Lumiste, Ülo G.

English

Lumiste Ülo G.

University of Szeged

Acta Sci. Math., 50 (1986), 51—57 Backlund's theorem and transformation for surfaces V„ in En 0 . L U M I S T E 1. Introduction and classical background. Backlund's classical transformation gives a way to generate new solutions of the Sine—Gordon equation d2ipldul du2= = sin 11> from a given solution. Its geometrical setting uses some basic propositions for surfaces V2 in E3, which are the following. A. If there is a diffeomorphism V2—V2, x<-+x*, between two distinct surfaces in E3 such that xx*€TxV2OTx*V2, |x**|=r=const and the angle (p between TXV2 and TX*V% is a constant, then both V2 and V2 have constant negative Gaussian curvature equal to —(sin2 q>)/r2 (the classical Backlund's theorem [1], [2]). B. This diffeomorphism, called a pseudospherical line congruence, maps asymptotic curves of V2 to asymptotic curves of V2 (i.e. it is a Weingarten con-gruence or (^-congruence). C. Asymptotic curves of a surface V2 with Gaussian curvature A"=const<0 (i.e. of an immersion of a piece of the Bolyai—Lobachevsky plane L2(K) into E^ form a Chebyshev net: in suitable net parameters u1 and u2 the metric of V2 can be given by ds2 = (duv)2+2 cos if/ • du1 du2 4- (du2)2. D. The net angle \]/ of a Chebyshev net of a Riemannian V2 satisfies the equa-tion d2\p/du1du2= —K sin \j/, where K is the Gaussian curvature of V2; in case if V2 is a piece of L2(—1) this equation is the Sine—Gordon equation. Due to C and D, every immersion of a piece of L2(— 1) into E3 gives a solu-tion i¡> of the Sine—Gordon equation and this correspondence is one to one up to rigid motion. Due to A and B, there is a transformation of such a solution to another, the analytical formulation of which gives Backlund's classical transforma-tion [2]. The aim of this paper is to give some generalizations of propositions A, B and C to the case of surfaces V2 in En, «>3. Note that D needs no generalization because it does not depend on the immersion. Received January 2, 1984 and in revised form August 17, 1984. 4« 52 U. Lumiste A generalization of the geometrical Backlund theorem and transformation in another direction, for the case of Vm in E2m_1, is given in [4], [5]. If m=2 this reduces to the classical one. 2. Main results. The next generalization of Backlund's theorem gives some additions to the classical case too. . . . . J Theorem 1 [3]. For two distinct surfaces V2 and V2 in En, let V2 — V*, x<-»x* be a diffeomorphism such that xx*^.TxV2C\Tx*V2 and |xx*|=/"t£0 for every point x(i V.y. Let <p be the angle between TXV2 and TX*V2, and let K and K* be the Gaussian curvatures ofV2 and V2 in the corresponding points x and x*. Then the fol-lowing four conditions are equivalent: (1) r—const and <p == const, (2) K=K*=-(sin2 <p)/r2 = const, (3) K=K*=-(sin2 (p)/r2 and r=const, (4) K=K*= -(sin2 cp)/r2 and cp=const. Under the assumptions of this theorem the diffeomorphism V2—V2 is called the line pseudocongruence (if n=3 "pseudo" is to be dropped); V2 and V2 are called its focal surfaces. They cannot be arbitrary surfaces, but necessarily must consist of planar points only. Tangent planes TXV2 and Tx* V2 in corresponding points x and x* lie in an Euclidean 3-plane (E3)x. Among the second fundamental tensors of V2 in normal directions to TXV2 we can distinguish the tensor h in the normal direction lying in (£3)^. A pair of null directions of the tensor h is called a pair of fc-asymptotic directions in TXV2 and corresponding curves on V2 are called /«-asymptotic curves. The diffeomorphism V2-~V£ is called/i-asymptotic if it maps /i-asymptotic curves of V2 to /i-asymptotic curves of V*. Theorem 2. Under the same assumptions as in Theorem 1 the next three con-ditions are equivalent to each other and also to each of the conditions (1)—(4): (5) K= — (sin2 <p)/r2=const and V2—V* is h-asymptotic, (6) K— —(sin2 <p)/r2, r=const and V2—V2 is h-asymptotic, (7) K= —(sin2 (p)/r2, <p=const and K2—V* is h-asymptotic. Here the Gaussian curvature K of V2 can be replaced of course by the Gaussian curvature K* of V*. Theorem 3. Under the same assumptions as in Theorem 1 let one of the con-ditions (1)—(7) be satisfied (and hence each of them). Lei the field of distinguished normal directions (i.e. belonging in each x£ V2 to (E3)x) be parallel along the curves tangent to directions of xx* with respect to normal connection ofV2. Then the net of h-asymptotic curves on Vs is a Chebyshev net. Backlund's theorem 53 Theorem 3, due to proposition D, gives a possibility to find a solution of the Sine—Gordon equation by the special immersion of a piece of the Bolyai—Loba-chevsky plane ¿¡¡(—1) into E„. Theorems 1 and 2 show how this solution can be then transformed. The well-known Hilbert's theorem [6] states, that there is no solution i¡/ : R2-*R of the Sine—Gordon equation, Which is different from 0 and % in every point (M1, u2)dR2:lt follows, that the class of surfaces V2, satisfying the assumptions of Theo-rem 3, does not include the Bolyai—Lobachevsky plane £a(— 1), globally immersed into E„ with regular ^-asymptotic net. That gives a contribution to the theorems about classes of surfaces V2 in E„, which does not contain a V2 isometric with L2{— 1) (see [7]). Here it is important that a surface V2, satisfying the assumptions of Theorem 3, can be defined by following conditions, without turning to V2aEn: 1) V2 con-sists of planar points only, 2) the field of normal curvature directions, corresponding to the lines of conjugated net family of V2, is parallel along the lines of the same family with respect to normal connection, 3) invariants r and <p (which can be expressed in terms of V2 only) are constants and r2=sin2 <p. In this paper we can-not give the complete explanation of the question about impossibility to realize L2(— 1) by such a V2. It needs a new publication. 3. Frame restriction. A local field of orthonormal frames will be choosen so that the origin is xdV2 and e,, e2£TxV2. In formulae dx—e¡01, de,=eK6f ; I,K, ... = 1, . . . ,«; dO'=eKAO^, dO'^e^Adl, 0?+6'K=0 for the field of ortho-normal frames in En we have then 03= = and hence 0xA0i+02A0£=O; a, p, ... =3, ..., n. By Cartan's lemma we may write 6"=b"j8j, b*J=baji, i,j, ...= = 1,2. From the assumptions of Theorem 1 it follows that the tangent planes TXV2 and TX*V.* lie in an Euclidean 3-plane (E3)x because TXV2 D Tx* V*Bxx* ^ 0. The frame can be choosen so that e3£(E3)x in each point x£V2 and e1=(]/r)xx*. Then the point x*^V2, corresponding to x£V2, has the radius vector x*=x+re1 and from (3.1) dx* = (01 + dr) e, + (02 + r62) e2 + r (dfe^ + 0fee), Q,<T,... = 4, . . . ,«, it follows that by such a choice of the frame we have 0f—0. Thus bf1=bf2=0. The linear span of normal curvature vectors baijX'Xiea with arbitrary unit vector X'EI^TXV2 is called the first normal space NX V2. Now it has dimension two because it is spanned on bii—3, bi2— b\2e3 and b22=bl2ex, the first two of which are collinear. We can finally restrict our choice of the frame by the con-dition that e ^ N x V 2 in each point x£V2. Then &aa=—= % = 0, 54 U. Lumiste and so we have 0f = fc,70'', h21 = h12, 0i = O, 9t = k220\ 0 ? = . . . = 0? = O, where the notations htj=b?j and k22=b\2 are used. 4. Gaussian curvatures. The above restriction can be done for the surface V2 choosing the frame vectors at the point x*£V2 in a similar way. Then et = «i, e2 = e2 cos q> + e3 sin q>, et = — e2 sin (p + e3 cos <p, C4, . . . , en cn , and dx* = etO^+etd*2 = e10*l+(e2 cos <p+e3 sin <p) 0*2. Comparing with (3.1) we have e ^ e i + d r , e*2cosq> = e2+rdi, 0*2sin<? = re?. Here sin <¡9 cannot be 0 because this would lead to 0J=O and V2 would be a torse with line generators JCC* and we had V* - V2 what is excluded by the assumptions of Theorem 1. Therefore (l/r)02+0? = cot<p-0f. From this, by exterior differentiation and using well-known formulae, (4.1) dd2 = -K91A8\ 0?A0i = KO^O2 we have (see [3]) (4.2) K = -((sm2cp)/r2)(l + r1)+(hi2(p1-h11cp2), where dr=r161+r202, d(p=(piei + <p2d2. For the surface V2, 0 f = det-et = ((sin <p)/r)e\ 0? = det • = dcp, and now the second formula in (4.1) for V* gives ([3]) /a K * _ sin2<p . /i^+fflx ( } • r* h^l + r j - h ^ ' These formulae (4.2) and (4.3) for the Gaussian curvatures K and K* will be used in the proof of Theorems 1 and 2, but they also have their own significance. Backlund's theorem 55 5. Proof of Theorem 1. If r=const and <p=const, then from (4.2) and (4.3) we obtain (2), (3) and (4) immediately. Conversely, let K = K * = -(sm2<p)/r2. Then the same formulae (4.2) and (4.3) give correspondingly (5.1) fc12 - hu <p2 = ((sin2 cp)/r2) r x , hisr1-hnri = q>1. In case of (2) we have iT=const and from & a + s i n a ç » = 0 it follows that dr=r cot q> • d<p and the last two equations give sin2 cp •<p1=0. Therefore <p1=r1=0 and hl2(p2=hnr2—0. Here / t u = 0 would lead to <p=0 what is excluded, and we have (1). In case of (2) or (3), when r=const or (p=const, the same equations give (1). Theorem 1 is proved. 6. Proof of Theorem 2. The /i-asymptotic curves of V2 are the null curves of the second fundamental form in the direction e3. This form is IP = 0i Of+02 03 = hi.eigj\ For V2 it is II*3 = 0* !0 f+ 0*20|3 = (01+dr) 0 2 + - A - 0Ï (Gl+dcp). r sin (p Using here that h^h22—h\2=K and (4.2), we have N * 3 = sin (p where 4> = 4 ^ - [ h 1 1 ( 0 1 ) 2 + 2/ii20102] + i i E ^ - ' - 2 + - ^ — fc12<p2)(02)2. smq> \ r sm<p ) If r=const and cp =const, then <P=0, and we have (5). Conversely, let F2—F2* be /z-asymptotic. Then <P must be proportional to IIs, and therefore - , sin2<p (6.1) fc22<Pi=—r2+h12(p2. In case of (5) we have r~r cot cp • (pt. Now (5.1) and (6.1) give ( h l 2 — S 1 ° = 0 ' sin 2(p\ h22(pi~\h12+—27") = 0. 56 U. Lumiste Here the determinant is : = * O 4 r j ri and hence (1) holds. In case of (6), from (5.1) and (6.1) it follows that h i 2 ( p i - h n ( p 2 = 0, h22<Pi~h12(p2 = 0, where the determinant is —t%2+h11h22=K^0. In case of (7) it follows analogously that (1) holds. Theorem 2 is proved. 7. Proof of Theorem 3. If the field of directions e3 is parallel along the integral curves of the equation 0 2 =0 with respect to normal connection, then (7.1) 0* = A02. Taking the unit vectors ex = e1 cos a + e2 sin a, e2 = — et sin a + e2 cos a in /i-principal directions, bisecting /i-asymptotic directions we have fi1 2=0 and besides this 8f = 0| sin a = k22 sin a • 02, (7.2) = 0| cos a = k22 cos a • 02, Bf = 0. The local parameters v1 and v2 on V2 can be choosen so that B1 = a1dvl, 62 = a2dv2, B\=b1a1dv1, 0 | = b2a2dv2, where b2=h22. Then from the formulae ddl = 6lASl, ddl = -dfABl, which hold due to (7.1) and (7.2), using the well-known expression fia 1 da2 1 dax 0? = k-t dv2 j-v dv1, di dv1 a2 dv2 we have 1 dbt ddnaj . . bi-bj dv>~ dv* ' l9"h The same computation as in [2] leads us to parameters w1, w2, in which for V2 dsa = cos2 x (dw1)2+sin2 x (dw2)\ II8 = sin x cos x [(¿w1)2 - (dw2)2] Backlund's theorem 57 and now by M1=W1-|-»V2, M2=W1—w2 we get ds2 = (du1)2+2 cos iA du1 du2+(du2)2, II3 = 2 sin iA du1 du2. The A-asymptotic net is a Chebyshev net. Theorem 3 is proved. References [1] A. V. BÄCKLUND, Om utor med konstant negativ krökning, Lunds Univ. Arsskrift, 19 (1883). [2] S.-S. CHERN, C.-L. TERNG, An analogue of Bäcklund's theorem in affine geometry. Rocky Mountain J. Math., 10 (1) (1980), 105—124. [3] Ü. LUMISTE, L. TUULMETS, On Gaussian curvatures of local surfaces V2 of a line pseudocon-gruence in E„, Tartu Ülikooli Toimetised, Acta et Comm. Univ. Tartuensis, 665 (1984), 55—62. (Russian) [4] K . TENENBLAT, C . - L . TERNG, Bäcklund's theorem for «-dimensional submanifolds of R~n~1i Ann. Math. I l l (1980), 477—490. [5] C . - L . TERNG, A higher dimensional generalization of the Sine—Gordon equations and its soli ton theory, Ann. Math., I l l (1980), 491—510. [6] D. HILBERT, Über Flächen von konstanter Gausschen Krümmung, Trans. Amer. Math. Soc., ,2(1901), 87—99. [7] S. B. KADOMCEV, Impossibility of some special isometric immersions of the Lobachevsky space, Mat. Sbornik, 107 (1978), 175—198. (Russian) FACULTY OF MATHEMATICS TARTU STATE UNIVERSITY 202400 TARTU, USSR 

Bäcklund's theorem and transformation for surfaces V2 in En

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Bäcklund's theorem and transformation for surfaces V2 in En

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