

lhll,=:L-lh.k,hijflk=-L-1(htklJ.;,hJ.kli), lil,,.,:L-lhiic,

hhjuk=2ChjkHL-i(hjkih+hhkl) ,

RiLk;=I-'*hij[lk-Chik,,J+ChtrPk'J

,

and

(1.11)g),

we obtain

(5.12).

Q.E.D.

Immediately we can state

Corollary

5.3.1.

Wlth

respect to an AMR-connection, the curvature tensor Pniiicis

skew-symmetric in the

first

two indices.

Next, we consider _the inducedTM-connection and the inducedAMR-connection, that

is,the connection on MA-i inducedfrornan AMR-connection on Ml,. First,_we _can _state

Proposition5.4. JVithrespect to the induced TMLconnection, the curvature tensor

jli.pT

iserpressible in

(5･13)

Rapr=RtikBk'pkr-[(QNap+Hlrp+gjkthBj.'hpNk)H}'PIr]

,

where

QN.s:=QkijN'Bkj-.

Proof. Transvecting

(4.26)

by Bdi,we have

(5.14)

R"pr=R`,,Be"iic,+BeiPk`j(BjpH}-PIr)Nk+B',(H}Niip-51r).

On the other hand,trnsvection of

<3.22)

by Bn,_yields

(5.15)

B6,IV{F:p=:-(HLpgEa+g,,,.Bg{gealVk).

Substituting

_(5.15)

into

(5.14),

transvecting the resulting equation by _g.oand noticing

Picw=

-Qkij, _we _can _show

_(5.13).

_Q.E.D.

From

Proposition

5.4we can state

Corollary5.4.1.

With

resPect to the induced AMR-connection, the curvature tensor

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reec=* Jki\va ee za 19g M 1 e

Raprisexpressible in

.v AJ -- e

(5･16)

Rapr=RijkB:'ikr+[( TNh.p+ZfLptap+qap-HL,p)Hle-Blr]

,

where

Tni

: =L;lililV)', haB:=gap-lalp, l.:=g.plP, IP:=YpfL,

f(ua,

y.)=f<x`(u"),Bi.y.)

,llnr.p=4tkNYB2kp ･

Proef. From

(5.4)

we can easily see

(5-17)

QNap=-(71btap+fLYas+4ap),

e-(5.18)

Hlp=Hl,p"fLptap･

Substituting

_(5.17)

and

(5.18)

into

(5.13)

and noticing giJTk=O,we have (5.16).

Q･E･D･

Moreover, we have the followingtwo corollaries.

Corollary5.4.2. PVithresPect to the induced Cartan connection, the curvat"re tensor Raprisempressible in

cb

(5'19)

Rapr=Rt,kB:'picr+[(Pnrafi-Hlrp)H}-Plr]

_(cf･

_[5],

_(5･14))･

Corollary 5.4.3. With resPect to the induced Hlxshiguchiconnectien, the curvature

tenser R.prisetpressible in

h b

(5-20)

R.pr=Ri,kB2'n'kr-[(HLtp-2RArap)M-Rlr]

･

By the way, forthe induced AMR-connection H. Yasuda

([13],

Theorem 7.6)_proved

Theorem

5.2. The induced AMR-cennection is the RTA4Lconnection en Ml,-,

deter-mined LLythe h･torsiontensor 7p"r with

(s.2n

a) Tkp"r=Tpar+(ptapH}.-P[r),

rp"r :=r,`,Bfpjk, =f(tpaar-Plr) _,

b)

and thenon-linear connection and h-connectienare etpressible resPectively in

(5.22)

r"r==Gar+Tar, Tar:=rar-G"r=-p"rHL+fLhar,

(5.23)

rp"r=r*par+Hlrppr-ptarH]3+f(lhhar-l"hfir-LCp"r)

+Hl,(Cr",g'p+Cpa,Fter-Cp'rrt",)

c L

=rper+f(lhh"r-l"hpr-LCper) _,

where Gcrr:=G"iir,

G":=:B"i(B,`.+2Gi),

B,i.:=B,`7:yr,har:==6"r-l"lr,r*p"r is the intrinsic

h-c

connection

of

Cartan and rp"ristheh-connection

of

theindtccedCartan connection, i.e,

c

(5.24>

rfiatr=B",(Bh`r+r",`kB'pk,+Cp`,2VkHl･).

We shall express the form _of the curvature tensor

R.prb

in terms _of fapr, Tapr･

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Curvature 71ensorson H]yPersut:f12cesofa FlinslerSPace endowed with TM-connections

Proposition 5.5. With resPect to the induced AMR-connection, the curvature tensor

fiapro

is

empressed as

(5･25)

R.pre=Rapre-fL(4prevR,finr)+f2L2Sapri+f(laTNTpn-lfiT-'ra6)

+(f-iT.rp6nf+f2h.rhp6rr[6) ･

Proof. Remarking that with respect to our connection g,i･k=Oholds,we have from

the Gauss equation

(4.30)

(5･26)

R.pre=RltijkB2fo'rka+Pht,kB2'p'

(B'rH}-rlti)Nte+(HlrHboLrl6)

･

By means of (5.10),the firstterm of the right hand side of the above equation is

wrltten m

(5

.27)

Rhi

pmB: `p'IS= RltijkB:2p'"r'

S

LfL(Rtijk- Il,ik;')B2'p'Jr'

S

+f 2L2ShijkB: Zi'r'

}

+(fMit.rp6kf'Bta+fT.ofilr+f2h.rhpa-rlfi)･ Similarlyby means _of _(5.12),_the _second _term iswritten in

(5.as)

Ph"kB:`p(B'rHk-r16)Nk=Il,ijkB:YrATteHi-fLShijkB:ifrlVteHla

+flivtlVin(lahpr-lphar)H}+f(laFtpr-lpstar)H}-rl6･ Moreover by virtue of

(5.18),

the third term iswritten in

cc .-c c

L-(5･29)

Hl,rHi36rr16=Hl,rHbi-fL(Hltrstpa+Hlsest.r)+f2L2tt.TFtfi6"rla･

In this case we have

c e c a

<5}30)

4rptps+H)ifiptarTr16=='

Hlr"pb+Hkepar-alP

･

Substituting

_{(5.27)fv(5.29)}

into

(5.26)

and making use of

(4.32),

(4.36),(4.37),(5.30),

(3.24)

and

(5.21),

we obtain

(5.25).

Q.E.D.

From

(5.25)

we have

Corellary 5.5.1. With resPect to the indteced AMR-connection, the curvature tensor Raerbisskew-symmetric in the

_first

two indices.

Next, we shall express the

form

of the curvature tensor P.pr6.

Proposition 5.6. JVithsesPect to the induced AMR-connection, the curvature tensor

4prs

isetpressed in

(5･31)

Paprs=IlrBrb-fLSapro+(fEialahfir+fL"ihaffgpr+flaCprs-alP)･

Proof. Remarking that with respect _to _our _connection _gv:k=O holds,we have from

the Gauss equation

(4.31)

Paprb;

B;ijkB"a

Yr

nk+ShijkB"ap`

tsArv"+

(HLrptpom

cr

IP)

.

Substituting

_(5.12)

and

(5.18)

into the above equation and making use of

(4.32)

and

(4.37),

(18)

NecrX rt\reaj eg19 # rg1 e

we obtain

(5.31).

Q.E.D.

In

virtue of

(5.31)

we have

Cerollary5.6.1. JVithresPect to the induced AMR-connectien, the curvatesre tensor Il,preissleew-symmetric in the

first

_two indices.

S6.

The IS-connectionand the induced IS-conneetion

The IS-connectionisdefinedas an r-metrical TM-connection _satisfying

Rf`h=O.

For

this connection, itfollowsthat the followingequations hold:

(6.1)

eji.=o,

g,",=o.

H, Yasuda

([13],

Theorem 2.6)proved

Theorem 6.1. IlheIS-connectionis characteriaed

by

the

_five

axioms

(TMI),

(TM3),

(TM4),

(TM6)

and

(ISI)

71hehv-curvaturetensor vanishes, i.e.PN.`d,==O.

Also,itisknown

[9]

that under the IS-connectionany two of indicatricesconsidered as Riemannian spaoes are isometric.

The connection on M.-t induced frornthe IS-connectionon M. iscalled the induced

IS-connection.In this case, from Proposition5.4,the equations

(4.30)

and

(4.31)

we can

state with

(6.1)

and the axiom

(ISI)

in mind

Theotem 6.2. Eor theindaced IS-connection,the.following relations

hold:

(6･2)

'

Rapr=:R"kB2'ikrH(HltpH}-Plr),

(6･3)

Raprb=RntjkB#aYrki+(HLrrHlae-r]fi),

(6･e

Papre=ShtjicB"._p`

_{3N'H}+(Hltrptpa-alP)}

･

Similarly,we have

Proposition6.1. llFlor the induced IS-connection,we

have

(6･5)

HI,.p]r-rslr=:Albljlii,".B'._fit

_{?-pta.R-Jofir-Hi,a7per}

,

(6･6)

Hltplr-pt.rlp+Hl"Cper+pt.ept"rHk=SLijkB".krNiATYHh.

Proof.

(6.5)

isobtained from

(4.33)

by using

(6.1),

(6.2)

and the axiom

(ISI).

(6,6)

is

obtained from

(4.34)

by using

(6.1),

(6.2)

and the equation

Pp"r=psapH}-Op"r

Qp"r:=QjikBgfirk

([13],

(6.9))

foraTM-connection.

Q.E.D.

Moreover, making use of

(4.31)

and

(5.31)

we obtain

Theorem 6.3. A necessary and suLOicient condition that the induced TMLconnection be

the

rs-connectien

on M.-i is

(19)

CurvatureTlensorson H)rPersudeces ofa FinslerSPaceendowed with TM-connections

(6-7)

AijkB:ipcr6+ShijkB".fi'iAUH+HitT"fibLpae(H]ir+gij]kiVblB"ikr)=O･

Corollary6.3.1.

A

necessary and su17icient eondition that the induced

RTM-connee-tionbe the rs-connectionon M.-i is

(6

.8) ilint,kB2 `p'rS+SntikBk. 'pknAPH)'+<Hlrgpb-al

B)

=O

Corollary6.3.2. A necessary and suficient condition that the induced

AMR-connec-tionbe the rs-connectionon M.-t is

(6･9)

RrpTowufLSafire+(fi]olahpr-FfL-ih.ngpr+fL,Cprb-cvlB)=O･

Acknowledgement. The author would like to express his hearty_thanks _to Professor

H.

Yasuda

forhiskindly_guidancesand constant encouragements.

References

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seminar on Finsler spaces, Brasov,Roumania, (1984),55-75.

[2] M. Hashiguchi: On Wagner's generalized Berwald space, J,Korean Math. Soc.,12(1975},51-61.

[3]

H. Izumi and T. Sakaguchi: Identitiesin Finslerspace, Memoirs of the NationalDefense

Academy, Japan,22 (1982),7-15.

[4]

M. Matsurnoto: Foundations of Finsler geometry and special Finslerspaces, ₍₁₉₇₇₎

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[5]

M. Matsumoto: The inducedand intrinsicFinslerconnections of a hypersurface and Finslerien

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[6]

R. Miron: A nonstandard theory of hypersurfacesin Finsler spaces, Analele Univ. "Al. _I.

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[8]

H. Rund: The differential_geometry of Finsler spaces, Springer Verlag, (1959).

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1-!1.

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[11]

H. Yasuda: On TMA-connections of a Finslerspace, Tensor, N.S., 40 (1983),151-158.

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