I. '5‘.“ l I II I l 1|”! WM! 1 I IHIIHIHIIW t | WI (”—3 03—: ._1 I “JD->4 THE DEFFERENTIAL GECMETRY (3? As swam!“ SURFACE EN 534 :1”. ‘ °.. " r~_, ‘ .‘ fl 3: (>1: :er is pagru 01 I‘ve. A {’1 ‘Eaxaies i H eyda 1937 MSU LIBRARIES “ RETURNING MATERIALS: P1ace in book drop to remove this checkout from your record. FINES wil1 be charged if book is returned after the date stamped below. ACKHOILED KENT To Professor Vernon Guy Grove without whose suggestions, aid and encouragement this thesis would have been impossible. TIE DIFFEREZTIAL GECJETRY OF A A Thesis Submitted to the Faculty of EICHIGAH ST;TE COLLEGE AGRICULTURE AHD APPLIED 5013303 In Partial Fulfillment of the Requirements for the Degree of master of Arts by James Francis Eeyda I. Introduction...................................l II. The Defining Differential Ecuations............3 III. Integrability Conditions.......................o IV. Poner Series Expansions Fsr the Surface.......10 V. A Canonical Form of the Defining Differential ECjue.tionSOOOO.IOOO0.....00..O°_OOOOOOOOOOOOOOCO.ll+ VI. OtEler Unique :‘TOI‘Z‘flalso.....o....o.o..o.o......olg VII. Conditions for Sx.to be Innersed in Three DiznenISiOI-LSCO0.0.00.0...O...IOOOOOOOCOOOOOOOOO O \N O VIII.TLe Relation R and the Conjugate Hornel....... \N U] IX. Change of Net Upon he Surface................ \u \1 18.53.33“: I T‘Vn‘? t‘_ TTflm ‘A'Y . .& &:‘-V I \— U; Kid... The purpose of this paper will be to study a general surface in a space of four dimensions by means of an orthog- onal net upon it. he first set up a defining system of oartial differ ial ecuations. Associated with each point of the surface is a unicue plane con aining all of the normals to the surface. we define certain unique normals and pairs of normals to the surface and characterize them geometrically. For this purpose we study the sustair ins surfaces of the orthogonal projections of the given net onto certain geometrically defined spaces of three dimen— sions. We call these surfaces norms grolection surfaces.‘ A normal determines a unique normal projection surface. Among the normal projection surfaces there are two, one possess ng maximum total curvature, the other minimum total curvature. The normals determining these as rt icular pro— ‘ jection surfaces are perpendicular. They are called the * nV. G. Grove, Differential Geometry of a Certain Surface 84, Transactions of the American Lo thematical Society, Vol. 39 (1936), p. 02. hereafter referred to as Grove, Geometry. 2. t Erincigal normals. A necessary and sufficient condition that the given surface be immersed in three dimensions is deduced, followed by theorems relating to change of net upon the surface. * Grove, Geometry, p. 64. 3. II. THE DEFINING DIFFEREKTIAL EQUATICKS Let the parametric equations of the given surface Sx be xizxgun‘r) .L=|,2,3,4. We assume that the non-homoEeneous cartesian coordinates ( X., X. ,X,,Xfl> of a point Px on Sx are analytic functions of the two variables (u, v) and that it is not possible to express the x as functions of a single variable. i The square of the element of are for a curve C lying upon the given surface is given by 4. 3 1 <1) 4322a. = Ed...‘ +1FdaAr+GoLv , £=I where the first fundamental coefficients E, F, G are defined by the relations (2) 5:.th , F=Zx..xv, G=Zx5. Let the curves of the given orthOgonal net Nv be taken as the parametric curves. We will then have (3) F: quxvzo We shall call the plane containing all of the normals to 8x at x the normal plane to 8x at x. Select in the normal plane two perpendicular lines l and [u with direction cosines (X, a x3.3x! 3 x7) and (Play! ;FJ,F¥) IeSPeCtivel-YO It follows that the functions A and ’LL satisfy the equations Zxx“=o> th‘fio) ZHXE=O>ZFXV=OJ (4) 273'“ ,_ Yuk-I, Zer-o. - - / I I! u The second fundamental coefficients D., 91’ D| , D1, D‘, D; are defined by 19‘: XXX.“ , ‘D:=2 XX“, D.”=Z 7&va (5) 19,; stm, D,‘_=prw, 33:53)“... From equations (2), (3),(4), (5), we obtain the following relationships (80 Zthé—Ev. vaxuwéGt. Z xxw= D.’ . EN...— 17;, (b) 2x..X....=-,£Eu. vaxm-«gev, 2).x....=D. , 2 ax...— 9:. (C) EXAM—lie... EXAM-Em, Zxxw-wffi Xi‘xvv‘ Dan, (.6) «Dingo, Eula-D.) 21m)”.- -1>.') z ,4 he -2 >‘Pu.’ -A,, (e) Ellyn), Z‘xv her-Dr, Xx. Kw:- .49", Zrkve‘ ~2Apv=-B., (f) Erma, 2x. avatars—a. 22M... -:M.= -A‘, (a) 23 r rm, waw —-:D.“, Saw-v.2, mm. - mm -B.- 5; The relationships (6) constitute seven sets of four linear equations each, in the unknowns X“, X“, XW , X“ , xv, er and Hw respectively. The determinant of the coeffi- cients in each of the seven sets is the same, namely XML 3 xzu, x3“, xii-u X“! ) XIV J x3V : x4v (7) H = )‘u ) XI.) )‘32 X, P”? #1) y’af‘fi From equations (2), (4), (7), we may write wabxuu x3“: x4“- Xmfitw, xn Pi E O O O H... *m mum». . G 00 (8) .- 1.,11,1 X4. = =EG#O 3’ “afl‘awnlni‘: O 0 lo 4 PI 1 PL 3 P33 f“! X4“ ,va , 1.5”“! O O 0' Since, for a real surface with real values for u, v, the determinant of the coefficients H is not zero, we may solve uniquely for the unknowns X“, X“, X“, A“ , ,1, , lug. , icy to obtain the system of defining differential equations for the surface Sx’ namely, qu=otxu+va+mx+9ly~ ) Kn," mlxu++|xv+AtP a (9) Xuv=axu+bxv+°:7\*9£#’ Av: 7.,x“+ “Ava-8,}; va=yxu+8xv+n:‘).+‘b:fi, pq=mtxu+lhxv+ A1)” ’4‘: $1xq+nlxv+8115 wherein d=i%>@=‘i%:D.=Z7~qu.91=zl*xw’ af'JiEE-x’ b=é 9a}: % ”(=23ka 3 D;=2wa: u (10) 1:419:22, psi—gr, =2“... Jar-2pm, u 0| D. n _ i in D r e’MP-e‘” m- a" “*“i‘ ' I I I _D. .._ D _ _‘D _ D' “fr: "3-6- )“f’t‘ 6) 3H" E!- w 31"“?- The equations (9) represent twenty-eight equations since x itself represents (x|, xt, x3, x9). The derivation of the equation: XHN= at xm+ pxw+ D.X‘+‘Dz H. is given below, the other twenty-seven being derived similarly. Referring to (7), let us multiply the first column of H by xn‘, and add to its elements the products of the elements of the second, third, and fourth columns by x1 , x3“, and xyu respectively. We obtain E , xtu, x3e») xfill. (11) H4“ = 0, X“. XIV: xw o, l“ 1 0, FL. ’13: (‘4 Hence By precisely the same process it is easily shown that (13) x1! *3, A“ =‘H x—é! ) xI-Vo X3V) xvv = Hx. 3 XIN) X3V|x¥v PL: PS: ”'I xzv a xiv 3 XQV x1) 133A? Pa.» F); f“! Kim, XII.) x95 Pt: PM [‘4 3: )‘V 2E. :H. x&&) XJK ‘I x'“~ X... E xzv ) va 7 X4" x1: 3 x3 ‘ 1’ FL ) P3 ) ”V X“ 3 x30»! X“ 1;. 1,, X‘ Formally solving equations (6b) for)hw , we obtain (14) ‘1: tin.) x‘u axlvux'nt l baa-'5') x‘v’ Xav’ x“v ‘D. J x" A): x, )3. ’ PM (‘33 P? *Hb, x1.“ 3 XIII.) XQ‘ xw. X“ , "Jv. xw 1: t X“ X), 1’ P! : PM Ha: F9 xiv 3 va : x'fll =‘Hf‘w XML J XJK ) x‘Iw l l X 1 x x t s 3 9 TE“ 1 ’ +Ji Ev ‘ ’ 1| 9 P10 F! a P" ”I. 3 F3 ) #3 X‘K) xaugx‘ib x‘“’ X,“’X~u 'D" X“, X" : XW ~31 xu. xiv. xw + [‘33 Flay”! x1, X., X; + H 8. Substituting from (12) and (13) into the right member of (1“). sives E -¢-x...g (15) xmu" 1?. .X\V+P,X.+P1H. “IL“ If the notations in (10) be used equation (15) may be written in the form quz- (XXN1- px‘v+‘n.l,+v1y1 III. INTICRAEILITY CCZIDITICIJS The coefficients in system (9) are not independent, but satisfy certain integrability conditions. he must have, for example (a) xuuv = giu J (16) (b) xvvu '3 xuvv With the aid of equations (9), (16) may be written (a) X...“ = («Vtau’r {31+ 1'.v.3,+19,5¢3,,)x.,L + («b +Fv+ f8 +a.n.+m'\.) "v +(«9.'+p9."+nw+el‘p,)x + (dD;-\-pv: +e‘v,+o,,v) I" , (b) qu = (audaddr ab+v:m.+v"_m‘\ Xu+ (4P+L“+b" +v.‘+'+ 7.17:.)Xv (17) I (a P.+b'9'.+1):w+ A2991 * (“pa *5”; “I ”in. +5.9” 1“ ’ (C) XVVU. (1“4' “14’“8 +‘p:.m.+ 9:“3)x“+ (P1+8.~+LS +,'~+.* 92") Xv +(1?.+ SDHvL-t Alba?) X + (int-\- 8 a; 1-7:“ + Ant") p. (d) X... (a, + o."+ 51 + 3.1”"; 1‘) x, + (ab-t bv-H-s +v\'n.+u"mg) xv + (a v,'+ Ln,"+ v,'v+e,v;) x + (an; 1- be: + 19;”... 3'19")“ 9. Equating coefficients of'XM Xv) ).,andf&,first in (a) and (b), then in (c) and (d) we get: H 0" |‘ (a) G..(-E5+ +C-‘G—“)+E (5+Ex)..2 ”(EVV+G1111+4(3 33+9191-‘b1-91), t v" _G . . (18) (b) 35 we'- +€>=ivé< -G“,1)+T>..+B o1 ~A,1:,1-1>,11 , 1 : c.u _ - (C) iEv(?‘ '53): to '(J- G>+9V+:Hv Adv ‘p' , (d) as. .(-§~ .3 Wu...“ %")D:-1>'.v~31°; . (e) t3 “C?" ’"=)v111+A,1‘i-'Ev-r év)v;- v;,-B,v.' . In addition to equation (16), there is also the condition (19) xU-V = XV“ 0 Using equations (9), we may write (19) as follows: X v=(“1w+am.++,‘7’+A.g,)Xu, + (+,v+w.+ 1.: HM.) X. 0 I J In (20) +(v1m-rv‘f +Ac,)}\+ (on +9,+11-Aw)r~, )Lw=(1m+d3,+n.,a+8 m.)X +(mlu*P7|+“|M+1)xv +(o 3.+b:n.+A 3.)} +05 3.4.91.1. +3.“)u . The only new integrability condition obtained is found by equating coefficients offs, namely, (21) Aw‘Bw“ (E an”- "G ‘6)D ' Also from relations (6 d,e,f,g) is obtained the set of conditions A‘+A1= BI+BZ=O o .l (22) Equations (18), (21), (22) constitute the integrability conditions for the system of equations (9). 10. IV. POKER SERIES EXPANSICNS FOR THE STTRFACE If we use the tangent lines to Cu and to CV, and the lines K.andfl for the axes of a local system of reference, we find that the coordinates of a point 1 with general coordinates (%.,%. ,1; ,34) will have local coordinates (E. , E, , E, , R.) defined by the expression (23) t3=x+i£j+ ‘ygvarsnwm- Let x be a point on the surface 8:: with curvilinear coordinates (n+au,v+4v) where (u,v) are the curvilinear coordinates of x. The coordinates of 1 are of the form (24) ‘1: x + an‘A‘t XVAV + to“, Au‘+ 2.x“Au4v-4r Kw AV‘)+--~ , If use be made of equations (9), (24) may be written t}: x + {nu+-,L_(¢Au‘+ 10.1w. 4v + i4v”)+- u} xm— {4v +-;-_(9Au“+zbuav + San/9*”) "v (25) +{Ji(D.Au‘+1D"AuAV +0,"AV")+---) l +{Ji(9,1Au"1’1°;4“AV +1’:A"‘)+") I" ‘ Hence the local coordinates of 1 are ‘5": E [Au+&£“4“t+ laAuAv-f 'YAV‘)+”‘] (26> 31=JE)_AV+-‘;(pau‘+iuum+Sav‘hm] , 33: J9: [19, Au" + an: AuAV + D." “I‘ll-inn“ , §4=t[D1AULI'+19LAuAV+D£Av‘] 4..-... . 11. We may express 3; and E, each.as power series in E and £1 obtaining D. ‘L D. ‘3" 1 ...._ E3 $1.8— ;‘+_J—EE g‘;3+‘i-él'g,_‘r 7 .i E: ‘ ‘9‘ '9 2.. E ¥'+fé‘:G-E‘;a+i— '6'? Equations (27) may be interpreted as follows. The first (27) r 4 £2...- ..... equation and Fq::o are the equations of the sustaining surface of the orthogonal projection of the given net onto the S3 determined by the tangent plane and the normal X.. A similar statement holds for the second equation with §3=.) , (Amr Bm‘ x“+(Af1~sf..)xv + (Au-'ALB))\ 'i' (AA? 3“) I‘- > R1 x11+ {91m +0.1; max/Chaim) + (“pang-151+ AP.) , = {.1 x11 +1T1xv+(-AA.s+aau-tAAi—ABA.)3\ +(A‘A.-AB.1+A..B 1,119 )1 , = fi1xu+ {,xv + {-ABLA.+A.)+41(A‘+3‘)..§1 +{A.(A‘+n‘)-Ao,+na} P . Referring to (22), (38), we note that A1+Al:o (A‘+B‘)11=0 Hence (40) becomes ) (41) i =~Tn1Xw+/i;,xv+(A1-AB..+A..B)P u. Therefore "7““ ANNE“; ’ 1'! = A‘h-Bf‘ I 7“ = Ac‘ABu+AwB‘ The transformed coefficients 51, h .... may be found in l, a similar manner. We list here all of the transformed coefficients for convenience of reference. &=N,F={3,a=a, b=b.:>’7-7”3:5 it: A‘D‘B‘P‘ ) 5;: "|+A‘Da. ) 5:: 531‘3"; ! it = B‘D.'+AD; : I If" Av:""a.' -:=”:.+AD;: fi,=Am,-ha_ : $1=A+‘-Bf;, (42) fit=BM.+A.M;; $t=3+g+Afti it‘Aju-Bil. ’ fi| ‘ A"|-B“1 " i“ Big-A3,, mason-rmv K. = A.+A.~'B «A15u , §.:B,+AvB-ABV,A._=A,1+AB“-Au3 , §1=st+A3v—AVB The total curvature Kl of the surface of normal projection S X is determined by the expression (Ln) as R. = (AD.-BD.)(AD."-BD£)-(ADJ-“0‘ / This surface has maximum or minimum total curvature if cm __L - (44) 41A ‘ 0 Now I 3| u 'I _ "_ " B _ '- ' '_ ' d . :IA— 2‘ E1;{?+ ’-‘-¥>-’-<”I"”U<’: * is“) di From (45) we see that -:~’ will be zero if (’46) (AD,_+BD.)(Av1‘-BD,1')+(AD.-sp,)(AD,'_'+ 331')—z(Av.'-sv;)(Av,’-+ 3191) = o Rewriting (#6), it follows that the surface of normal pro- jection 8'1 has maximum or minimum total curvature if and only if A and B satisfy the quadratic equation (47) LA‘+1MAa-Ls‘=o , wherein 'I- u . _ u to .‘_~ (“-3) L: ‘D‘ZD: +20. ‘Da-Z'91'Dt , M " 319. "Da 91 +132. DI 17. Equation (47) determines two values for-%-, each of which fixes a particular normal line a. . Since the product of these two values is minus one the two normals determined by them are perpendicular. These normals are called the principal normals* of 8x at x, and the normal projection surfaces determined by the principal normals, the principal normal projection surfaces. The equation corresponding to (47) for the surface S p is (49) LA‘—2MAs-LB‘=o. Since the roots of (49) as a quadratic in 134 are the negatives of the roots of (47), and since their product is minus one, the normals determined by (49) coincide with the normals determined by (47). We may sum up our results with the following statement: Through the point'x there exist two normals with the property that the normal pgojection surfaces determined py them have maximum and minimum total curvatures. Let us suppose that the transformation (38) with values of A and B determined by (47) has been effected on the system of equations (9). Then using (42) we can easily verify that —--u -..- —v L=D1Dz+ZDD‘1‘Di; is identically zero. ' Grove, Geometry, p. 64. 18. The resulting differential equations assume a canonical form in which I! u (50) D.D1 +‘D‘Dz : 19.13; For this form 1 and {L are the principal normals. VI. OTHER UNIQUE NCRKALS The general coordinates of the principal centers of normal curvature of the surface of normal projection 31 determined by 1 = A x - B m are (51) X+Rmi , 'mzl,z. If i be replaced byAA-Bfw , equation (51) may be written (52) X+A§merRmIA . m=\,i. The local coordinates of the points (52) are (53) §.=°. Aw. €,=Afi.. From (53) we obtain (54) E" E:+§;+§:=A‘§;+B‘R;=(A‘+B")-R;=—R,: , M=|,2. 19. Hence "‘ a. a. (55) Rm: ‘J;3+£4 , M=l,1. It follows immediately from (53) and (55) that / ‘1 g’... " t 1. (9°) Am ’ B" , were, J§;+S; Referring to equation (33) we see that the principal radii of normal curvature Rh at the point Px for the surface 83. are given as roots of <57) (ear-arm: - (2-5? + a5.) a... + e c. = o . Using (42), eouation (57) becomes (5‘3){(A171—BD‘XAD."-BD£)-(AD:~BD,1)‘}§;—{E(Av:'-‘Bv£)+G(Av.-Bv,)& RN-t E G = o . We may obtain the locus of the centers of principal normal curvature for all normal projection surfaces 83 by eliminating A, B, and Rm from (58) by using (56). As the equation of our locus we find S|=°) 3$=02 (-9) (aw-v?) :; + (v."1>.+t°.vl-n.'91)§, t. + t-cnrmmu. + as = 0 Exactly the same equation (59) is obtained when the surface S- is considered. Hence we may state that the locus of the p. _..__ centers of principal normal curvature for all normal projection surfaces is a conic in the normal plane with equation given by (59). We shall call this conic the central conic of 3x at x. 20. 9) may be written in the form W H 0 WY '0 u 0 A _u M H 4. ”v M i s m >< 51 W? U *- 3'“; a" '3 I I A '8 an + va- «7 ‘I v P II 0 If our given orthogonal net is a conjugate net‘, that is if I 0 201-31:0 , then (60) becomes ('81) 5:0, 51:0 1 (91:34-91 £.,,‘E)(171"£31'D:E4“G): o . J | Hence if Sx sustains an orthOgonal conjugate net, the locus pf1the principal centers pf normal curvature for all normal projection surfaces isms pair pf straight lines lg the normal olane" whose equations are E|=O, Ez=O , D,E,+‘D,E,:E, (62) c . u h Z‘:O, !‘=° , Jag-t1), E, The form (60) of the equation of the central conic shows clearly that the lines (62) are tangent to the conic at their points of intersection with the line I l (63) Enzo: §L=° D Digz+pt {1:0, The line (63) lies, of ourse, in the normal plane and passes '3 through the point Pr’ and is thus a normal to the surface S ~ x at the point Px- The normal line perpendicular to (63) has ‘ E. P. Lane, Projective Differential Geometgy of Curves and Surfaces, Chicago, University of Chicago Press,_1952, p. 122. g; *‘ Grove, Geoaetry, p. 64. for its equation (61+) E20) 31:0, Dtga-D"§,’=O . The normal line (64) passes through the point with local R . ‘ l o 7 . coordinates (O,O,D,,Dl) and the pcint (0,0,0,0), or, spearing in terms of general coordinates, through the points x... ‘ I (0')) X+ ”D. A+93F > X ° The direction cosines of the normal (64) are therefore preportional to the differences of the coordinates (65), namely 0 l ('66) D. I + '9‘ ,1. Referring to the transformation (33) and equations (42), we recall that — I g -— , l (57) ID| = AD.'~BP,, , 1:: BD,+AD,_ . If we should. choose % = :fl , then from (67) and (38) we see that r - —l - —I ‘- ‘ - 31- D. (05) D.-o, D,_=«(-p:+u; ,A-——————- , B: \)1:,‘"+ of Jpg‘w 7;” The transformation (38) then becomes 3x = 17,11 ‘2‘ F4 1 «(bf-kw." D,‘ x 4.13;}; P‘Ff—I v,+-v‘ , A‘+B‘=\. (\3 (‘J 0 From (64), (66), (69) we readily observe that the normal line, with direction cosines proportional to D" A + D; H. , is the normal ‘5. for ,5. given by (69). It is easily shown that the normal line (63) has the direction cosines i.. Since, from (67) it is plain that not both D. and.5; can be made zero simultaneously, and since EG=O ihplies that the second fundamental coefficient 5; for the normal pro- jection surface S? is zero, we have the following character- ization for the normal line aiven by (63): The normal ling .Léll is unicuely determined as the only line through Px in the normal plane which has ppon its normal prgiection sur— £§£§_gn orthogonal coniugate net. Hence the normal line (61+) with direction cosines proportional to ‘D,'>.+D,_'y. is characterized uniquely in that it is perpendicular to (63). We shall call it the coniugate normal 59 SK 2; g. We also note that the pole of the unique normal line (63) with respect to the conic (59) is the point of intersection of the lines (62). Let us define a geodesic as an extremal curve of the integral (70) 5(Ew"+Gv"’)/1€u > u'= % ’ Vt: Si . Ecuation (70) is easily changed to the form (71) 5(E+Gv")y‘da , V'=j—i 23. Euler's eeuation for the extremals of the integral in (71) is ‘1 - SQuVV')—( Tr")"‘ (75?) 4:5'7'_5V’ ’ ’ " E+G ,‘V=v Performing the indicated differentiation in (72) we obtain (73) he: 935 +5.1, $3: + fwv’ = fv CalCulating the various partial derivatives in (73) we have 5’ - EG ) 5’ ‘ 1(E 6‘5“}. E“G)V'+ G G‘s-v.1 T'T. - “ 3/ “iv. — 7 5-? = EY+ va.‘- S'v'v: 1(E6v‘iEvG)v'1-GG-vv 3 1(‘iG-V")y‘ 1(E+GV“)% By making the necessary substitutions from (74) into (73) and reducing the result, we obtain r- H- _G__; _. EL... 9: ' E1 (79) V "‘ "V a?“ “’lv *(2. E e)-V+ 2.6 Using (10), eouation (75) may be written (76) ”\T = v-v"+ (ia-s)'v"+(a-15)V‘- {3 Remembering that v’= % , it is easily seen that I: I. H‘. ulvn‘uuv' .- l_ dAI‘ (77) V u.,‘\T_ w: ,u-%,v-3 From (76) and (77) we find that the differential ec uation of the geodesics on Sx is (78) u'v"- u."v" = o’v"+(1.a-S)u'v"' + (ct-zb)u"v'- ’Gu." 24. Consider now the curve u: uU) , 1r: U'Uc) on S . Let us find the equations in local coordinates of the osculating plane of the curve at the point Px‘ The equations of any plane in hOmogeneous coordinates are A.'x.+ a.'x,,+c:x, + v,‘x.,+s,'x, =o , (79) , . . , , A,x,+ B,x,+c,_x3 + Iraqi—E; x5 = 0. Since the plane (79) must pass through (0,0,0,0,l), we have T E.=E,=o. Taxing suitable linear combinations of (79), we may write them in the form Alx\+ leb+ CIXJ I D (80) t 0 Azx|+ Baxl‘) ’1 x'l - In addition to the point (O,C,0,0,l), the osculating plane passes through the points I \ I H J c o n" u u X=qu#xvv , x =xmu +zxwuv +xwv +xuu. +xvv. The local coordinates of these points are Q64? , V“J§', C), o, 0,) 2 (81) {<«ufi+1““'v'+yv't+ u"), (Pufi‘tzbu‘vWSvfi-r Y") ) o I t , I ‘1- (v, u"+2.1>,'u.'v +17: v"), (1’, u' +zv,_'u‘v +vz'v')) o) . m U] 0 Substituting the coordinates (81) in equation (80) we obtain Aprgu“? B.\‘EV':O : A‘ (du.t+zau¢vl+yv.3+uu)+B‘@u|1+lbwgv5+ s v.1-+vu)+c.(p'u.t+zb|lucvt+p:lvct)= O , A 0‘1 m V AfJE'u3+ B, a V'= o , Add u'”+uu.'v '+ yv"+u")+8,@w‘+1bu'v '+8v"’+v")+ 9,, (9, arm»; u'v '+—D,:'v‘”) : 0 Assuming 53:0, 83:0, we may solve (82) for the ratios , 9.: , is , Pa , B! 81 31. as and make the proper substitutions in equations (80). We thus obtain as the equations of the osculating plane, (after replacing the x's by; 's to preserve the local coordinate notation,) the eouations \E‘.‘ v'(:0.u"'+2v.'u'v‘+v."v")¥.—JE “'(D.“‘“"*1’.'“'V'+’."V")§z + JEG J 5: 1 ° ) (53) \5'6' V'(91u“+zv£u'v'+p:v'r) §.-\5Eu'(nzu"+2.1>; u'v'+v:v") 5+ @7ch E9 = o , wherein (54) J = u'V"- U~"V'-1(v""+(8-=a)u‘v"‘- (u-it)w‘v'+ Pa" The equations of the normal plane to 3x at x are We may think of eeuations (83) and (85) as four homogeneous linear equations in the unknowns 3,, 3,, I, and 3' . The condition therefore that these four equations possess a solution other than (0,0,0,0), or, in other words, the con- R") OK 0 dition that the osculating plane (83) and the normal plane (85) intersect in a line is (86) \ 0 O o o | o o s ’ I GEv'(n.u"’+av.'u'v'+1>."v"), —J'Eu'(p.u"’+zn: u'v’+x>,"v"),d?a 3' , 0 dz v'(vzu"+zv,'_ u‘v'+'9;‘v" , - E u'(v,_u“+2.p;_ u‘v'+.v,:'v"'), O ; 0E6 J Simplifying (86) we find the condition to be (87) From equations (73), (87) we readily see that the osculating {I =. O . 2.1.522 13.9. the curve i1=u(tL; V=V(tl 91; S a; 3 point P intersects the normal plane to the surface §t_P O X and only i: the curve EE‘E geodesic. It follows from (78) by putting v==constant that the curve Cu is a geodesic if and only if Similarly the curve Cv is a geodesic if and only if v/z‘i-E-izo. Hence we may state that the parametric curves C , Cv are __ 11 geodesics if and only if Eszu=O x in a line f 27. Moreover we see from integrability condition (18a) that if then (g3) (nor—inf) = — (D1 D;._D£2) . Using (88) we notice that the discriminant of the central conic (59) becomes H H I I t "“D'L 2 (go) (D'D1+D.D1—1D‘Dz) + 409,3). . . Since the expression in (59) is definitely positive it follows that the central conic is an hyperbola if the curves of the given net Nx 23 SX are geodesics. The equations of the osculating plane to the curve Cu at the point Px may be written immediately from equations (83) by putting v=const. They are Riz- (JOE—£3 =0 . 1:151- 90? L, Each of the equations in (90) represents a three—specs :Jw (90) O . ’ their intersection being the plane in cuestion. Any linear combination of these equations also represents a three-space containing this plane. If we should multiply the first of (90) by —Dz , the second by D. and add, we obtain (91) Da£3"£+=°° [0 U) 0 which represents a space of three dimensions passing through the plane (90). The equations of the tangent plane to the surface Sx at the point Px are It is evident from (92) that the tangent plane to Sx at Px lies in the three—space (91). Hence we may say that the osculating plane to the curve on at x and the tangent plane to 8x at x determine the space of three dimensions (91). This space and the normal plane intersect in the line (93) E.=o, §,,=o, D,€,~p,§.,=o. The direction cosines of the line (93) are proportional to D,A+ 131». In exactly the same fashion we may show that the tangent plane to S at x and the osculating plane to the curve C x v at x determine a space of three dimensions which intersects the normal plane in the line (91+) E'=o, §1=o , D;'E,-1>"§.,=o, The direction cosines of the line (94) are prOportional to H I. ‘D‘ A + £5 rL . The two normals (93), 94) to 8x at x with directions defined by D|x+ D2?" and :9? A -r Ififft respectively we shall call the intersector normals of the Cu and Cv curves. The intersector normals will be orthOgonal if Z(’.X+31P)(D?l +13: V) = O ' Using (4) this condition reduces to r‘ \O \Jl V t, From (93), (94) we see that the intersector normals will coincide if (96) D.‘D - D 31‘:- O . Again from (64) we can say that the conjugate normal coincides with each of the intersector normals if I I " I a u (97) D'Dz-D.Dz=0 > D\ 31.13: pa. = O Combining eouations 96 (97), we may state that the con— “ J, 1 ’ l _..__ __ jugate normal and the two intersector normals all coincide if. the matrix '0. v: 19? (98) A = (I): D; pal-I is 2: rank one. O. \rJ VII. C NDITIOKS FOR 8 TO BE IILERSED IN THREE DIJEESIOKS Let us suppose that in the system of defining differen- tial equations (9) for SX the following condition is satis- fied: the matrix A in (98) is of rank one. If we make the transformation (99) i=““°"‘ "= min.» VDTI'V: v9.1"? 9: on equations (9), we obtain immediately from (42) with v i . A: ‘ ) B: T,“ V D|t+ 9: VP: 1- P: ’ the following transformed system: XII“: “XK+PXV+JD.‘+v-gt i I I - ' . Xuv = “xx“. va+ pins+vtvz A + 9.91-9‘01 - 3 473+ “1‘ VD,‘+ of P I " '.' . Xw= qu+ 3xv+ "I" +°‘°‘ 'i + "’1‘“: ”I ‘ V D.‘+ P: V D.‘+ v: - x + u E GVm‘+v‘ "' v.v'+v v' in“ on" - ' - Avg—#xu_ \ g I. xv+[B‘+v|ptv gaunt)", (100) _ JD‘+D‘ 9'» v _ x " #X,‘ - + D; [A + 9.9“.- D10.“‘}F3 It a: + 9!. E"D:’+ 7;. D... + D:- - 9:91-D 0:: ‘D ‘D —‘D D - P“... +[A1" \ :u. 3‘ us] 1 ’ @119" + v: D. + P; I Fv - v" D‘- ”I p; x - 9‘ 9‘ “—9." D: ”[8 D'Dtv-D‘ D‘v] _ —- —_—-——-— “ . E¢9f+ 9: 64 of+ 9,} v,‘ 1- p;- 1. \JJ If we make use of the fact that the matrisz is of rank one, we see immediately that in the expressions for xuv and xvv the coefficients of F. vanish; likewise in the expressions for Pu_and flvthe coefficients of xv and xu also vanish. We shall show presently that the coefficients of F for the last four of ecuation (100) are zero. Referring to the integrability conditions (18), p. 5, let us multiply ecuation (c) by D. and subtract from it the result of multiplying equation (b) by Dz_. We obtain I a (101) 'D\‘Dw-D*Dw+3.(9:'+v:)-A.(p‘n.'+v._o"_)+ 9‘ ”up"? D = O . I.“ Since A is of rank one it follows that DI" " 'DD ” _vuv 9‘9 'H' 9.: —-\—— , v.=_'_;—. °.i-—L——,2—+_~_::_-°_._v___..ufu . Substituting into (101) and simplifying we obtain I I I 9" II o I I . (102) (Vim-Pi °..)-§ + A. (ah-91")?- + vivw- 9. Pu ~13. (9.543) = o . I. I. I .ZLEL in (102) and simplifying again, there I Putting ‘3‘: t results II 1- I '91 . . 7' 1, 1. D I. ‘ _- (103) “-9 D‘Dlu-‘Dl 9w,“ .5: A'(D‘ +vt) +—;!'C 'Dt‘bw- D. 91v. 3‘ (p. 4. pt) .- O , 3. 1 a By (101), the second bracket is equal to ' I I "A‘ (v’p’ +vgpi) + Ipzpu‘- D‘ D1“ \)J P0 Hence (103) may now be written in the form 9" D:- Dr. 9' c c (101+) o“ A.(°?‘+P$)-;‘— + (mm: 91°... = 0 - I. I. . u 1 . . . . Since 9191-1fl’ is not invariant under rotation we may assume it not zero. Finally let us substitute into the second bracket in (104) to obtain I a. 'D D (105) A| (Uri-91)??? + 33(0'171‘: Dl D“ = O . From (105) it follows that A/ vaLu-DLDWL (100) A‘+ 1. L = O 0.1- v1. The coefficient pi Dav " pa. Dw 8+, I L 1- ta + v; is shown to be zero by a similar demonstration. Equations (1C0) under the conditions imposed now become XML otx“+FXv+xJDc-+n: A , x“: axu+ bxv+ 3".*°‘31 i . vf+9: Xw = vxa+8xv +5v;'+vz°_=': 1, 493+D: (107‘ X =_. ”H": ”WU-”1.9; / IL x“ Xv ’ E G J of-r v: ‘ I I AV : ._ 'D|D|+st3. x _ \) D:‘L+V£‘ X V 33- The equations (107) are the ordinary "Gauss Equations" of a surface in three dimensions.* Hence, a necessary and sufficient condition that Sv he itmersed in a soace 2: three dimensions * 'is that the matrix ' II ‘D. D, ‘D. b ‘D; D: be 2: rank one. From 08 we ma state the above result as follows: J I» necessary and sufficient condition that Sx.§§ immersed in'g soace pi three dimensions is that the conjugate normal and the two intersector normals all coincide. From eouations (36) and (42) we note that the surface of normal projection S; defined by (35) is a minimal surface“ if and only if - ‘ .3" D'BV «- vol (1083) M.=-E-3+-é-'- A ‘E " LL69): AM.-BM1=O , Since M‘ and.M2 represent the mean curvatures of the surfaces of normal projection S) and S“it follows that if two surfaces 2£_normal projection taken in perpendicular directions are both minimal surfaces, all surfaces g£_normal projection are minimal. if not both M and V are zero, there exists just .._.._. I—“a one normal to the sur ace at‘g which determines a minimal surface of normal projection. ‘ Eisenhart, Geometry, p. 154. I""‘Eisenhart, Geometry, p. 129. 34. ’V If the surface bx is not immersed in a space of three dimensions, then from the point x there may be drawn two normals each tangent to the central conic. nach of these .c'r, normals determines a surface of norm which d H 0 D H) O H {.0 l projec the two principal radii of normal curv§ture are equal. Hence the point x is an umbilical pointzfor each of these surfaces of normal projection. 1 Equation (43) may as rewritten (1-09) (any-19:") A1;- (9.12: +v‘“v1-2 9:21) A8 + (avg—of) 3": E G R. , The surface of normal projection S will be developahle if Hence if ‘ g u I I II S 1 (11¢) (9\D,LD“)A1-_(D.D:+P. lat-2‘0, v&)AB 1- (35151 ~ D; ) 3 = o . For the canonical form of equations (9), the coefficient of AB is zero. Equation (110) may therefore be written: " '1' t n (111) (19.2-1). ) A + (p.91-19,?) 81: o, The latter form indicates that there are exactly two surfaces of normal projection which are developables. The normals determining these developable surfaces are called the devcloo- able normals.** It may be readily verified that the developahle * Eisenhart, Geometry, p. 120. *‘Grove, Geometry, p. 07. \r] \1] o normals separate the principal normals harmonically. VIII. THE RELATICK R AKD THE COKJUGAIE ICELAL If we should construct tangents to the curves Cu at the points where they meet a fixed Cv curve, we obtain a . (v) . . ruled surface which we may denote by R . Similarly by constructing tangents to the curves ” at points where they Uv (u) meet a fixed C curve we obtain a ruled surface R . I! '4-"fl4 u Let .1, be any line lying in the tangent plane to 8x at x, but not passing through x. Let «A be a line passing through x but not lying in the tangent plane at x. The line.lL intersects the tangents to the Cu and Cv curves in . w) points 3 and g respectively. If the tangent planes to R Em» and at r and s respectively intersect in the line 1, the O given lines.1, and 1; will be said to be in r lation E} with respect to the parametric net Nx' The points r and s are defined by expressions of the form (112) Azx- x, S: -—‘—x.,. _\_ 1' The tangent planes to R and R at r and s intersect in a * E. P. Lane, Projective Differential Geometry_of Curves and Surfaces, Chicago, University of Chicago Frees, l 32, p. 32. line joining x to y defined by (113) t3: X+ K (Xw~jxu-'fxv) 2 o where k is preportional to the distance between the points x and y. Using the second of eouations (9) we may rewrite (113) in the form (114) %=X*K{(a-7)xj+(L-|.)xv +D")\+v;y.} . Since % — xiere proportional to the direction cosines of the line xy, the line xy will be a normal line to 8x at x if (115) thtg—xha-gw . va(}-x)=b-r =0 . Hence if a=§ , bzk. The direction cosines of this normal will then be proportion- al to ' l D‘A+sz From (64) we readily verify that this normal is the conjugate normal. We may therefore characterize the conjugate normal _a_s_ the only normal line t_o_ fix §.__t_ lg which is in relation B with a given line i_ the tansent plane with respect to the N . x :3 d e I X. CHAZTGE C ’11 "1 {T1 *1 C: (2 5‘: 0-] :1: [1| m (:1 OJ) ‘11 tx> 0 {1} Any net of curves on Sx may be defined by a differen- tial ecuation of the form‘ (11K) (dAr-edh)(d“‘wd”)= O a \J 01' 9d.:—(\+0w)d.ucl4r +coobv"=o, ,‘g‘ vhere 9 andcu are functions of u,v. If the discriminant of (116) vanishes, that is, if |-9(0 is zero, then there is only a one parameter family of curves on the surface and not a net. If 1+6!» = o , the net is a conjugate net. If 9==h3= o , the net is our given parametric net. Hence we shall suppose that (117) 90) (I- e‘w‘) at o The net (116) will be orthOgonal if the harmonic invar- iant of the two forms 040:“ (|+OW)dut£/U’ + wd~t= O , Eda." +Gd4rt=0, ‘ V. G. Grove, §_6eneral Theory 2: Nets on a Surface, Transactions of the American KatnematicsT‘Society, Vol. 29 ‘*Eisenhart, Geometry, p. 80. -. A. . . U I 4’ T is condition is (115) GG+wE .0. Let us ”she t-e net (113) parametric by tre transfor- M p u : f o 1" ‘ C A I Q n nation 01 varia-les ceiine; by A H .4 \O :3 H 43 f" S < V (I II VS A E < V 4.4.,__3:_6 £__£5—=J- d“- TV- ’ du..- vv ‘0 thence ceme the relationships (12C) ¢u=‘e¢v ) )évz-wy“ From (119) we may now write, after formal differentiation, the equations xu= xa?u*xe?v » xv=?v"a*y’vx\7 . X“: ?: xii *1?“ Wu xw+ 1’; xv; + fl... “a + Van X a . X“ = Pu?vxai*( Pv= Q’vFa +va‘v \ r! KL) 0 Substituting from (121) into equations (G) we find x. Q: X“ ”Tick“? + 1”: KW: (“QU’F Yv'cPuu) X; +- («¢“+pVV-¢uu)xv +D.X+v,u , Q“¢vxfii“’(¢v¢d¢u WJXW + 3h )5». XW a (aqua-la cry-em.) x a + (d-VHWV-mvhwvfhfip. <03 "au +14". 1‘. xi: +¢5Xw = (Yen ivy-sow) M: + (VW.+8¢v-¢w)xe + ”Swipe . @Xa’r CPA; = Ln.q..+1v.ev)xa *(m.¢v+1v.¢v)x<7 thr‘. e. Km 14. A; = (3.e..+~v.?.)xa: + (1.¢.+m¢v) Xv +31?" Q“ 114;.“va = (m,qu+f,qv) x; +(N;?v+1.1’v)xv + A17) » (Pv Hi +Y’v F; = (11QH+“1QV) X; + (7.1%! '1'“: iv) XV ‘1' BL X ‘ Consider the first three equations of (122) as linear equations in the unknowns Kan, X13, Xw . The determinant of the coefficients may be written, by using (120), in the form GtQJ a ’“QV ions ) V’: , 3 ~64": , OHMWWKQK = {QV Va. (ecu-In . 4;: -w ?v 50.. . NV: 3 By (117) we note that this determinant is not zero since Gw-I $0. Ln. ‘r’vo Hence we “1.3V solve for X3; 3X“, XW obtaining a 12,61» 21:3.» + of} 1 {1’1 «9+ 2.9,: w + D; } ) X-..==ELX-+ —X- ‘1’ A H m \N v x :1 <1 u _ ._ 1: w+n:(u+0w)+v:'61 Kali-iv; (1+ew)+17£0\ X-+ b X' +{ ‘ 2. x +{ ‘L i a . V m. (ow-n) q». w. (as...) 7 F .. .. I'D.+1‘D.’O+D‘"9t} {Uzi-19:61.1): 6"} vxa+ s x-Q +1” (MM). 1 + V: (ow-0“ P 1 where 0L,{3',....., 8 are new functions of u,v corresponding to on, p ,....., 8 in (9). Solving the last four equations of (122) for M; ,lv‘ya,yvthere results _ AV—B hazm‘xa+$,x¢,+ \V WW I". V“ )oV‘ ‘Pvt )gz- Xa‘va'Xq‘f A'QV-B'?“ v 3" ¢vk‘ ‘7“ WV P. ) (124) W .. A - B ‘P _ 2 1&1‘ x-a + XV + 1 V ,‘ V x , F“. $1. QuV’v" Qvt -- — A -a .. [Ag-33x;+n,x9+ ”V i? 1, Equations (123) may be written Xi: = axa+§xv+5.x+5.e, X113: ixa+1xv+iit>‘*;=-P7 (I <1 wherein a... (new; war) , 5.= i. (v.~*+zv;w+v:), 14¢ [Wu—0]", (125) 5": {11mm +D.'(i+ow) Hare], i:="1[‘pzw+ pz.(”°‘°)+’£9] , ‘Az=[¢?v)"u(9w")u]-: 5‘": (‘3 (we-13:0 +D:'o‘), 5,112+, (91+.1vfl ”£99 1 *3 = [30, (904)14- For the surface defined by equations (123), (124), the intersector normals will coincide if (126) v. D .. i5."1'>1L = 0 From (125), we may write (126) as follows (61:32:09.4n:‘)(o‘n;'+zot;+v,)-—,- (95319 v,‘+v&(u‘v,+zwv;+ 12,"): o .' Upcn expanding and collecting terms the latter form becomes (?._v,:'—'D:'D,_).w‘o‘+1(v,v£-v,'n,)w‘e + 2. (v: 15:51:." 17,3030“ + (127) ‘ 1(9,'vz—D,D,f)w + 1CD." ”2’79: 17:)9 + (17." ’1’ 9'1): : O . Imposing the condition (118) that the net (116) be orthog- onal, enuation (12?) becomes (128) [G (”192‘3:.Dz)] 9131' 1[G baffled-{(1): v2-v,"v,f)] 6 - E (v..v:- w,"v,_) = O. If in addition to having the intersector normals coincident, the net were conjugate, then the matrix A becomes — H - n 2. O ”L ‘ which is of rank one. Referring to our previous results, 51:” we see that 8x would then be immersed in a space of three dimensions. We shall suppose therefore that -1 ..: 3.4-0 , 171*0. Since (128) is a quadratic equation in 9 , there exists gg_§ surface not sustaining a conjugate net two orthOgonal nets whose intersector normals coincide. We shall call these nets the intersector nets 2£'8x° After dividing out ear—t which is not zero, equation (127) becomes (“"0 (1w:— D."v.) + z «9 (1°. Di- 115v.) + z 6 (DJ 9.3- P." v.‘ ) = 0 . The latter equation when solved for 9 gives (129) w zfiifl. ‘BO'+I> wherein A '2 2. (9,"31" pl. D1” ; B : (yuuvz-p'vlu) a D = 1(D,'D;-'D,' v3) ' From the form of (129) we may state that the tangents £2 the curves gf'non—orthogonal nets whose intersector normals coincide are projectively related. If in (129 we should a_| put 9 ~‘6’ , we obtain a):= B€9+-A ’ ne'— 8 which is the equation of an involution. Let us refer the surface 8x to one of the two inter— sector nets. In addition let us make the transformation (130) 1:5,).4-5311 , fi=-5’-X+S‘F IVfr+v, v.+ v: on equations (123), (124), giving us equations of the form C (100) but with Hence the surface Sx (not sustaining a conjugate net) may he'defined byflg system 2: differential equations 9: the form x“: exu+va+ml lw=m(xu++.xv+ Aw. , 7 I l (131) x“: 4xj+bxv+ql+vzh Av: %‘x“+n.xv+ B.» , xw=yxu+8xv+nyk (Au: inflatfzxv't A1). 7 P‘v: glxu-q-vHxv-t-sz , where the letters d,p,..-.,DHD:,....,m., H...,A B “A”... ., ".. represent the transformed quantities corresponding to the letters in ecuations (123), (1211-). The matrix A for the system (131) is Hence the surface Sx represented by (131) will b§_immersed ig_§ space g£_three dimensions if and only if I 131:0 . For the system (131) the parametric net on SX is one of the intersector nets, and the normal 3K represents the two coincident intersector normals for curves of that net. The mean curvature M2 for the surfabe of normal projection sf" whereft represents the normal perpendicular to the normal X , is D " (133) M.=f+%ao. Hence the normal projection surface S is a minimal surface. 'A. The equation of the central conic for the canonical system (131) now beeches (134) $5 0, If 0 , CD.'D:LD:‘) §:-Z‘D:‘p;_ is 5—3;" £:-(E‘p"|+ GE”; +EG =0. 1+5. The condition that the quadratic rxpression in E5, E4 in (134) be factorable, in other words the condition that the central conic be degenerate is ,I- a I ”tutu-3| > “1% D2. 1 -y1(Ep‘"+GD.> E G -‘/..(E 1.3.5»), 0 8 Equation (135) may be written in the simple form n L / C p. D‘ (130) {E G pa. 3" 1-)} 3 0. Referring to integrability condition (21), we may write (136) as follows (137) {E6 (A.v-Bm)}z = 0 . Therefore, a necessary and sufficient condition that the central conic be degenerate is ()9 (13) A -B.:O. \V us From (133) two cases arise: Case I : ‘D; = O . CaseII: EL—il—zo E (3 ‘\ If ‘D,’=o , we see from (132) that SX is immersed in a space of three dimensions. To investigate the geometric meaning of Case 11, let us find the principal directions' in the tann gent plane to the surface S x at the point Px' They are given by (139) (Exp)? F; Dx)d.u.1+(E1'D;-Gkbx)duc94r + (F313;; GAD‘A)°LVL= 0. Using (29), (31) equation (139) may be written 1:" (140) E on» +(%'1’—_é.)atww- (iv-"=0 ETG The principal directions given by (1M0) separate hart onice.11y the tangents to the curves of the intersector net (141) 1 cu car on 87, if and only if ‘5 II D, :9, ——--—-=0 G E Hence when the latter relationshig obtains the central conic lé degenerate. lgLsummary Eg_may therefore state that the central conic _i_s_ degenerate L1; “213° _i_r_1_ which case the 3.:_ 2;_ o , when the principal directions in the tangent (a E “ clane to the surface fi‘AE-E the Eoint Px separate hermonically the tangents to the curves 2; the intersector net on 8x' ‘ Eisenhart, Geometry, p. 121. o- Eisenhart, L. P., Differential Geometry of Curves and Surfaces, new York, Ginn and Company, 1903. Grove, V. G., Differential Geometry 2f.a Certain Surface in S , Transactions of the American hath- ematical Society, Vol. 39 (1930), pp. 60-70. Transactions of the American lathematical Society, Vol. 29 (1927), pp. sci-s14. Lane, E. P., Projective Differential Geometry of Curves and Surfaces, Chicago, University of Chicago Press, 1932. Tilczynski, E. J., Projective Differential Geometry of Curved Surfaces, Transactions of the American hathematical Society, Vol. 8 (1907), pp. 233- 200. fi' . l l} 5: CO. WAC E NVOOPD E STATE UN llllllllHllHlll 1293 03 E_RS|TY LIBRARIIES