トップページ - 横浜国立大学学術情報リポジトリ

全文

(1)Theory of surfaces of Euclidean 3-space from an alternative point of view Isuke SAT6 and Masao MAEDA (Revceived April 30, 1983) ,3. gl. Introduction. In the present paper we will exclusively deal surfaces in Euclidean 3-space.. hl. In case of the study of surfaces, we have usually for our object to obtain the geometric properties issued in reference to the surface and its ambient space. Then, it is well known that Euclidean metric of Euclidean 3-space plays a very important role.. On the other hand, Euclidean 3-space admits naturally a normal contact metric structure such that its associated metric is just one of our space. The pur-. pose of the paper is to study surface of our space in connection with the normal almost contact metric structure.. In g2, we consider normal almost contact metric structure of Euclidean 3-space which is naturally induced from an almost complex structure of Euclidean 4-space. In the next section, we deal with (f g, u, v, 2)-structure induced on a surface of Euclidean 3-space. In g4, we find differential equations which .f; g, u, w, and Z satify. In g5,. we prove some lemmas which are valid for later use.,. In g6, we consider the special cases where R=O and 1-Z2==O everywhere on surfaces. g7 is devoted to the study of surfaces with normal (7C] g, u, w, 2)structure. In the last section we study surfaces with anti-normal (.fl g, u, w, 2)structure.. g2. Normal almost contact structure in E3.. ]. We consider a linear map in Euclidean 4-space E` rererred to a rectangular coordinates (xi, x2, x3, x4) defined by. Xl -X3. ii. (2.1) F: - . N X2. orrepresentedbymatrixform -. -X4. X3 Xl X4 X2. (2.2) fi=[IO, '--8] (h;unit matrix of order k). NN. from which F2= -I4, that is, F is an almost complex structure on E`. It is easily.

(2) I. SATo and M. MAEDA. 16. seen that lv F is a member of the quaternion structure on E`. Now, we consider a hyperplane V in E` defined by. (2.3) x,=rv,,x,=O, (h,･i,7'=1,2,3), which is nothing but an Euclidean 3-space E3 with rectangular coordinates (ori, bl2, or3). If we denote by X the position vector from the origin to a point of V, then V is represent,ed by. (2.4) X=X(or,, or,, or,).. The. (2.5) Xle==O,X (O,==a/OY;,),. pt. i. e.. )G=. 1. o. o. o. 1. o. o. , X2=. o. o. ,&==. ts. 1. o. are orthonormal vectors tangent to Y and N=:t(O,. v. iO o, o,. The transforms fiXl of XZ･ by # can, be represented. (2.6) iXl==Fi'Xlf+ai2V,. 1) is the normal of V.. as. from which (2.7). OOI ooo. F=(.l7,h) . --. , a=(ab a2, a3)=(O, 1, O),. 1 O O. o. A =ta =. 1. '. o. where F, a and A denote a tensor of type (1.1), a row vector and a column vector in V respectively. We can easily see that. a(A)=1, F2Y== - Y+a( Y)A,. (2.8) 0(FY, FZ)==0(Y, Z)--a(Y)a(Z), N(Y,Z)==O (Normality), where Y and Z are arbitary vectors on V and 0 is a Riemannian (Euclidean) metric induced from E` and. IV(Y, Z) =[Y, Z]+F[FY, Z]+F[Y, FZ]-[FY, FZI "-'{Y'a(Z)--Z･a(Y)}A. In other words, the set (F, g, A, a) gives a normal almost contact metric structure on E3 ([2]).. Next, making use of the rotation of an almost contact structure ([1]), we shall construct some almost contact structure in E3.. For the almost contact structure (F, A, a), putting ei= A, we take a constant vector e2==t(p, g, r) of E3 which is lineary independent upon ei, i.e. one of p and r is at least non-zero. Then, e3=Fe2=t(r, o, -p) does not vanish and ei and e2 are perpendicular to e3,. E,. i ( `.

(3) Theory of surfaces of Euclidean 3-space from an alternative point of view. 17. Denoting the dual of the basis ei=t(eii, ei2, ei3), (i=1, 2, 3) by ei*=(di`, d2i,. di we have ei* =(-Pq/s, 1, -'gr/s),. (2.9) e2"==(p/s, O, r/s),. e3"= (r/s, O, -p/s), where we have put s=:p2+r2. If we put. (2.10) pmh==di2eih+diie2h+di3e3h, then, taking account of (2.9), we have s. --p2g+r2 ps -pr(1+q) 1. '. (2.11) pt=(pah)=ge -Pg2+P qs (1-g2)r inpr(1+q) rs p2thpgr. b. Proposition 2.1. The follozving facts holds good: (i) pt gives an almost product structure on E3, (ii) ei--pte2 and e2--ptei.. Proof. (i) Cleary pt ¥I3. Using (2.11) and by slight long computation, we find. 100 rf=(padpah).. O 1 O ==l,,. OOI Thus, (i) is proved. (ii) Making use of (2.11) and (i), the assertion follows immediately.. Now, define a tensor g, a row vector 6 and column vector rp on the almost contact structure (F, A, a) by. (2.12) g=pt-iFpt, 6=pt-iA, rp=aopt. Proposition 2.2. The set (g, 6, rp) is an almost contact structure on Euclidean 3-space.. Proof. Taking account of Proposition 2.1, we find, rp(6) =a(ptpt-iA)=a(A) =1, g(6) = pt-'Fpa-'A = pt-iFA = o,. and X being an arbitary vector on E3, we also find, e･. i l l.. g. rp(gX) =a(pa-iFptX)=:a(FptX)= O, g2(X) =:(pt-iFpt)(pt-iFpt)X :pt-iF2ptX. =:pthi(-yX+a(ptX)A) =--X+a(ptX)pt-iA= -X+pt(X)g. Hence the set (g, e, rp) is an almost contact structure.. From the option of the vector e2 and Proposition 2.2, we see that Euclidean 3-space admits a lot of almost contact structures..

(4) 18 I. SATo and M. MAEDA '. g3. Surfaces with (f, g, u, v, Z)-structure. We consider an orientable surfaces S covered by a system of coordinate neighborhoods {U; x', T2} and immerced differentiably in an Euclidean 3-space E3. referred to a rectangular coordinate sytsem. If we denote by X the position vector starting from the origin of E3 and ending at the point P Qf S, then the surface is represented by. (3.1) X==X(T"), (a,b,c･･･=1,2).. pm. The. Bb=O,X, O,=O/OTb are two linearly independent vector fields tangent to the surface and gbc=Bc'Bb are the local components of the tensor representing the Riemannian metric on S. h. .i. N. e. induced from that of E3. We choose the unit normal vector C to S in E3 in such a way that the vectors C, Bi, B2 defined along the surface S give the positive orientation of E3.. Since E3 admits naturally almost contact metric structure (F, g, A, a), the. transforms FBb of Bb by F and FC of C by F are expressed as linear combination of Bb and C as follows:. , (3.2) FBb :.1(b"B.+ubc, FC==-uaB., (ua=ubgba), A=vaBa+2C. respectively, where .fba are components of a tensor field of type (1, 1) and ub, vb=gbcvC are those of 1-forms of S and 2 is a function on S and a(X)== 0(A, X) for any vector X of E3. From (3.2), by the index notation, we have .ICba=BblFdZB%,. (3.3) u,= B,jLiCi, vb==Bbdad, R=CiAi,. where we put B%= g"bOihBbh; 0: metric of E3,. (3･4) c, :o,,ch.. Now, applying the operator F to the first equation of (3.2) and taking account of (3.2), we find. F2Bb=fbaFBa+ubFC,. s. or. (3.5) --Bb+a(Bo)A==.]f}"(.](ZtCBc+uaC)-ubuCBc.. On the other hand, we have a(Bb)= 0(A, Bb)=0(v"B.+C, Bb) =vag(B., Bb)+20(C, Bb)=vagab=vb･ Taking account of this, (3.5) becomes. --Bb+vb(waBa+RC)=(Jrb`VhC-abuC)Bc+fbauaC, or (--"6ba+vbv")Ba+2vbC=(fi'LIC;･a-ubu")Ba+JFb"uaC,. from which. f". i. i.

(5) Theory of surface of Euclidean 3-space from an alternative point of view. 19. fb'lf;a= -6ba+ubua+wbva, .f5aua= Zwb. Applying the operator F to the second equation of (3.2), we find. F2C = - ubFBb, or. (-I2+a(E9A)C=-ub(.fb"B.+ubC), ny-C+a(C)(v"Ba+RC)==-.fbaubB.-ububC, that is,. -(1-R2)C+ZzJ"Ba==-ublfb"Ba-ububC, ･). from which, by means of a(C) =R,. fBaub= -2wa, f. ubub == 1 - 22.. tts,. Applying also the operator F to the last equation of (3.2), we find. O=FA =vaFBa+RFC. Using (3.2) again, we find. va(.EvbBb+uaC)-2ubBb==O, or. (f},bv"-2ub)Bb+2wauaC=O, from which we have. fhbva-2ub=O, uawa == O. By virtue of a(A)=1 and. a(A)==g"(A,A)+(v"Ba+ZC)(vbBb+2C)=vbwb+R2, we have vbz7b=1-R2.. Thus summing up, we have. fBwa=:-6ba+ubua+vbwa, (3.6). ur:fLr==Rzib, wr.7(5r= -Rub,. f;aur :-2va, f;avr :zua, uru' =wrwr=1--R2, urzi' =O.. Now, substituting the first equation of (3.2) into the third equation (2.8), we have. 0(FBb, FBa)==0(Bb, Ba)-a(Bb)'a(Ba), from which 0(.fbdBa+ubC,.f}teBe+uaC)=:gba'-vb2L,a,. if,. l. or. (3.7) Y7bdLIF}vCgdc==gba"""-ubua-vbva.. aL. If we put. (3.8) f}b= fl,'g,b,. we find, from the first equation of (3.6). .1(b'Lf;-a=-gba+UbUa+fVbWa and by (3.7) and (3.8) j(b `7bd = gba ' ubua - vbVa.. From these two equations, we find. i. of.

(6) 20 LSATo and M. MAEDA (3･9) .fl)'(.f;･a+.f}tr) =O. Transvecting (3.9) with jF}b and taking account of the first equation of (3.6),. we find (-6cr+ucu'+vc?y')(jC;･a+.](}tr) =O. from which. (3･10) j(La+j`}tc=O,. because of the second and third equations of (3.6). Thus the tensor .1(}b defined by (3.8) i's skew-symmetric.. The structure defined on S by such a set (.1(] g, u, w, Z) of a tensor field f. of type (1, 1), a Riemannian metric g, two 1-forms u and w and a function 2 satisfying (3.6) and (3.7) is called an (f g, u, v, 2)-structure induced on S. ". l. ([4, 5]).. itL.. Example 1.. Let x==f(u), z=g(u), ueI be any curve which is parametrized by the arc length and the domain of definition I is any open interval of real numbers including zero. We define a surface of revolution in E3 with generating curve (7`k(u), g(u)) by X(u, w) =(f(u)cosz,, .f(u)sinv, g(u)). uE!L OSvS2n. By the regurality of the surface we may assume 7C<u)>O on L Then, we have Bi= (f'cos v, f' sin v, g'),. B2==(-fsinv, fcosv, O), Bn == (f" cos w, f" sin v, g"),. Bi2=B2i :(-f'sinw, f'cosv, O), B22 :(--fbosw, -fsinv, O),. where f' =dfldu, g'= dg/du, Bb==0X/Oub, Bcb= OBb/OuC, =OB,/Oub,. u- 1u, u2 =vt from which we have C == Bi × B2/(IBi['IB2 D =(-g'cosv, - g'sin w, f').. The first and second fundamental quantities of X are. (gcb)==[5 fi(Z),]･(g'a):[5 i/Lf?.)2]･. k ,. }. i. and. (h,,):[f'g"-6"g' .f<￡)g,(.)],(h,")=[f'g"o--f"g' g,(.)O/f<.)]. respectively. . Next, we consider (.7C; g, u, v, R)-structure of X. From the first equation of (3.4), we find.

(7) TheoryofsurfacesofEuclidean3-spacefromanalternativepointofview 21 (B"i)-['-'8,P.Sg,,. gl.'8.inZii %']. Taking account of the equations of (3.3), we find the following series of equations, For a sta'rt from the first equation of (3.3),. .ICIi=:B,dF"B,i==B,3F,iBi,+BiF,3Bi, ==g"(-ri1)'f'cosv+f'cosve1･g'==O, by similar computation .1(>i= 'fo'sin v, 7`}2= (g'sin z,)Lfl fi2=O,. 7. that is,. (i) (.fsa)=[21 'fifii]=[...fo9,i.. (g'Si"oV)/f],. .1. (.f}b)==(.fi'g,b)=[..-fo9,i.. fo'8inV].. From the second equation of (3.3). ui==BidL℃i==BiiFi3C3+Bi3F3iCi ==f'cosv･1･f'+g'･(-1)･(-g'cosw)=cosw where we have used (ft)2+(g')2==1, similarly, u2 =-ff'sinv. thatis, '. (ub)=(ub u2)==(cosv, '-ffisinv), (2) (eca)=:(u,g,")==(-cosw, ff' sinv)e By the third and fourth equations of (3.3), we find (27b) =(f' sin v, fcos v),. (3)(w"). == (f' sin ?y, g'cos wlf),. and. (4) R=-g'sinz,. Finally, making use of (1),""(4), we see that (3.6) and (3,7) hold good and then we find (jC; g, u, z,, R)-structure of surface of revolution.. N.B. (i) Making use of the quaternion structure of E` and by analogous method mentioned above, we can get a so-called normal almost contact metric. f i". 3-structure on E3 ([2]).. (ii) Since the surface under consideration is orientable, it is common knowledge that the surface admits a structure of Riemann surface i.e., the surface is a one dimensional complex manifold.. g4. Differential equations which f, g, u,v and Z satisfy. We denote by (cab) the Christoffel symbols formed with gcb and Vb the operator of covariant differentiation along S with respects to (cab]. Then the equation of Gauss of S is.

(8) ' 22 I, STAo and M. MAEDAi･･ '. (i41A.i,)e h,,=h,, are compo7nCeA s08i7btlliiCs"ebc)oftk =fhuCA[li5mentai tensor with respect to. the normal C. The equation of Weingarten is. (4･2) 7cC ='--hc"Ba,. where hc"= h,.gra.. Now applying the operator 7, of covariant differentiation to the first equation. of (3.2) and taking account of V,F==O and (4.1), (4.2), we find. F7cBb=(V,V(5")Ba+jCLa7,B.+(VcubC)+ub7cC, or. ,F7icbC=(V(z.IFia)B.+B,uCiahcaC+(7cub)C-"ubhc"Ba,. t. 1. 1 'k. putting in order,. -hcbecaB.=(7,z.7Cba--ubh,a)B.+(h,`if5"+Vcub)C,. from which Vqfba== -h,bua+h,aub, Vcub :--hc.fb". Finally, applying the operator 7c to the third equation of (3.2) and taking. account of 7cA =O, we find ･ O=(7,wb)Bb+vb7cBb+(7cR)C+R7cC, or. O=(7cv')Bb+wbhcbC+(VcR)C-ZhcbBb,. from which l7cvb= Rhcb, 7cR= -hcbwb.. Thus, summing up, we have ' (4.3). V,tfba= -h,bua+h,aub, 7cub = k-hcrLfb', 7evb= Rhcb, J7cl= -hcrvr.. Employing the Ricci identities,. 7d7cBb--7cVdBb=-KdcbaBa and. 7d7cC-7c7dC=O, where the Kd,ba are the components of the curvature tensor of g,b, we obtain the equation of Gauss,. Kd,ba=hdah,b-h,ahdb,. (4.4) Kdcba==hdahcb-hcahdb,. and those of Codazzi,. (4.5) 7dhcb --"7chdb =O･ g5. Some lemmas on surfaces. In this section we shall prove some lemmas on surface with (f g, u, v, 2)structure for later use.. 't". l. 1. i '.

(9) TheoryofsurfacesofEuclidean3-spacefromanalternativepointofview 23. We now compute. (s.D, s,ba=Mba+(7,ub-vbu,)ua+(7,vb-7bv,)va where ,zNhba=fir7.fia-firv,,f,a-(7,fir-7,ptr)Aa is the Nijenhuis tensor formed with .fb". When the tensor Scb" vanishes identitically, the (f g, u, v, 2)-structure is said to be normal ([6]). Substitutingi (4.3). into (5.1), we obtain. } inl. (s.2) s,,a == (f}rh,a-h,71.f;a)u,-(fbrh.a-h,7a)u,, Then we have Lemma 5.1. in order that the induced (JFI g, u, v, R)-structure on a sur:7Case ojC E3 be normal it is necessary and sufiicient that f ccmmutes zvith h.. Proof. See K. Yano and M. Okumura's paper [6] in which they are considered in a Sasakian manifold. In our case its proof is entirely same as their one.. Now, it is well-known that. (5.3) Rd,b.==K(gdag,b-gdbgca). where K is Gauss curvature (total curvature). Differentiating covariantly the second equation of (4.3) and taking account of the first equation of (4.3), we. have 7dVcub=-'(7dhcr).fbr+hc,hdbu'---h,.hdrub.. Empolying the formula of Ricci and (4.5), we have -Rdcbaua=7dVcub-Vc7dub=(hc.hdb-hd,hcb)u', from which, substituting (5.3) into the above equation, we have. (5.4) K(udgcb -'" u,gdb)=hd,u'hcb '- hcru'hdb. Similary, differentiating covariantly the third equation of (4.3) and taking account. of the last equation of (4.3), we have ･ (5.5) K(vdgcb "- v,gdb) =hd.v'h,b nt h,.`v'hdb. Transvecting (5.4) with ud or vd and using the fourth equation, we have (5.6). K((1-22)gcb'-ucub)=.llheb"-m'hcsuShbru', Kucvb = -- Qhcb + hcsuShbrv',. where we have put P==hcbuCub, Q=hcbuCvb. Similarly, transvecting (5.5) with va, we have {t. (5.7) K((1-R2)g,b-v,vb) == Rh,b-h,swShb,vr, where we have put R = hcbvCvb.. Transvecting the first equation of (5.6) with vC and the second one of (5.6) with vb, we have (1 ･-- R2)Krvb == - (?hb.u' + .l 7ibrv',. (5･8) (1-22)Klu,=Rhb,u'-Qhbrv',. from which, transvecting the first equation of (5.8) with vb, we have. (5.9) (1-22)2Kr=PR-Q2.. If PR--Q2#O or if (1--22)2K)FO, (5.8) and (5.9) give us the following. i.

(10) 24 I. SATo and M. MAEDA hb,u' =Pub +qwb, hbrv' =gub+rvb,. (5.10). where we have put p=P/(1-R2), g= Q/(1-Z2), r=R/(1--R2). By virtue of the equation of Gauss (4.4) and (5.3), we find K(gdagcb--gdbgca)=hdahcb--hdbhca,. from which, transvecting the above equation with fd'=:gdVb and noticing the skew-symmetricness of .fktb, we find. (5.11) Kl7(}te=hadhcVkeb,. 'i. or. (5.12) Kfia=h,WSh,a.. ("-. g6. Speeial caseg. (I). Case in which the vector A is always tangent to surface. In that case, from the third equation of (3.2), the structure vector A of almost contact metric structure (1!7, g, A) is tangent to surface S if and only if Z= O. Then, by vir'tue. of the Ricci identity with respect to wb and 7cvb= O which follows from the third equation of (4.3), we find. Rdcbava=O. Taking account of (5.3), the last equation becomes. K(gaagcbdgdbgea)v"==O from which va being unit vector field, we find. K=O. Lemma 6.1 ([3]). A complete surfoce of Gaussina c"rwature O in E3 is a aylinder.. By Lemma 6.1 we have Proposition 6.2. Su21bose that the structure wector A is tangent to complete orientable sur:7Cace everywhere, then the surfoce is a cylinder.. Now, from (3.6), we have, because of 2 :O .1`}tbub==O and .f},bvb=O,. from which we have jFh,=O.. f'"'/. s. Conversely, suppose that .7C}tb vanishes identically, then we see easily, by the first equation of (3.6) that 2=O holds.. Summing up the conclusions obtained above, we have Proposition 6.3. The following three conditions are eguivalent to each other: (i) the structure vector A is tangent to surface, (ii) Z=O ower the zvhole sui:flace, (iii) JFb"=O over the whole surfoce, (ll) Case in which A is always perpendicular to surface. In this case, by the third equation of (3.2), we have. .A.

(11) TheoryofsurfacesofEuclidean3-spacefromanalternativepointofview 25. (6.1.) fva=O. and. (6"2) l2 == le. Also from the fourth equation of (3.6) and (6.2), we have. (6,3) ua =o.. From the first equation of (3,6), (3.7), (6.1) and (6.3), it follows that. ･LfbVa= -a,a and fiSLICbrgsr =gcb.. 'g,. t /t. Fl. These two equations show that (f; g) define an almost Hermitian structure on surface. Moreover from the third equation of (4.3) and (6.2), we have. h,b =O, ･ which shows that surface is a portion of a plane or a plane.. Thus, we have Proposition 6.4. Sumpose that the structure vector A is perpendicular to complete orientable suryCace, then the suilfrace is a plane.. Example 2. In the surface of revolution of example 1, we consider the case where R vanishes identically. In that case we have g'= O by (4) and consequently g is constant. Then, the generating curve of surface of revolution is parallel to x-axis. Therefore, surface is a portion of a plane.. Next, if in example 1, the function 1--12 vanishes everywhere on surface, functions f and g both become a conatant. Then the generating curve is reduced to a point on!y and consequently, surface of revolution becomes a circle which is exceptional case.. The axis of revolution of surface in example 1 was z-axis. But here, we take y-axis as the axis of revolution. Let. x =j(1(u), or =g(u). be any curve which is parametrized by the arc length, A surface of revolution in E3 with generating curve (.f(u), g(u)) by X =(J`<u)cos w, g(u), j`<u)sin z,).. Then the unit vector C is given by C== (g'cos?y, -f', g'sin w). If the function 2 vanishes identically, the function f must be constant, be･". 1. cause of R=AiC`=--f'=O. Then the generating curve is reduced to a straight line parallel to y-axis and consequently, surface of revolution is a right circular cylinder with ov-axis as the axis of revolution.. .F,.. g7. Surfaces with normaE (f, g, es, v, R)-structures.. Suppose that (.f] g, u, v, R)-structure on surface S is normal. Then, by means of Lemma 5.1, f and h commute, i.e.. (7.1) firh,a--h,wa=:o. which is equivalent to.

(12) 26 I. SATo and M. MAEDA (7.2) hc..tir+hbr,f}'=O that is, hc.L7C5' is skew-symmetric with respect to the lower indices c and b. Moreover, in section 7 and 8, assume that the function 2(1-22) dose not vanish almost everywhere on S. Proposition 7.1. Let S be a surface oLIC E3 such that (7.1) is satisheld and junction Z(1-R2) is almost everorwhere non-zero in S. Then we have. (7.4) hbau"==Pub, hba22"==Pwb. Proof. Transvecting (7.2) with uCub, we find. 2(hcbueub+ hcbuCub) = O,. from which. ,s 1･. (7.5) Q= hbcuCvb=O.. Transvecting (7.2) with uCvb, we obtain. l .M. -.. Z(hcbuCubmhcbwCwb)=O,. from which. (7.6) h,bueub=h,bwCvb, i.e. P==R. By (5.10), (7.5) and (7,6), we have (7.4). Consequently, Proposition is proved.. Proposition 7.2, Let be (1-22) ¥･O almost ez,e7zywhere in normal S, then S is of non-negntive curvature. Proof. It follows easily that, taking account of (5.9), (7,5) and (7.6), we can verify the assertion.. Next', differentiating the second equation of (7.4) covariantly, we find, using (4.3),. (7chb.)v"+2hc"hab=(l7cp)wb+iphcb, from which, (l7 ,p)vb == (V bLlt))Wc･. Transvecting this with vC, we have. (7.7) 7bP=avb,. where a is some function in S. From (7.7), it follows that. (7.8) . (Vb,zb)ub==O. Transvecting (7.2) with JFhb and using (7.4), we have (7.9) h,b=h,.]ISIfbr+p(ucub+vcwb), and from the second equation of (4.3) and (7.2), we see that ua is a Killing. vector field on S. '. '1. Differentiating the first equation of (7.4) covariantly, we find, by (4.3),. (7.10) (7chba)ua+hcShb'LAr=(7cp)ub-phcr7`5'. Subtracting the equation interchanging the indices b and c in (7.10) and using (4.5), we find. (7.11) 2hcShbXr==(7cP)ub'in(7bP)ue-2Phcr.fir･ Transvecting this with uC, we find, using (7.10),. 7,p ==O. from which we have. (7.12) p=constant.. i.

(13) TheoryofsurfacesofEuclidean3-spacefromanalternativepointofview 27. Thus (7.11) becomes hcShbXr"= -Phcr7Cbr, or, using (7.2),. h,Sh,.fLr= ph,,Lfbr, from which, transvecting this with .fltb, we find. (7･13) hbrhar=phba, or hb'hr"=phb". The equation (7.4) shows that vector fields ua and w" are eigenvectors of hb". and the constant p is the corresponding eigenvalue of which the multiplicity is ･i. j. two. Then we have haa=2p and consequently the mean curvature is a constant too. Thus we have Proposition 7.3. Under the same assumption as stated in Proposition 7.1,. then the mean curz2ature of su7VCace is a constant. . We can prove ' Proposition 7.4. Under the same assumption as stated in Proposition 7.1,. wehave ･. (7.14) . 7dh,b==O.. Proof. Differentiating (7.13) covariantly, we find. (7.15) (7chbr)ha'+hbr7char =P7chba, from which, being symmetric with respect to c and b, we have (7char)hb'-(7bhar)hcr=O, and, interchanging the indices c and a,. (7･16) (7ahcr)hb'-(7bhcr)ha'=O. Adding (7.15) and (7.16), we find. (7.17) 2(7cha')hrb =PVchba･. Transvecting (7.17) with hdb and using (7.13), if p¥O, we have (Vcha')hrd = O･. Thus, (7.17) implies that. (7.18) 7chba == O･. On the other hand, from (7.13), we have h,bhCb==phbb. Thus, if p =O, we have hcb =O and hence 7dhcb=O. This completes the proof of the proposition.. Next, taking account of (5,9), we have. (7.19) K=p2.. As p is a constant, K=O, if p=O. Consequently, by Lemma 6.1 if our surface +t. l. is complete, it is a cylinder. Thus we have Proposition 7.5. Let S be a complete orientable surface such that (7.1) is. satisy6ed and junction R(1-R2) is almost ewerorwhere non-2ero in S. Then S is a aylinder when p= O. Next, from (7.19) we see that if p ¥O, the Gaussian curvature K is a positive COnstant p2.. Lemma 7.6 (Hopf and Rinow). A complete sur:7`Tace whose Gaussian cur℃ature is everywhere greater than or equal to a positive constant p2 is closed and has a diameter note exceeding n/lpl. Lemma 7.7 (Liebmann). A closed surface whose Gaussian cur27ature is con-.

(14) 28 !. SA'ro and M, MAEDA stant is a sphae.. By two lemmas above we have Proposition 7.8. Let S be a complete orientable surLRzce of .E3 such that' (7,1) is satistfied an`l junction 2(1--22) is almost everyzvhere non-2ero in S. Then S. is a sphere when p#O. g8. Surfaees with anti-normal (f, g, ee, v, A)-struce"xe. In this section, we assume that f and h anti-commutative, i.e.,. (8.tt) fi}'h."+hbVa==O,. whieh is equivalent to. tr. t'. (8.2) hbrfh' -'m har.fbr == O,. l '. 4･. that is, hb,jtC},' is symmetric. (8.2) shows by the second equation of (4.3) that the vector field u is a harmonic one.. Proposition 8.1. Let S be a surface of E3 such that (8.1) is satisy7ed and the junction R(1-otR2) is almost everyzvhere non-2ero in S. Then we hawe. (8.3) p+r=O,. Proof, By transve￠tion (8.2) with ubwa, we find. 1(hebecCub+hcbvCvb)=O, from which we have (8,3). Proposition 8.2. in an orientable surface S wz'th (.IC g, u, v, R)-structecre of. E3 such that R(1-R2) is almost eweryxvhere non-zero, (8,1) and the follozving (8.4) are eguiwalent to each other,. (s.4) s,,a :2(u,7,ua-u,7,ua).. Proof. suppose now that (8.1) is satisfied. Then we have, by (5.2) and. (8,1),. S,ba=--u,(fbrhr.-h,7Llga)+ec,(f}rh,a-h,7a). ==2(u,h,wa-u,h,va) --2(ucebua--ubecua), (bythesecondequationof(4,3)). Thus (8.1) gives (8.4). Conversely, suppose that (8.4) is satisfied. (8,4) is written as -u,(firh.a---h,77;a)+ec,(ptrh.a--h,'1fia) :2(u,hbWa-u,hbVa), or. -u,(firh.a+hb7a)+u,(rtrh,a+h,Wa)==o, from which, transvecting ub to the last equation, we find, taking account of (5.10) and (8.3),. 't. 1 'l. k. (1-R2)(f}rh,a+h,va)--o, from which f}rh,a+h,'1fia :O and' we have (8.1). Thus the Proposition is proved,. Since'the commutativity of f and h and the condition S=O are equivalent for a surface of E3 and an (.f; g, u, v, R)-structure satisfying S :O is said to be norrma1, we say that an (.f; g, u, w, 2)-structure satisfying (8.1) or (8.4) is anti-.

(15) TheoryofsurfacesofEuclidean3-spacefromanalternativepointofview 29 normal ([7]). Proposition 8.3. Let S be a surface ojC E3 such that (8.1) is satisield ana the junction 2(1-R2) is almost everywhere non-2ero in S. Then S is of nonpositive Gaussian curvature. Proof. By means of Proposition 8.1, (5.9) becomes. (1-22)2K=:-(P2+Q2), from which the proposition follows immediately. Now, transvecting (8.1) with jFb' and using (4.3), we have -<. i. -h,a+u,h,aur+v,h,awr+fish,wa=o, from which, contracting a and c and noting P+R=:O, we have -h.a+.fhsLf}ah,r=o. k. or. (8.5) h." =O. and consequently V.?yr==Rh.'==O, from which, we see that vector field w is a harmonic one. Proposition 8.4, Under the same conditions as stated in Propositions 8.3, the surfoce is a minimal sui:face. Exarnple 3. Taking account of the consequence of example 1 in g3, we shall calculate the components of .IC)Vi±of. First of all, we deal with the case where J6ii-. h,f. Suppose g'#O, then (f!)2+(g')2= 1 gives us flf"+g'g" =O or g'== --f7"/g'.. Substituting this into f'g"-f"g', it is equal to mf"/g' which will be used hereafter. fbrh,a: .JFi'h.i = (g'/f) sin v･O= O,. .1`>rh,i=-fo'sinw･(f'g"-f"g')=ff"sinv, fl'h.2 == (g'Lfl) sin w･(g'/71) == (g'/LIC)2sin w,. .7`> 'h,2=fo'sin v'O -- O,. By the same method, we also have hb'Lfia:hiX'--O, h2'LIC'=--(g')2sinv. hiX2==-(f"11f)sinv, h2V2=O.. Summing up, we find . ' "IChM]if=[(ff"+(gO,)2,i..)/:7,, ("fZf"+(8')2)Si"W] i I,. from which we see that jF;ei--hh=:O holds if and only if ff"+(g')2=O and (f')2+(g')2==1. It is not so dificult to see that the solutions of the differential. equation are f==cosu, g=sinu. Then the equation of surface of revolution is given by X== (cosucos v, cosusin v, sin u), which is nothing but the equation of a sphere. Next, similarly, we find". ' fh+]ijC=[-(.1:tr"-(Ogr)2),i..lf, (ff"--(g')2)si""] from which we see that fa+1ijC=O holds if and only if ff"-(g')2=O and (f'2) +(g')2==1. Eliminating g', we have.

(16) 30 I, SATo and M. MAEDA ffM+(f')2-1=O, or(ff')'-1=O, from which by integr' ation,. 1. 7(f2)'" ff'=u+c. and again integrating,. "=u2+2cu+d, or f=Vu2+2cu+d, where c and d denote constant of integration. Consequently, g'= ±Vi'(f')2 =:±V .2 +d2--,-.C2+d. For brevity, we may put c=O, d==a2 (a>O) unless loss of generality. Then, the generating curve is given by x= f(u) == u2 +a2,. sf. 1 '(. 1". 2Fg(u)== ±Jg" vt¥d+t.2 == ±asin h-'-lf,. or eliminating parameter u, by. x =a cos hGL. a This is the equation of catanary. Therefore, the surface of revolution is a catenoid which is only a minimal surface among surfaces of revolution, Conversely, it is not so dithcult to see that a minimal surface of revolution satisfy .fh+Zijfr= o.. References [1]. BLAiR, D.E.B. and LuDDEN, G.D.: Hypersurfaces in almost contact manifolds. T6hoku. Math. Journ. 21 (1969), 354-362.. [2]. Kuo, Y.Y.: On almost contact 3-structures, T6hoku Math. Journ. 22 (1970), 325-332.. [3]. MAssEy, W.S.: Surfaces of Gaussian curvature zero in Euclidean 3-space. T6hoku. [,4 ]. YANo, K. and OKuMuRA, M.: On (.7(; g, u, v, R)-structures. K6dai Math, Sem. Rep.. Math.Journ.14(1962),73-79. ' ･. 22 (1970), 401-423. , ･. YANo, K.: On (.L g, u, v, Z)-structures induced on a hypersurface of an odd-dimensional sphere. T6hoku Math. Journ. 23 (1971), 671m679. YANo, K. and OKuMuRA, M.: Normal (.IC; g, ec, v, 2)-structure on submanifolds of [6] 'codimeusion 2 in an even dimensional Euc]ideau spaces. K6dai Math. Sem, Rep, 23. [5]. (1971), 172--179.. [7]. e/. YANo, K. and Ki, U.H.: Manifolds with anti-normal (L g, u, v, R)-structure. K6dai Math. Sem. Rep. 25 (1973), 48-62.. IsuKE SATO 5654-418, Hino-cho, Konan-ku, Yokohama MAsAo MAEDA Department of Mathematics, Faculty of Education, Yokohama National University. l. ". .di..

(17)