フィンスラー空間の超曲面の第2基本H-テンソル

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(1)Title. フィンスラー空間の超曲面の第2基本H-テンソル. Author(s). 北山, 雅士; 佐藤, 明; 柴田, 鋹光. Citation. 北海道教育大学紀要. 第二部. A, 数学・物理学・化学・工学編, 37(1) : 17-21. Issue Date. 1986-10. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/6124. Rights. Hokkaido University of Education.

(2) :)b?@Mtftir:^Ne^ (US 2 Â) ^37^ ^l^- [}gftl61^10J!. Journal of Hokkaido University of Education (Section II A) Vol. 37, No. 1 October, 1986. The Second Fundamental H-tensor of a Hypersurface of Finsler spaces.. Masashi KITAYAMA, Akila SATO * and Choko SHIBATA Mathematics Laboratory, Kushiro College, Hokkaido University of Education, Kushiro 085 Kushiro East High School, Kushiro. 7 ^ >^7-SI^®/t§ft®®^2S^H -7-yy^. ^m±-^m w* .^BaM ^mm^^±sf-mw^^.&^m "'ÎIS&Xit^-^^. Abstract Let J[;'"==(Mn, L(x, y)) be an n-dimensional Finsler space with fundamental function L(x, y). The second fundamental H-tensor of Finsler hypersurface is an important quantity for enriching the Finslerian hypersurface theory. In this paper, we study the properties of the second fundamental H-tensor and give an example of the minimal hypersurface.. § 1. Second fundamental H-tensor.. The fundamental tensor field 9u(x, y) of the Finsler space F"=(M", L) is defined by gu(x, y) ==31(Ll/2)/3y'3yJ. A hypersurface M"-I of the underlying differentiable manifold M" maybe parametrically represented by the equations x'=x'{ua), a=\, •••••., n—1, where u" are Gaussian. coordinates on M"~\ Using the notations B'a=3xt/3ua, we shall assume that the matrix of these projection factors is of rank n-1. The following notations are also employed ; B'ae= 31x'/3u"3u'1, ~Bta'e=BlaBi. If we assume that the supporting element y' be tangential to M"~\ we may then write yl=Bla (u)v°, so that v" is thought of as the supporting element of Af"-l at the point (u°). Since the function L(u, v)=L(x(u), y(u, v)) gives rise to a Finsler metric of M "-1, we get an (n-l)-dimensional Finsler space F"-I=(M , L(u, u)). At each point (u°) of F"-l, the unit normal vector Bt(u, v) is defined by guBW=Q, g,jB'BJ=l. If (Ba, Bi) is the inverse matrix of. (17).

(3) 18 Masashi KITAYAMA, Akila SATO and Choko SHIBATA {Bi, B'), we have (1.1) B^BB=SS, BiBt=Q, BtBa=0, B'B^l. We are concerned with a Finsler space (F", FD equipped with a Finsler connection FF== (F/K, N'j, C/k). We are not concerned, however, with any relation between the Finsler metric L(x, y) and the Pinsler connection FF. Putting the induced Finsler connection IFF=(Fia~,, N"», Ce°y) on a hypersurface Fn-l of F", we have. (1. 2) B^+B^F^Bf+ C/,BkHy)=F,ayB^H^Bt, where Hais (resp. Ha) is called the second fundamental H-tensor (resp. the normal curvature vector), and these are given by. (1.3). Hey;=B,\B^+B^F,\B!}+C,\BI:Hy)\, Ha •,=Bt(B^+N'jB^, B»=B^v0.. We now treat of another Pinsler connection Fr*=(F*fc', N*'j, C*j\) to satisfy the relation (1.4) FA.=F*\.+DA., NJ,=N*i,+D1,, CA=C*(.+AA, where D/n (resp. A/^) is a (0) p-homogeneous tensor (resp. a (—1) p-homogeneous tensor). If we assume that both of the deflection tensors of FF and FF* vanish identically, we then have. (1.5) D^=D^. Moreover we assume that the two Finsler connections FF and FF* satisfy the Ci-condition. As examples of (1. 4), we find the following: The Cartan connection Fr"=Cr=(F/Ai N j, Cjn) of a Finsler space F" with an (a, ft) -metric and the Riemannian connection Fr*=(IA.I, to\-t, 0) are. related by FA-=iAI+D/,, JV<,=to',!+Doi,, C,',=A/,,. and the difference tensors Dj „ and Aj \ are respectively given by ([6]) (1,6) D/,= N'E^+F'.Nj + F'jN, +B{j b,, + B\ bw - b^g"°B^- C,(ôm.- C,^D,mo+ C^D.m.gw. It follows from (1. 4) and the Ci-condition that Aic=0.. Putting (1.7). H*^;=B,Wr+B^F*/^+C*,\BKH^, H^=B,(B^+N*',B^,. from (1. 4) and (1. 7), Ha and Her in (1. 3) are respectively rewritten as. (1.3/). H^=HSy+D^+Af,W+Ar)+MSAr, * Ha=HS+D,, '00',. (18).

(4) The Second Fundamental H-tensor of a Hypersurface of Finsler spaces. 19. where we put Ae=BiA;^BK, Dey=BiD^B^, MS=BiC*I^B>:. Contracting Ha by v°, we get (1.8) ffo=//o*+0oo. If each path of hypersurface F"-l with respect to the induced connection IF F is a path of the enveloping space F" with respect to the FF, then Fn~' is called a hypersurface of the first kind. It is well-known [3] that a hypersurcace F"~l is a hyperplane of the first kind if and only if the normal curvature vector Ha vanishes identically, and Ha=0 is equivalent to Ha=0. Thus from (1. 8) we can state. Proposition 1. Suppose that two Finsler connections FF and FF satisfy the relation (1. 4). A hypersurface F"~t is a hyperplane of the first kind if and only if H5°+Dw:=0 holds good.. We shall treat of a Finsler vector field X [x, y) of the F" which is regarded as the covariant component of a normal vector field ofF"-I. Then, along F"-l we get (1.9) ! X,B<$=0, X,yl=Q, X,(x(u), y(u, v))=^X~2B, where X2=:g'JX,Xj, from which we have X,^+XiB^=Q. Since Bme=HSeBl. this yields. (1.10). H^ - (X^B^X^BJBiHS)/^XS HS=-X,^'B^ /(!+-/>), Ho*=-Z,n^y/(l+(*),. where </>=Xi^y'BJ /^X1 and X,n,=5-,X,-XrF*r,.. (i.ii) H^-Xi^ytyj/^+<f')+D», D«,=x,D^/Vx1. From Proposition 1 and (1. 11) we have. Theorem 1. Let a Finsler vector X of the F" be normal to the hypersurface F"~\ Suppose that two Finsler connections FF and FF satisfy the relation (1. 4). Then the Finsler hypersurface F"-I. is a hyperplane of the first kind if and only if the vector X satisfies the following condition along F"~l (1.12) X^ytyJ-{\+<t»X,Di^y]yli/^=Q. In particular, we deal with the vector field Xi which is a function of the coordinate only. Then. (1.12) leads to ' (1.13) X^tyj-X,D/,yJyl!/VXl=0. We here are concerned with a special Randers metric L=a+fi, where as=a,f(x)yy^ and ^=6,(a;)i/ with a gradient vector 6,(3;)=3,fo for a scalar function, and consider a hypersurface F"~'(c) which is given by an equation b(x)= c (constant). Such a hypersurface has been treated by M. Matsumoto [3]. From the parametrical equation x'=x'(u) of F"~l, along the F"-1, we. (19).

(5) 20 Masashi KITAYAMA, Akila SATO and Choko SHIBATA. get bi{x)Bi =0. This b, is nothing but the Finsler vector Xi in (1. 9), which is a function of the coordinate only. Then from (1. 13) and the difference tensor £>A of a Randers space ([!]), we get bi.,j=biC,+bjC,,. where (;) means the covariant differentiation with respect to the Riemannian connection FF*. Thus from the above Theorem we get. Proposition 2 CMatsumoto [3]). Let F" be a Panders space with a gradient vector bi{x) \=9{b(x) and let F"~t(c) be a hypersurface of F" which is given by b(x) =c (constant), provided that the Riemanndin metric aij{x)dx'dxj is positive-definite and bi is the non-zero vector field. Then the condition for Fn-l(c) to be a hyperplane of the first kind is bi,j=btCj+bjCi. Next, we consider the fundamental function L=L+ft, where L=(az7)I/3, 7=nn(x)yl: and P=bi(x)yt, with a gradient vector b,{x)=3,b(x), and a hypersurface Fn-I(c) which is given the equation b, (x) =c (const.) and 6,ni=0. Let CF==(F, N, C) (resp.' pr= CF=(F, N, 0) be the Cartan connection of F"=(Mn, L)(resp. F"=(Mn, D). Then, for these connections, it is well-known [4] that (1.14) FA.=FA.+DA, FA.=F*'.+DA, where F*J'K=\J'^, £>A and Djn are the difference tensors. This vector bi (x) is also nothing. but the Finsler vector Xi(x) in (1. 9), which is a function of the coordinate only. By a similar argument with the Randers space, we can state Proposition 3. Let F" be a Finsler space with a fundamental function L=(a •y}"3+f3, where a2=au(x)y'yj is non-degenerate, 7=bi(x')yl and t3=ni(x}y', with a gradient vector bcix) =3(6(a;) and rii y=0, and let Fn-I(c) be a hypersurface of the F" which is given by b{x)=c (constant). Then the condition for F"-l(c) to be a hyperplane of the first kind is bi..j= biCj +bjCi.. .. .,. § 2. Minimal hypersurface. A hypersurface F"-l of a Finsler space F" such that Hw9M vanishes identically is called the minimal hypersurface, where g =g Bw. We shall deal with a Randers space in § 1. And we shall give a condition for the hypersurface Fn-l(c) of the Randers space F" to be minimal.. Theorem 2. Let F" be a Randers space with a gradient vector bi(x)=3tb(x) and let F"~l(c). be a hypersurface of the F" which is given by b{x) = c (constant). Assume that the Riemannian metric au(x)dx'dx is positive-definite and that bi is non-zero vector field. Then , the hypersurface F"~ (c) t's minimal if and only if &„»= Cia,;i+ c^bibn, where we put. (20).

(6) The Second Fundamental H-tensor of a Hypersurface of Finsler spaces. 21. c,=-4/(n-2)bs(-brr+bw/b1), c,=(l/((n-2)64)) ((-4brr+W bl+(n-2)) b^. Proof. By virtue of Matsumoto's monograph [3], we already know that (2.1) â2brD/t=-(2abw-bmbl)hu+2a(bjabi+bt,bj)-2b«,btbj, bu=brj.. Putting Xi (x) =bi (x), from (1. 10) and (2. 1), Hw in (1. 3') is rewritten as (2.2) ^blHw==-buB^+brD,T,B^.. Contracting this by gw we get (2.3) Vb2H^go0^-brr+bw/bl-(n-2)(2a/3w-blb«,)/4a2, where we put bm=bubly'. If we assume that F"~l(c) is a minimal hypersurface, that is Hwgae =0. Then (2.3) gives (2.4) (n-2)b^=0, 4:(-brr+bw/b2)a!+(n-2)b2b«,=0. Differentiating the latter by v" and ve, we get (-'ibrr+b^/bl)aw+(n-2)blb,,BÔ.. Further, contracting this by BSS and referring to (2. 4), we have (2.5) 6,»=ciai»+ Cibib,,,. where ci=-4(-&rr+6,,/62)/(n-2) b\ c,=(-46r.+(4/6z+(n-2))^)/(n-2)64. Conversely if (2. 5) holds good, we immediately get (2. 4), so that we observe that Haeg vanishes.. References. [1] Matsumoto M. (1974), On Finsler spaces with Panders' metric and special forms of important tensors, J. Math. Kyoto Univ. 14, p. 477-498.. [2] Matsumoto M. (1977), Foundations of Finsler geometry and special Finsler spaces, Kyoto, 373p. [3] Matsumoto M. (1985), The induced and intrinsic Finsler connections of a hypersurface and Finslerian projective geometry, J. Math. Kyoto Univ, 25, P. 107-144. [4] Matsumoto M. and Numata S. (1979), On Finsler spaces with a cubic metric, Tensor, N. S., 33, P. 153-162.. [5] Shibata C. (1984), On invariant tensors of P -changes of Finsler metric, J. Math. Kyoto Univ. 24, P. 163-188.. [6] Shibata C. (1984), On Finsler spaces with an (a , /? )-metric, J. Hokkaido Univ. of Education 35, p. 1-16,. (21).

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