Svoboda, Karel

English

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeKarel SvobodaOn surfaces in E3 with constant Gauss curvatureCommentationes Mathematicae Universitatis Carolinae, Vol. 19 (1978), No. 4, 755--761Persistent URL: http://dml.cz/dmlcz/105890Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1978Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCOMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAÄ 19,4 ( Ш 8 ) ON SüШåCЖS IN ü3 ЯTH CONSTâNf ШLÜSS CÜШŁWm Kвrel SfOBODà, Brne Abstract: A global characterization of surfaces ia r with constant Gauss curvature. Key words; Surface, Gauss and mean curvatures, integ-ral formula. AMS. 53045 H. Fath el Bab introduced in CI] the conditions imply-ing H -* const on a surface M in ! . In what followst we ap-ply the method used in Cl] and prove an analogous theorem for the Gauss .curvature K ef M. Let M be & surface in the 3-dimenaional Euclidean spa-ce Jr and iM its boundary. On Mf consider fields of ortho-normal frames 4Mfv,fV2fVn1 with •,,•* *T(M)tT(M) being the tangent bundle of M. Then we have (1) <Ш = đv x = co-v^ + « 2 v 2 , »Ъз* đv 2 = - " l r l + »ІҺ» đv 3 = -tíЬl - W 2 V 2 anđ (see t-з, P . 8) - 7 f Ç -(2) o * | « a c * 1 + bct>2f 4 * | * be**1 • e 6 * 2 | (3) 4 a « da - 2bd*2 -eic**1 • ^c** 2 , Ab « db + (a-c)c**2 * fcw1 * T&>29 4 c « de * 2 f e ^ i s T ^ 1 * «f*>2j (4) AeC= d«* - 3/Sct*f » 4 A * 1 • (B-bK)c«*2f Afim d(l + (a t - 2 y ) 6 * 2 « (B •bK)***1 • (C*aK)e<*2f A f « d r + (2ft -€T)c«*| « (C+cK)^*1 • (J>*bK)c«*2f Aef« d / • 3 y a*2 « (B-bK)c,*1 • 1&*2 where (5) K « ac - b 2 i a t he Causa c u r v a t u r e of II • The e o v a r i a n t d e r i v a t i v e s K^jK^. ( i f j « 1,2) of Kf d e -f i n e d by (6) dK • K-j**1 + Kg**2 , dK2 - Kg**2 - I ^ * * 1 -> K ^ * * 2 , dKg • t ^ \ . K ^ * * 1 + • K 2 2 o*2 a r e g i v e n , accord ing t o (3) and ( 4 ) f by (7) % « § y - 2 b / | • ecc f Kg « aeT - 2 b ^ • e / S } (8) K ^ « aC - 2bB + c4 • 2 ( c i y - ft2) • (ac - 2b2)K, 1^2 « aB - 2bC • cB + («ccf-/J«f) - b(a+c)K, Kgg « a l - 2bD * cC • 2(|3<f- y 2 ) • (ac - 2b2)K. Now, we fo rmula te the - ?5S -Theorem 1» .Let M be m surface in 'Sr with K> 0 and ^ M its boimdary. Let ?lf?2*T(M) be orthonormal vector fields on M such that (9) ?XK = 0, ?gK = 0 on #M and (10) V 1 V 1 K s 0 , ?gK = 0 on M. Then K = const on M. I¥oof.» Consider a 1-fora y = Rj&l1 + B ^ 2 on M# The cova r i an t d e r i v a t i v e s of M^ ( i = l f 2 ) being de -f ined by m% - % « * ! s H i ! 6 * 1 + H i 2 0 2 » dig + H i ^ i = ^i^1 + a 2 2 f t ? 2 we have , according t o £13, p , 247-250, the i n t e g r a l formula (ID k-ciyi,! - RaRiP^1 + (a-a-., - V^*-* 3 -" i t2(BllR22 " *12B21> " <4 + 4 ) K D W1A »*' Now, l e t tis choose the tangent frames a s s o c i a t e d t o M i n such a way t h a t v^ = ?.,, v^ = ? 2 # Then i t fo l lows from (6) ?XK « K l f ?2K « Kg and W ^ i * V*4(V-Thus we have, using ( 9 ) , ( 1 0 ) , - ? S 7 -кx « o, к2 * o on дм, ^ ш Of Kg « 0 on If and hence the integral formula (11) f re-wr i t ten fer the 2 iCd f y ields 4 i ( 2 K i 2 * *lK} **1/s **2 * °* 1-forю K-,01 + Kg д2., i l  ISms eepeeiaЦy % = v i к on M, i . e . K s const on M. Bemark that the surf aces with K » const depend on 4 functions of 1 variable. Following C13, we are going to prove that there are. locally, aurfacee M iair possessing two orthonoraal tangent vector fields Vl f V^ such that VgK « 0f V ^ K = 0 and with K not constant on M. For this purpose, we shall prove that the surfaces satisfying the preceding conditions depend on 4 functions of 1 variable. The considered surfaces are defined by the system (4) and (12) V2K • acf - 2b|* + e/I = 0 , V ^ K » aC - 2bB + cA + 2(*f- /I 2) + (ae - 2b2)K » 0. Becauee of K+const f we have TjK -» % + D . By exterior dif­ferentiation of (4) we obtain (13) tkkAm1 + 4 1 A02 » (4flK + b K 1 ) & i1 A ^ 2 i ABAdl 1 + ACAiil2 » U 3 y - 2*)K - aKx + bKgl^A&J2, AC A&1 + A l A C i 2 • t ( 2 ^ - 3$)K - bK1 + eK^t t*1 *^ 2 , ADActf1 + A l A W 2 s - ( 4 r ^ • bKj,)**1* ftl2 where (14) A4 * dA - 2(2B + bK)6>ff AB m m + fA - 3C - (2a+e)K3 c^f, AC • dC + 2{B-D)w2f AD • dD + I3C - I + (a+2e)K3 m\% A» * d l + 2(2D + .bK)&>|. Differentiating (12) and applying (3) f (4) f (14) we get (15) aA«f- 2bAf • eA/3 + cTA a - 2 /Ab + 0Ae - %6if » 0 t aAC - 2bAB + eAA + + tC + e(2K - b2)3 A a - 2 tB + b(3K - b 2)3 A b + + tA + a(2K - b 2 ) 3 Ae + 2 ( e t A r - 2(1 A(I + r * * > • With regard to the second equation (15), the closure (13) f (15) of the system (4)f(12)" contains q » 4 l inear ly indepen-dent forms and s-̂  s 4 l inear ly independent exter ior equa-t ions , so that ag = ° a n d $ * *• Applying the Cartan'a lem-ma we obtain from (13) A A « F^c**1 + F t6^2f AB « (F t • 40K • b.^)***1 + FjC*i2f AC • tF j • ( 3 r ~ 2*}K - aJ^ + tfl^la*1 + X^2* AD » t F 4 + (2cf - 3/l)K - bl^ + e K g ] ^1 + F ^ 2 , - T 5 9 -ill - (F5 - 4 r ^ - bl^}**1 + Fgui2, the functions F^t•••,#£ satisfying two independent rela-2 tione obtained from (15) by elimination of &>£. Thus, Jtf « * 4 and the general solution of the coneidered system de-pends ©n 4 functione of 1 variable. Finals notice that the heorem 1 and that ©»e due to H. Fath el Bab can be generalized to thie form: Theorem 2. Let M be a surface in r with K^O end ^ M ite boundary. Let F(HfK) be a. non-zero function defin-ed on M. Let Vlf?2i T(M) be orthenormal vector fields such that ?1F(HfK) - 0, V^Ftt^K) = 0 on BM and ? l v l f ( H f K ) a ° t V2F(HfK) = 0 on M. Then F(HfK) » cons t on M. The proof of t h i s a s s e r t i o n i s analogous t o t h e above mentioned one. R e f e r e n c e e [1] H. FATH IL BABs On surfaces with constant mean curva-ture, Comment. Math. Univ. Carolinae 16(1975)9 245-254 111 A. g?EC: Contributions to the global differential geo-metry of surfaces, Rozpravy CSA? 1, 87, 1977, 1 - 9 4 - ГêO Katedra matematiky FS VUT Gorkého 13 60200 Brno Československo (Oblátům 6.6. 1978) -761-

On surfaces in $E^3$ with constant Gauss curvature

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On surfaces in $E^3$ with constant Gauss curvature

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