Complete classification of parallel Lorentz surfaces in

Complete classification of parallel Lorentz surfaces in four-dimensional neutral pseudosphere Bang-Yen Chen Citation: Journal of Mathematical Physics 51, 083518 (2010); doi: 10.1063/1.3474915 View online: http://dx.doi.org/10.1063/1.3474915 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/51/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Fourdimensional Osserman metrics revisited AIP Conf. Proc. 1093, 35 (2009); 10.1063/1.3089205 Erratum: “Classification of marginally trapped Lagrangian surfaces in Lorentzian complex space form” [J. Math. Phys.48, 013509 (2007)] J. Math. Phys. 49, 059901 (2008); 10.1063/1.2929664 Four-dimensional indefinite Kähler Osserman manifolds J. Math. Phys. 46, 073505 (2005); 10.1063/1.1938727 Classification of the pseudosymmetric space–times J. Math. Phys. 45, 2343 (2004); 10.1063/1.1745129 On the classification of type D space–times J. Math. Phys. 45, 652 (2004); 10.1063/1.1640795

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JOURNAL OF MATHEMATICAL PHYSICS 51, 083518 共2010兲

Complete classification of parallel Lorentz surfaces in four-dimensional neutral pseudosphere Bang-Yen Chena兲 Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027, USA 共Received 21 November 2009; accepted 9 July 2010; published online 23 August 2010兲

A Lorentz surface of an indefinite space form is called parallel if its second fundamental form is parallel with respect to the Van der Waerden–Bortolotti connection. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in general relativity since extrinsic invariants of such surfaces do not change from point to point. Parallel Lorentz surfaces in four-dimensional 共4D兲 Lorentzian space forms are classified by Chen and Van der Veken 关“Complete classification of parallel surfaces in 4-dimensional Lorentz space forms,” Tohoku Math. J. 61, 1 共2009兲兴. Recently, explicit classification of parallel Lorentz surfaces in the pseudoEuclidean 4-space E42 and in the pseudohyperbolic 4-space H42共−1兲 are obtained recently by Chen et al. 关“Complete classification of parallel Lorentzian surfaces in Lorentzian complex space forms,” Int. J. Math. 21, 665 共2010兲; “Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space,” Cent. Eur. J. Math. 8, 706 共2010兲兴, respectively. In this article, we completely classify the remaining case; namely, parallel Lorentz surfaces in 4D neutral pseudosphere S42共1兲. Our result states that there are 24 families of such surfaces in S42共1兲. Conversely, every parallel Lorentz surface in S42共1兲 is obtained from one of the 24 families. The main result indicates that there are major differences between Lorentz surfaces in the de Sitter 4-space dS4 and in the neutral pseudo 4-sphere S42. © 2010 American Institute of Physics. 关doi:10.1063/1.3474915兴

I. INTRODUCTION

Let Em t denote the pseudo-Euclidean m-space with the canonical pseudo-Euclidean metric of index t given by t

g0 = − 兺 dx2i + i=1

m

dx2j , 兺 j=t+1

共1.1兲

where 共x1 , . . . , xm兲 is a rectangular coordinate system of Em t . We put Ssk共c兲 = 兵x 苸 Esk+1兩具x,x典 = c−1 ⬎ 0其,

共1.2兲

k+1 Hsk共− c兲 = 兵x 苸 Es+1 兩具x,x典 = − c−1 ⬍ 0其,

Ssk共c兲

共1.3兲

Hsk共−c兲

and are complete pseudowhere 具 , 典 is the indefinite inner product on Riemannian manifolds with index s and of constant curvatures c and −c, which are called pseudo-k-sphere and pseudohyperbolic k-space, respectively. These Esk, Ssk, and Hsk are known as Ek+1 t .

a兲

Electronic mail: [email protected].

0022-2488/2010/51共8兲/083518/22/$30.00

51, 083518-1

© 2010 American Institute of Physics

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083518-2

J. Math. Phys. 51, 083518 共2010兲

Bang-Yen Chen

indefinite space forms. In particular, Ek1, Sk1, and Hk1 are called Minkowski, de Sitter, and anti-de Sitter space-times in relativity, respectively. The needs of physics aside, Lorentzian geometry is of mathematical interest for its unique causal structure. One may wonder whether any other indefinite signature warrants special attention beyond the general theory of pseudo-Riemannian geometry. Neutral signature is indeed of special interest for, while an indefinite geometry, it retains many interesting parallels with Riemannian geometry. Such parallels are particularly evident in four dimensions, where the Hodge star operator is involutory for both positive-definite and neutral signatures. Thus, both signatures possess the decomposition of two-forms into self-dual and anti-self-dual parts, without the need to complexify as in the Lorentzian case. Parallel surfaces are those which have parallel second fundamental form. Such surfaces are locally invariant under the reflection with respect to the normal space at each point 共cf. Refs. 1, 10, and 16兲. Moreover, extrinsic invariants of a parallel surface do no change from point to point. Hence, parallel surfaces form a natural and important family of surfaces in geometry as well as in general relativity. For the classification of parallel surfaces in Riemannian space forms, we refer to Refs. 1, 10, and 4. Some special families of parallel surfaces in indefinite space forms were studied in Refs. 1 and 11–13. The full classification of parallel Lorentz surfaces in four-dimensional 共4D兲 Lorentz space forms was achieved in Ref. 9 共see Theorem 3.2兲. Recently, parallel Lorentz surfaces in the 4D neutral pseudo-Euclidean 4-space E42 and the neutral pseudohyperbolic 4-space H42共−1兲 are classified in Refs. 7 and 6, respectively. Moreover, parallel spacelike surfaces in indefinite space forms are classified in Ref. 5. In this article we completely classify the remaining case; namely, parallel Lorentz surfaces in the neutral pseudosphere S42共1兲. Our main theorem states that there exist 24 families of parallel Lorentz surfaces in S42共1兲. Conversely, every parallel Lorentz surface in S42共1兲 is obtained from one of the 24 families. By comparing the main theorem of this article with Theorem 8.2 of Ref. 9, we see that there are major differences between Lorentz surfaces in the de Sitter 4-space dS4 and in the neutral pseudo 4-sphere S42. II. PRELIMINARIES A. Basic notations and formulas

Let Rsm共c兲 be a pseudo-Riemannian m-manifold of constant sectional curvature c and with index s. The curvature tensor ˜R of Rsm共c兲 is given by ˜R共X,Y兲Z = c兵具Y,Z典X − 具X,Z典Y其.

共2.1兲

A vector v tangent to Rsm共c兲 is called spacelike 共respectively, timelike, or lightlike兲 if 具v , v典 ⬎ 0 共respectively, 具v , v典 ⬍ 0, or v ⫽ 0 and 具v , v典 = 0兲. A surface is called Lorentz if its metric is Lorentzian. Let ␺ : M → Rsm共c兲 be an isometric ˜ the Levi–Civita connections on immersion of a Lorentz surface M into Rsm共c兲. Denote by ⵜ and ⵜ m M and Rs 共c兲, respectively. For vector fields X , Y tangent to M and vector field ␰ normal to M, the formulas of Gauss and Weingarten are given, respectively, by 共cf. Refs. 2, 3, and 14兲 ˜ Y = ⵜ Y + h共X,Y兲, ⵜ X X

共2.2兲

˜ ␰ = − A X + D ␰, ⵜ X ␰ X

共2.3兲

˜ Y where ⵜXY 共−A␰X兲 is the tangent component and h共X , Y兲 共DX␰兲 the normal component of ⵜ X ˜ ␰兲. These formulas define the second fundamental form h, the shape operator A, and the normal 共ⵜ X connection D of M.

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083518-3

Parallel Lorentz surfaces in 4D pseudosphere

J. Math. Phys. 51, 083518 共2010兲

For each ␰ 苸 T⬜ x M, the shape operator A␰ is a symmetric endomorphism of the tangent space TxM at x 苸 M. The shape operator and the second fundamental form are related by 具h共X,Y兲, ␰典 = 具A␰X,Y典

共2.4兲

for X , Y tangent to M and ␰ normal to M. The mean curvature vector H of M in Rsm共c兲 is defined by 1 H = trace h. 2

共2.5兲

The equations of Gauss, Codazzi, and Ricci are given, respectively, by R共X,Y兲Z = c兵具Y,Z典X − 具X,Z典Y其 + Ah共Y,Z兲X − Ah共X,Z兲Y ,

共2.6兲

¯ h兲共Y,Z兲 = 共ⵜ ¯ h兲共X,Z兲, 共ⵜ X Y

共2.7兲

具RD共X,Y兲␰, ␩典 = 具关A␰,A␩兴X,Y典

共2.8兲

¯ h is defined by for vector X , Y , Z tangent to M , ␰ , ␩ normal to M, where ⵜ ¯ h兲共Y,Z兲 = D h共Y,Z兲 − h共ⵜ Y,Z兲 − h共Y,ⵜ Z兲, 共ⵜ X X X X

共2.9兲

D

and R is the curvature tensor associated with the normal connection D, i.e., RD共X,Y兲␰ = DXDY ␰ − DY DX␰ − D关X,Y兴␰ .

共2.10兲

A normal vector field ␰ on M is called a parallel normal vector if D␰ = 0 holds identically. The surface M is said to have parallel mean curvature vector if the mean curvature vector of M ¯ h = 0, i.e., h is parallel with satisfies DH = 0 identically. A surface M is called parallel if we have ⵜ ¯ respect to the Van der Waerden–Bortolotti connection ⵜ. A Lorentz surface M in a pseudo-Riemannian manifold is called quasiminimal if the mean curvature vector H is lightlike at each point in M.

B. Lemmas

The following is an easy consequence of Ricci’s equation 共see Ref. 5兲. Lemma 2.1: Let M be a Lorentz surface with parallel mean curvature vector H in a pseudoRiemannian m -manifold Rsm共c兲 . Then 具H , H典 is constant. Moreover, either AH is proportional to the identity map or the normal connection is flat. In particular, if 具H , H典 ⫽ 0 and m = 4 , the normal connection is always flat. The following lemma is useful to check whether a flat submanifold in an indefinite space form is parallel. Lemma 2.2: Let L : M nt → Esm be an isometric immersion of a pseudo-Riemannian manifold M nt into Esm and 兵x1 , . . . , xn其 be a coordinate system of M nt , such that gij共1 ⱕ i , j ⱕ n兲 are constants. 3 Then M nt is a parallel submanifold in Esm if and only if Lijk ª ⳵⳵i⳵ jL⳵k 共1 ⱕ i , j , k ⱕ n兲 are tangent to M nt . ¯ h = 0 if and only if Proof: Under the hypothesis, we have ⵜ⳵i⳵ j = 0. It follows from 共2.9兲 that ⵜ D⳵kh共ei , e j兲 = 0 for i , j , k = 1 , . . . , n. Thus, we find from 共2.2兲 that Lij = h共ei , e j兲. Hence, by Weingarten’s formula, we have Lijk = −Ah共⳵i,⳵ j兲⳵k, which is tangent to M for each i , j , k. Conversely, assume that Lijk are tangent to M nt , then by Lij = h共ei , e j兲, we have D⳵kh共⳵i , ⳵ j兲 ¯ h = 0. = 0. Thus, we get ⵜ 䊏

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Bang-Yen Chen

C. Moving frames

Let M be a Lorentz surface in a 4D neutral indefinite space form R42共c兲. Let 兵e1 , e2其 be a local tangent frame and 兵e3 , e4其 a local normal frame, such that 具e1,e1典 = 具e2,e2典 = 0,

具e1,e2典 = − 1,

共2.11兲

具e3,e3典 = 具e4,e4典 = 0,

具e3,e4典 = − 1.

共2.12兲

It follows from 共2.5兲 and 共2.11兲 that the mean curvature vector of M is given by H = − h共e1,e2兲.

共2.13兲

We define one-forms ␻ and ␾ by equations ⵜXe1 = ␻共X兲e1,

ⵜXe2 = − ␻共X兲e2 ,

共2.14兲

DXe3 = ␾共X兲e3,

DXe4 = − ␾共X兲e4 .

共2.15兲

III. PARALLEL LORENTZ SURFACES IN E24 AND IN S14

The neutral pseudo-Euclidean 4-space E42 can be identified with the flat complex Lorentzian plane C21 whenever E42 is equipped with the standard complex coordinates z1 = x1 + ix2 , z2 = x3 + ix4. The following result of Chen et al. in Ref. 7 completely classifies parallel Lorentz surfaces in E42 ⬵ C21. Theorem 3.1: A Lorentz surface M with parallel second fundamental form in the complex Lorentzian plane C21 is congruent to an open portion of one of the following nine types of surfaces: (1) (2) (3) (4) (5) (6) (7) (8) (9)

a Lorentzian totally geodesic surface; a Lorentzian product of parallel curves; a complex circle, given by 共a + ib兲共cos共x + iy兲 , sin共x + iy兲兲 , 0 ⫽ a , b 苸 R ; a B -scroll over the null cubic in E31 債 C21 ; a B -scroll over the null cubic in E32 債 C21 ; −iy a surface given by e冑2 共i共1 + a兲 − x − ay , i共1 − a兲 + x + ay兲 , a 苸 R ; a surface with lightlike mean curvature vector given by 共q共x , y兲 , x , y , q共x , y兲兲 , with q共x , y兲 = ax2 + bxy + cy 2 + dx + ey + f and a , b , c , d , e , f 苸 R ; a totally umbilical surface given by a共0 , sinh x , cosh x cos y , cosh x sin y兲 , a ⬎ 0 : a totally umbilical surface given by a共sin x , cos x cosh y , cos x sinh y , 0兲 , a ⬎ 0 .

Conversely, each of the surfaces listed above is a Lorentzian surface with parallel second fundamental form in C21 . The following classification of parallel Lorentz surfaces in de Sitter 4-space S41 was obtained in Ref. 9,Theorem 8.2, by Chen and Van der Veken. Theorem 3.2: If M is a Lorentzian parallel surface in S41共1兲 傺 E51 , then M is congruent to an open part of one of the following two types of surfaces: (1) (2)

a totally umbilical surface given by 共a sinh u , a cosh u cos v , a cosh u sin v , b , 0兲 with a2 + b2 = 1 ; a flat surface given by 共a sinh u , a cosh u , b cos v , b sin v , c兲 with a2 + b2 + c2 = 1 .

Conversely, each surface defined above is a Lorentzian parallel surface in S41共1兲 . Remark 3.1: There is a typo in the statement of Ref. 9, Theorem 8.2. The above one corrects the error.

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083518-5

J. Math. Phys. 51, 083518 共2010兲

Parallel Lorentz surfaces in 4D pseudosphere

IV. CLASSIFICATION OF PARALLEL LORENTZ SURFACES IN S24„1…

Let ␺ : M → S42共1兲 be an isometric immersion of a Lorentz surface M into the neutral pseudo 4-sphere S42共1兲. Let L = ␫ ⴰ ␺ : M → S42共1兲 傺 E52 be the composition of ␺ with the inclusion map ␫ given by 共1.2兲. Denote by ⵜˆ the Levi–Civita ˆ the normal connection of M in E5. connection of E52 and by D 2 ˆ Let h and h be the second fundamental forms of M in S42共1兲 and in E52, respectively. Then we have hˆ共X,Y兲 = h共X,Y兲 − g共X,Y兲L

共4.1兲

ˆ L = 0 holds, we have D ˆ ␰ = D␰ for any normal vector field of M in for X , Y tangent to M. Since D S42共1兲. Moreover, it is easy to verify that M is a parallel surface in S42共1兲 if and only if it is a parallel surface in E52. Thus, one may apply Lemma 2.2 to determine whether a flat surface in S42共1兲 is parallel. Main Theorem:There are 24 families of parallel Lorentz surfaces in the neutral pseudo 4-sphere S42共1兲 傺 E52 : (1) (2)

a totally geodesic de Sitter space-time S21共1兲 傺 S42共1兲 傺 E52 ; a flat surface in a totally geodesic S31共1兲 傺 S42共1兲 defined by 共冑a2 + b2 − 1,a sinh u,a cosh u,b cos v,b sin v兲,a,b ⬎ 0,a2 + b2 ⱖ 1;

(3)

a flat surface defined by 共a cos u sinh v + b sin u cosh v, 冑a2 + b2 sin u sinh v, 冑a2 + b2 sin u cosh v, a cos u cosh v + b sin u sinh v, 冑1 − a2兲,a 苸 共0,1兴.

(4)

a flat surface defined by 共a cos u,a sin u,b cos v,b sin v, 冑1 + a2 − b2兲,a,b ⬎ 0,b2 ⱕ 1 + a2;

(5)

a flat surface defined by

冉 (6)

ku,pu2 +

冊

共1 − b2兲␸ k2 共1 − b2兲␸ k2 2 − ,b sin ,b cos ,pu + + ,b,k,p, ␸ ⫽ 0; v v k2 4␸ k2 4␸

a flat surface defined by 共冑b2 − a2 − 1,a cosh u,a sinh u,b cos v,b sin v兲,a,b ⬎ 0,b2 ⱖ 1 + a2;

(7)

a flat surface defined by

冉 (8)

pu2 +

冊

共b2 − 1兲␸ k2 共b2 − 1兲␸ k2 + ,b sinh v,b cosh v,ku,pu2 + − ,b,k,p, ␸ ⫽ 0; 2 k 4␸ k2 4␸

a flat surface defined by 共a cosh u,b sinh v,a sinh u,b cosh v, 冑1 + a2 − b2兲,a,b ⬎ 0,b2 ⱕ 1 + a2;

(9)

a quasiminimal surface of constant curvature one defined by

冉

冊

2 x − y 2 + xy xy , , , ,0 , x+y x+y x+y x+y

x + y ⫽ 0;

(10) a flat surface defined by 共x + xy , y − xy , x − y + xy , 1 + xy , 0兲 ;

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083518-6

J. Math. Phys. 51, 083518 共2010兲

Bang-Yen Chen

(11) a surface of positive curvature c2 defined by

冉

冊

xy − c2 2冑1 − c2y xy + c2 c2共x + y兲 − 2y , , , ,0 ,c 苸 共0,1兲,x + y ⫽ 0; c2共x + y兲 c2共x + y兲 c2共x + y兲 c2共x + y兲

(12) a surface of positive curvature c2 defined by

冉

0,

冊

xy − c2 xy + c2 c2共x + y兲 − 2y 2冑c2 − 1y ,c ⬎ 1,x + y ⫽ 0; , , , 2 c2共x + y兲 c2共x + y兲 c2共x + y兲 c 共x + y兲

(13) a surface of negative curvature −c2 defined by 1 共cosh u − sinh u tanh v,sinh u tanh v,sinh u − cosh u tanh v, 冑1 + c2,0兲,c ⬎ 0; c (14) a flat surface defined by

冉

冉 冊

1+v 4c2 − 1 1 + 8c2 + 2v v cos u + sin u, cos u + c + sin u, 4c 2c 4c 2c

冉

冊

冊

1 + 2c2 + v 1 v sin u 4c2 + 1 v + 2c + , cos u + sin u,0 ,c ⬎ 0; cos u + 4c 2c 2c 4c 2c

(15) a flat surface defined by

冉

eu −

冊

共2c − v兲e−u veu e−u u 共2c − v兲e−u veu e−u , − ,e + , + ,0 ,c ⬎ 0; 8c 4 2c 8c 4 2c

(16) a flat surface defined by

冉

x+

冊

y 2c2y 3 c2y 4 c2y 4 y 2c2y 3 2 + ,c ⬎ 0; ,xy + ,x − + ,cy ,1 + xy + 3 6 3 6 2 2

(17) a flat surface defined by 共av sinh u + b cosh u,av cosh u,av cosh u + b sinh u,av sinh u, 冑1 + b2兲,a,b ⫽ 0; (18) a flat surface defined by 共a sin u − bv cos u,a cos u + bv cos u,bv cos u,bv sin u, 冑1 + a2兲,a,b ⫽ 0; (19) a flat surface defined by

冉

v cos u +

冊

sin u cos u sin u cos u , v sin u − , v cos u − , v sin u + ,1 ,c ⬎ 0; c c c c

(20) a flat surface defined by

冉

cos u cos v −

sin u sin v sin u cos v sin u sin v ,cos u sin v + ,cos u cos v + , c c c

cos u sin v −

冊

sin u cos v ,1 ,c ⬎ 0; c

(21) a flat surface defined by

冉

ev cos u +

冊

e−v sin u −v ev sin u v e−v sin u −v ev sin u ,e cos u − ,e cos u − ,e cos u + ,1 ,c ⬎ 0; c c c c

(22) a flat surface defined by

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083518-7

J. Math. Phys. 51, 083518 共2010兲

Parallel Lorentz surfaces in 4D pseudosphere

共eu + ae−uv,euv − ae−u,eu − ae−uv,euv + ae−u,1兲,a ⫽ 0; (23) a flat surface defined by 共eu − ae−u,ev + ae−v,eu + ae−u,ev − ae−v,1兲,a ⫽ 0.; (24) a flat surface defined by 共a cosh u cos v,a cosh u sin v,a sinh u,cos v,a sinh u sin v, 冑1 + a2兲,a ⬎ 0. Conversely, every parallel immersion L : M → S42共1兲 傺 E52 of a Lorentz surface M into the pseudo 4-sphere S42共1兲 is congruent to a surface obtained from one of 24 families of surfaces described above. Proof: With the help of Lemma 2.2, we may verify that each of the 24 families of surfaces described above defines a parallel Lorentz surface in S42共1兲 by long direct computations. Conversely, assume that ␺ : M → S42共1兲 is an isometric immersion of a Lorentz surface M into 4 S2共1兲 and L = ␫ ⴰ ␺. We choose a frame 兵e1 , e2 , e3 , e4其 satisfying 共2.11兲 and 共2.12兲. Define the connection 1-forms ␻ and ␾ by 共2.14兲 and 共2.15兲. Suppose that M is a parallel surface in S42共1兲. Then M has constant Gauss curvature and parallel mean curvature vector H. Thus, 具H , H典 is constant according to Lemma 2.1. Hence, one of the following three cases occurs: 共a兲 共b兲 共c兲

M is a minimal surface in S42共1兲; M is a quasiminimal surface in S42共1兲; 具H , H典 is a nonzero constant.

Case (a): M is minimal in S42共1兲. If M is totally geodesic, we obtain Case 共1兲. Next, assume that M is nontotally geodesic in S42共1兲. Then, according to 共2.13兲, the second fundamental form h satisfies h共e1,e1兲 = ␣e3 + ␤e4,h共e1,e2兲 = 0,h共e2,e2兲 = ␭e3 + ␮e4

共4.2兲

for some functions ␣ , ␤ , ␭ , ␮, not all zero. After replacing e3 , e4 by ␣e3 and ␣ e4, respectively, 共4.2兲 reduces to −1

h共e1,e1兲 = e3 + ␥e4,

h共e2,e2兲 = ␦e3 + ␸e4

h共e1,e2兲 = 0,

共4.3兲

for some functions ␥ , ␦ , ␸. Hence, the shape operator of M satisfies A e3 =

冉 冊 0 ␸

␥ 0

,

A e4 =

冉 冊 0 ␦

1 0

.

共4.4兲

By using Eqs. 共2.8兲 of Ricci and 共4.4兲, we obtain 具RD共e1,e2兲e3,e4典 = ␦␥ − ␸ .

共4.5兲

Also, it follows from 共4.3兲 and Gauss’s equation that the Gauss curvature satisfies K = ␸ + ␥␦ + 1.

共4.6兲

Since M has parallel second fundamental form, Eqs. 共2.9兲 and 共4.3兲 imply that ␥ , ␦ , ␸ are constant. Further, we have

␾ = 共1 − ␦兲␻ = 共␥ − ␸兲␻ = 0.

共4.7兲

If ␻ ⫽ 0 holds, we get ␦ = 1 and ␸ = ␥. Thus, 共4.3兲 reduces to

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J. Math. Phys. 51, 083518 共2010兲

Bang-Yen Chen

h共e1,e1兲 = e3 + ␥e4,

h共e1,e2兲 = 0,

h共e2,e2兲 = e3 + ␥e4 .

共4.8兲

¯ h = 0, we find from Eqs. 共2.9兲, 共2.14兲, 共2.15兲, and 共4.8兲 that ␻ = 0, which Hence, by applying ␾ = ⵜ is a contradiction. Therefore, we must have ␾ = ␻ = 0. In particular, ␻ = 0 implies that K = 0. On the other hand, it follows from ␾ = 0 and Eq. 共4.5兲 that ␸ = ␥␦. Combining this with Eq. 共4.6兲 and K = 0 gives 1 ␸=− , 2

␦=−

1 , 2␥

␥ ⫽ 0.

共4.9兲

Because ␻ = 0 holds, there are coordinates 兵x , y其, such that e1 = ⳵⳵x and e2 = ⳵⳵y . It follows from Eqs. 共4.1兲, 共4.8兲, and 共4.9兲 that L : M → S42共1兲 傺 E52 satisfies Lxx = e3 + ␥e4,

Lxy = L,

Lyy = −

e3 e4 − . 2␥ 2

共4.10兲

ˆ ␰ = D␰ for any normal vector field of M in S4共1兲, we Moreover, by using ␾ = 0, Eq. 共4.4兲, and that D 2 obtain from Eqs. 共2.3兲 and 共4.10兲 that ⵜˆ ⳵/⳵xe3 = − ␥Ly,

ˆ e = Lx , ⵜ ⳵/⳵ y 3 2

ˆ e =−L , ⵜ ⳵/⳵x 4 y

ˆ e = Lx . ⵜ ⳵/⳵ y 4 2␥

共4.11兲

Case (a.1): ␥ = −k4 / 2 ⬍ 0. After solving system 共4.10兲 and 共4.11兲, we obtain y u = kx + , k

L = c1 cosh u + c2 sinh u + c3 cos v + c4 sin v,

v = kx −

y k

for some vectors c1 , c2 , c3 , c4 苸 E52. Thus, after choosing suitable initial conditions, we obtain a special case of Case 共2兲. Case (a.2): ␥ = 2k4 ⬎ 0. After solving system 共4.10兲 and 共4.11兲, we find L = cos v共c1 cosh u + c2 sinh u兲 + sin v共c3 cosh u + c4 sinh u兲, y u = kx + , k

y v = kx − . k

So, after choosing suitable initial conditions, we obtain a special case of Case 共3兲. Case (b): M is quasiminimal in S42共1兲. In this case, the mean curvature vector H is lightlike. Thus, for any pseudo-orthonormal frame 兵e1 , e2其 satisfying 共2.11兲, we get h共e1 , e2兲 = −H, which is lightlike. Hence, we may choose pseudo-orthonormal frame 兵e3 , e4其, such that h共e1,e1兲 = ␤e3 + ␥e4,

h共e2,e2兲 = ␦e3 + ␸e4

h共e1,e2兲 = e3,

共4.12兲

for some functions ␤ , ␥ , ␦ , ␸. For such a frame 兵e1 , e2 , e3 , e4其, the shape operator satisfies A e3 =

冉 冊

0 ␸ , ␥ 0

A e4 =

冉 冊

1 ␦ . ␤ 1

共4.13兲

Since the second fundamental form is parallel, Eqs. 共2.9兲 and 共4.12兲 imply that

␾ = 0,

d␤ = 2␤␻,

d␥ = 2␥␻,

d␦ = − 2␦␻,

d␸ = − 2␸␻ .

共4.14兲

From ␾ = 0 and the equation of Ricci, we get ␤␸ = ␥␦. By combining this with the equation of Gauss, we derive that 2␤␸ = 2␥␦ = K − 1.

共4.15兲

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083518-9

J. Math. Phys. 51, 083518 共2010兲

Parallel Lorentz surfaces in 4D pseudosphere

On the other hand, it follows from Eq. 共4.14兲 that d共␤␦兲 = d共␥␸兲 = 0. Therefore, both ␤␦ , ␥␸ are constant. Case (b.1): K = 0. We choose coordinates 兵x , y其 such that ⳵⳵x = e1 and ⳵⳵y = e2. So, we have g = − 共dx 丢 dy + dy 丢 dx兲,

共4.16兲

which gives ␻ = 0. Hence, ␤ , ␥ , ␦ , ␸ are constant according to 共4.14兲. Now, it follows from Eqs. 共4.12兲 and 共4.15兲 that Lxx = ␥共ae3 + e4兲,

Lxy = e3 + L,

Lyy = −

ae3 + e4 , 2a␥

a=

␤ . ␥

共4.17兲

Moreover, since ␾ = 0, we obtain from 共4.13兲 that ⵜˆ ⳵/⳵xe3 = − ␥Ly,

ˆ e = Lx , ⵜ ⳵/⳵ y 3 2a␥ ˆ e = Lx − L . ⵜ ⳵/⳵ y 4 y 2␥

ˆ e = − L − a␥L , ⵜ ⳵/⳵x 4 x y

共4.18兲

By applying Eqs. 共4.17兲 and 共4.18兲, we obtain the following differential equation: L共xv兲 + 2␥Lxxx + 共1 + 2a兲␥2Lx = 0.

共4.19兲

Case (b.1.1): a ⬍ 0 and ␥ ⬎ 0. If we put a = −b2 / 2 ; ␥ = c2 with b , c ⬎ 0, then after solving Eq. 共4.19兲, we obtain one of the following: L = H共y兲 + A共y兲cos共c冑1 + bx兲 + B共y兲sin共c冑1 + bx兲 + C共y兲cos共c冑1 − bx兲 + D共y兲sin共c冑1 − bx兲, 共4.20兲 L = H共y兲 + A共y兲x + B共y兲x2 + C共y兲cos共冑2cx兲 + D共y兲sin共冑2cx兲,

共4.21兲

L = H共y兲 + A共y兲cos共c冑1 + bx兲 + B共y兲sin共c冑1 + bx兲 + C共y兲cosh共c冑b − 1x兲 + D共y兲sinh共c冑b − 1x兲 共4.22兲 according to b 苸 共0 , 1兲, b = 1, or b ⬎ 1, respectively. Case (b.1.1.1): b 苸 共0 , 1兲. Substituting Eq. 共4.20兲 into systems 共4.17兲 and 共4.18兲 yields L共x,y兲 = c0 + c1 cos u + c2 sin u + c3 cos v + c4 sin v ,

u=

冑1 + b共bc2x + y兲 bc

,

v=

冑1 − b共bc2x − y兲 bc

.

Thus, after choosing suitable initial conditions, we obtain a special case of Case 共4兲. Case (b.1.1.2): b = 1. Substituting Eq. 共4.21兲 into systems 共4.17兲 and 共4.18兲 yields L = c0 +

u共c2c1 + c2u兲 + c3 cos v + c4 sin v , c4

u = c2x + y,

v=

冑2 c

共c2x − y兲.

This gives a special case of Case 共5兲. Case (b.1.1.3): b ⬎ 1. Substituting Eq. 共4.22兲 into systems 共4.17兲 and 共4.18兲 yields

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083518-10

J. Math. Phys. 51, 083518 共2010兲

Bang-Yen Chen

L = c0 + c1 cos v + c2 sin v + c3 cosh u + c4 sinh u,

u=

冑b − 1共bc2x + y兲 bc

v=

,

冑1 + b共bc2x − y兲 bc

.

Thus, after choosing suitable initial conditions, we obtain a special case of Case 共6兲. Case (b.1.2): a ⬎ 0 and ␥ ⬍ 0. Let we put a = b2 / 2 , ␥ = −c2 with b , c ⬎ 0. Then after solving systems 共4.17兲 and 共4.18兲, we get L = c0 + cos u共c1 cosh v + c2 sinh v兲 + sin u共c3 cosh v + c4 sinh v兲,

u = px − qy,

v = bc2qx +

py , bc2

p=

c冑冑1 + b2 − 1

冑2

,

冑冑1 + b2 + 1

q=

冑2bc

.

共4.23兲

This yields Case 共6兲 after replacing v by −v. Case (b.1.3): a , ␥ ⬎ 0. If we put a = b2 / 2 , ␥ = c2 , b , c ⬎ 0, then after solving systems 共4.17兲 and 共4.18兲 in a similar way as Case 共b.1.2兲, we obtain a special case of Case 共3兲. Case (b.1.4): a , ␥ ⬍ 0. If we put a = −b2 / 2 , ␥ = −c2 with b , c ⬎ 0, then after solving Eq. 共4.19兲 we get one of the following: L = A共y兲 + B共y兲x + C共y兲x2 + P共y兲e L = H共y兲 + A共y兲e

冑1+bcx

+ B共y兲e−

冑2cx

冑1+bcx

+ Q共y兲e−

+ P共y兲e

冑2cx

冑1−bcx

共4.24兲

, + Q共y兲e−

L = H共y兲 + A共y兲cos共冑b − 1cx兲 + B共y兲sin共冑b − 1cx兲 + P共y兲e

冑1−bcx

冑1+bcx

共4.25兲

,

+ Q共y兲e−

冑1+bcx

共4.26兲

depending on b = 1, b 苸 共0 , 1兲, or b ⬎ 1, respectively. Case (b.1.4.1): b = 1. Substituting Eq. 共4.24兲 into 共4.17兲 and 共4.18兲 yields L共x,y兲 = c0 +

c1 2 c2 2 冑2共cx+y/c兲 冑 2 + c4e− 2共cx+y/c兲 . 2 共c x − y兲 + 4 共c x − y兲 + c3e c c

So, after choosing suitable initial conditions and changing of variables, we obtain a special case of Case 共7兲. Case (b.1.4.2): b 苸 共0 , 1兲. Substituting Eq. 共4.25兲 into 共4.17兲 and 共4.18兲 gives L共x,y兲 = c0 + c1 cosh v + c2 sinh v + c3 cosh u + c4 sinh u,

u=

冑1 − b bc

共bc2x − y兲,

v=

冑1 + b bc

共bc2x + y兲.

After choosing suitable initial conditions, we obtain a special case of Case 共8兲. Case (b.1.4.3): b ⬎ 1. Substituting Eq. 共4.26兲 into systems 共4.17兲 and 共4.18兲 yields L共x,y兲 = c0 + c1 cos v + c2 sin v + c3 cosh v + c4 sinh u,

u=

冑b + 1 bc

共bc2cx + y兲,

v=

冑b − 1 bc

共bc2cx − y兲.

This gives a special case of Case 共2兲. Case (b.2): K = 1. We divide this into two cases. Case (b.2.i): ␤ , ␥ , ␦ , ␸ = 0. We may choose coordinates 兵x , y其, such that

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083518-11

J. Math. Phys. 51, 083518 共2010兲

Parallel Lorentz surfaces in 4D pseudosphere

冑2 ⳵ = e 1, ⳵x x + y

冑2 ⳵ = e2 . ⳵y x + y

共4.27兲

So, the metric tensor is given by g=−

2 共dx 丢 dy + dy 丢 dx兲. 共x + y兲2

共4.28兲

Hence the Levi–Civita connection satisfies ⵜ ⳵/⳵x

⳵ −2 ⳵ = , ⳵x x + y ⳵x

ⵜ ⳵/⳵x

⳵ = 0, ⳵y

ⵜ ⳵/⳵ y

⳵ −2 ⳵ = . ⳵y x + y ⳵y

共4.29兲

Thus, Eqs. 共4.12兲, 共4.27兲, and 共4.29兲 imply that L : M → S42共1兲 傺 E52 satisfies Lxx = −

2Lx , x+y

Lxy =

2e3 + 2L , 共x + y兲2

Lyy = −

2Ly . x+y

共4.30兲

Moreover, since ␾ = 0, we obtain from Eq. 共4.13兲 that ˆ e = 0, ˆ e =ⵜ ⵜ ⳵/⳵x 3 ⳵/⳵ y 3

ⵜˆ ⳵/⳵xe4 = − Lx,

ˆ e =−L . ⵜ ⳵/⳵ y 4 y

共4.31兲

After solving Eqs. 共4.30兲 and 共4.31兲, we obtain L共x,y兲 = c1 + c2y +

c4 + c3y − c2y 2 . x+y

共4.32兲

Thus, after choosing suitable initial conditions, we obtain Case 共9兲. Case (b.2.ii): ␤ , ␥ , ␦ , ␸ are not all zero. It follows from Eq. 共4.14兲 that ␻ is a closed form. Thus, the structure equation implies that M is flat, which is a contradiction. Case (b.3): K ⫽ 0 , 1. It follows from Eq. 共4.15兲 that ␤ , ␥ , ␦ , ␸ are nowhere zero. So, just as in Case 共b.2.ii兲, we conclude that M is flat, which is a contradiction. Case (c): 具H , H典 ⫽ 0. In this case, H is either spacelike or timelike. Since M has parallel second fundamental form, H is a parallel normal vector field of constant length according to Lemma 2.1. Let us put H = ␭e3 + ␮e4 with ␭ , ␮ ⫽ 0. Then after replacing ␭e3 , e4 / ␭ by e3 , e4, respectively, we have H = e3 + be4 for some real number b ⫽ 0. This implies that 具H , H典 = −2b. Thus, we have either b ⬎ 0 or b ⬍ 0, depending on H is timelike or spacelike, respectively. Now, it follows from h共e1,e1兲 = ␤e3 + ␥e4,

h共e1,e2兲 = e3 + be4,

h共e2,e2兲 = ␦e3 + ␸e4

共4.33兲

that the shape operator of M satisfies A e3 =

冉 冊

b ␸ , ␥ b

A e4 =

冉 冊 1 ␦

␤ 1

共4.34兲

for some functions ␤ , ␥ , ␦ , ␸, Because H = e3 + be4 is a parallel normal vector field, we have ␾ = 0. Thus, after applying 共4.34兲 and equations of Ricci and Gauss, we find 2␤␸ = 2␥␦ = K + 2b − 1.

共4.35兲

Also, since M has parallel second fundamental form, Eqs. 共2.9兲 and 共4.12兲 yield d␤ = 2␤␻,

d␥ = 2␥␻,

d␦ = − 2␦␻,

d␸ = − 2␸␻ .

共4.36兲

Case (c.1): ␤ = ␥ = ␦ = ␸ = 0. Equations 共4.33兲 and 共4.34兲 reduce to

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083518-12

J. Math. Phys. 51, 083518 共2010兲

Bang-Yen Chen

h共e1,e1兲 = 0,

h共e1,e2兲 = e3 + be4,

A e3 =

冉 冊 b 0

0 b

,

A e4 =

h共e2,e2兲 = 0,

冉 冊 1 0 0 1

共4.37兲 共4.38兲

.

Case (c.1.1): b = 21 . In this case, M is flat. So, we may choose coordinates 兵x , y其, such that ⳵ / ⳵x = e1 , ⳵ / ⳵y = e2. Thus, we have g = − 共dx 丢 dy + dy 丢 dx兲. Therefore, the immersion L : M → S42共1兲 傺 E52 satisfies Lxx = 0,

Lxy = e3 +

e4 + L, 2

共4.39兲

Lyy = 0.

Moreover, since ␾ = 0, we obtain from Eq. 共4.38兲 that ˆ e = − Lx , ⵜ ⳵/⳵x 3 2

ˆ e = − Ly , ⵜ ⳵/⳵ y 3 2

ⵜˆ ⳵/⳵xe4 = − Lx,

ˆ e =−L . ⵜ ⳵/⳵ y 4 y

共4.40兲

Hence, after solving Eqs. 共4.39兲 and 共4.40兲, we obtain 共4.41兲

L共x,y兲 = c1 + c2x + c3y + c4xy,

which yields Case 共10兲. Case (c.1.2): b ⬍ 21 . Since K = 1 − 2b ⬎ 0, we put K = c2. Because b ⫽ 0, we get c ⫽ 1. Let us choose coordinates 兵x , y其, such that

冑2e1 ⳵ = , ⳵ x c共x + y兲

冑2e2 ⳵ = . ⳵ y c共x + y兲

共4.42兲

So, the metric tensor is given by g=−

2 共dx 丢 dy + dy 丢 dx兲, c 共x + y兲2

共4.43兲

2

and hence the Levi–Civita connection satisfies ⵜ ⳵/⳵x

⳵ −2 ⳵ = , ⳵x x + y ⳵x

ⵜ ⳵/⳵x

⳵ = 0, ⳵y

ⵜ ⳵/⳵ y

⳵ −2 ⳵ = . ⳵y x + y ⳵y

共4.44兲

Thus, Eqs. 共4.37兲, 共4.43兲, and 共4.44兲 imply that L : M → S42共1兲 傺 E52 satisfies Lxx = −

2Lx , x+y

Lxy =

2e3 + 共1 − c2兲e4 + 2L , c2共x + y兲2

Lyy = −

2Ly . x+y

共4.45兲

Moreover, since ␾ = 0, we obtain from Eq. 共4.13兲 that 2 ˆ e = c − 1L , ⵜ ⳵/⳵x 3 x 2

2 ˆ e = c − 1L , ⵜ ⳵/⳵ y 3 y 2

ˆ e =−L , ⵜ ⳵/⳵x 4 x

ˆ e =−L . ⵜ ⳵/⳵ y 4 y

共4.46兲

After solving Eqs. 共4.45兲 and 共4.46兲, we obtain L共x,y兲 = c1 + c2y +

c4 + c3y − c2y 2 . x+y

共4.47兲

Therefore, after choosing suitable initial conditions, we obtain Case 共11兲 or Case 共12兲 depending on c 苸 共0 , 1兲 or c ⬎ 1.

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083518-13

J. Math. Phys. 51, 083518 共2010兲

Parallel Lorentz surfaces in 4D pseudosphere

Case (c.1.3): b ⬎ 21 . Since K = 1 − 2b ⬍ 0, we put K = −c2 with c ⬎ 0. We choose coordinates 兵x , y其, such that

冉

冊

冉

冊

共4.48兲

共dx 丢 dy + dy 丢 dx兲

共4.49兲

⳵ cx + cy = sech 冑2 e1, ⳵x

cx + cy ⳵ = sech 冑2 e2 . ⳵y

So, the metric tensor is given by g = − sech2

冉

cx + cy

冑2

冊

and the Levi–Civita connection satisfies

冉

冊

冉

冊

ⵜ ⳵/⳵x

⳵ cx + cy ⳵ = − 冑2c tanh 冑2 ⳵ x , ⳵x

ⵜ ⳵/⳵x

⳵ = 0, ⳵y

ⵜ ⳵/⳵ y

⳵ cx + cy ⳵ = − 冑2c tanh 冑2 ⳵ y . ⳵y

共4.50兲

Thus, Eqs. 共4.37兲, 共4.49兲, and 共4.50兲 imply that L : M → S42共1兲 傺 E52 satisfies

冉

Lxx = − 冑2c tanh

Lxy = sech2

冉

cx + cy

冑2

cx + cy

冑2

冉

Lyy = − 冑2c tanh

冊

冊

Lx ,

共␰ + L兲,

cx + cy

冑2

冊

Ly

共4.51兲

with ␰ = e3 + be4. Moreover, since ␾ = 0, we have ˆ ␰ = − 共1 + c2兲L , ⵜ x ⳵/⳵x

ˆ ␰ = − 共1 + c2兲L . ⵜ y ⳵/⳵ y

共4.52兲

After solving Eqs. 共4.51兲 and 共4.52兲, we obtain L = c0 + c1 sinh u + c2 cosh u + 共c3 cosh u + c4 sinh u兲tanh v , u = 冑2cy,

v=

cx + cy

冑2

.

Thus, after choosing suitable initial conditions, we obtain Case 共13兲. Case (c.2): At least one of ␤ , ␥ , ␦ , ␸ is nonzero. In this case, Eq. 共4.36兲 implies that ␻ is closed 1-form. Hence, M is flat. So, we may choose coordinates 兵x , y其, such that ⳵⳵x = e1 , ⳵⳵y = e2. Thus, we have g = − 共dx 丢 dy + dy 丢 dx兲.

共4.53兲

Therefore, by applying Eq. 共4.36兲 again, we know that ␤ , ␥ , ␦ , ␸ are constant.

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083518-14

J. Math. Phys. 51, 083518 共2010兲

Bang-Yen Chen

Case (c.2.1): ␤ = 0. It follows from Eq. 共4.35兲 that b = 21 and ␥␦ = 0. Thus, we have either ␥ = 0 or ␦ = 0. Case (c.2.1.1): ␥ = 0. In this case, Eqs. 共4.33兲 and 共4.34兲 reduce to h共e1,e1兲 = 0,

h共e1,e2兲 = e3 +

A e3 =

冢 冣 1 ␸ 2 1 2

0

,

e4 , 2

h共e2,e2兲 = ␦e3 + ␸e4 ,

共4.54兲

冉 冊

共4.55兲

A e4 =

1 ␦

0 1

.

Thus, we have Lxx = 0,

Lxy = e3 +

e4 + L, 2

Lyy = ␦e3 + ␸e4 .

共4.56兲

Moreover, since ␾ = 0, we obtain from Eq. 共4.55兲 that ˆ e = − Lx , ⵜ ⳵/⳵x 3 2

ˆ e = − ␸L − L y , ⵜ ⳵/⳵ y 3 x 2

ˆ e =−L , ⵜ ⳵/⳵x 4 x

ˆ e = − ␦L − L . ⵜ ⳵/⳵ y 4 x y

共4.57兲

Case (c.2.1.1.1): ␦ = 2␸. If ␸ ⬎ 0, we put ␸ = 2c with c ⬎ 0. Then after solving 共4.56兲 and 共4.57兲, we have 2

L = 共c1 + c2v兲cos u + 共c3 + c4v兲sin u, v = 2c共x + 2c2y兲,

u = 2cy,

which yields Case 共14兲 after choosing suitable initial conditions. Similarly, if ␸ ⬍ 0, we put ␸ = −2c2 with c ⬎ 0. Then, after solving systems 共4.56兲 and 共4.57兲, we obtain L = eu共c1 + c2v兲 + e−u共c3 + c4v兲, u = 2cy,

v = x − 2c2y.

After choosing suitable initial conditions, we obtain Case 共15兲. Case (c.2.1.1.2): ␦ = −2␸. After solving systems 共4.56兲 and 共4.57兲, we find 2 c3 L共x,y兲 = c0 + c1x + c2y + c3xy + c4y 2 + c1␸2y 3 + ␸2y 4 . 3 6

共4.58兲

So, after choosing suitable initial conditions, we get Case 共16兲. Case (c.2.1.1.3): ␦ ⫽ 2␸ and ␦ + 2␸ = −c2 with c ⬎ 0. In this case, after solving systems 共4.56兲 and 共4.57兲, we obtain L = c0 + 共c1 + c2v兲cosh u + 共c3 + c4v兲sinh u, u = cy/冑2,

v = 4c␸2y + c3共x + 2␸ y兲.

Thus, after choosing suitable initial conditions, we obtain Case 共17兲.

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083518-15

J. Math. Phys. 51, 083518 共2010兲

Parallel Lorentz surfaces in 4D pseudosphere

Case (c.2.1.1.4): ␦ ⫽ 2␸ and ␦ + 2␸ = c2, c ⬎ 0. Solving 共4.56兲 and 共4.57兲 gives L = c0 + 兵c1 + c2u其cos v + 兵c3 + c4u其sin v , u = 冑2c3关c2共x + 2␸y兲 − 4␸2y兴,

v=

cy

冑2 .

After choosing suitable initial conditions, we obtain Case 共18兲. Case (c.2.1.2): ␥ ⫽ 0 and ␦ = 0. Equations 共4.33兲 and 共4.34兲 reduce to h共e1,e1兲 = ␥e4,

A e3 =

h共e1,e2兲 = e3 +

冢 冣 1 ␸ 2

1 ␥ 2

,

e4 , 2

A e4 =

h共e2,e2兲 = ␸e4 ,

冉 冊 1 0

共4.59兲

.

共4.60兲

Lyy = ␸e4 .

共4.61兲

0 1

Thus, we have Lxx = ␥e4,

e4 + L, 2

Lxy = e3 +

Moreover, since ␾ = 0, we obtain from 共4.60兲 that ˆ e = − Lx − ␥L , ⵜ ⳵/⳵x 3 y 2 ˆ e =−L , ⵜ ⳵/⳵x 4 x

ˆ e = − ␸L − L y , ⵜ ⳵/⳵ y 3 x 2

ˆ e =−L . ⵜ ⳵/⳵ y 4 y

共4.62兲

Case (c.2.1.2.1): ␥ = c2 , ␸ = 0 , c ⬎ 0. Solving systems 共4.61兲 and 共4.62兲 gives L = c0 + 共c1 + c2v兲cos u + 共c3 + c4v兲sin u,u = cx, v = y. This gives Case 共19兲 after choosing suitable initial conditions. Case (c.2.1.2.2): ␥ = c2 , ␸ = b2 , b , c ⬎ 0. Solving systems 共4.61兲 and 共4.62兲 gives L = c0 + cos u共c1 cos v + c2 sin v兲 + sin u共c3 cos v + c4 sin v兲,u = cx, v = by which yields Case 共20兲. Case (c.2.1.2.3): ␥ = c2 , ␸ = −b2 , b , c ⬎ 0. Solving systems 共4.61兲 and 共4.62兲 gives L = c0 + cos u共c1ev + c2e−v兲 + sin u共c3ev + c4e−v兲,u = cx, v = by. Hence, after choosing suitable initial conditions, we have Case 共21兲. Case (c.2.1.2.4): ␥ = −c2 , ␸ = 0 , c ⬎ 0. Solving systems 共4.61兲 and 共4.62兲 gives L = c0 + eu共c1 + c2v兲 + e−u共c3 + c4v兲,u = cx, v = y. Hence, we find Case 共22兲 after choosing suitable initial conditions. Case (c.2.1.2.5): ␥ = −c2 , ␸ = b2 , b , c ⬎ 0. This gives rises to Case 共21兲 too. Case (c.2.1.2.6): ␥ = −c2 , ␸ = −b2 , b , c ⬎ 0. Solving systems 共4.61兲 and 共4.62兲 yields L = c0 + c1eu + c2e−u + c3ev + c4e−v,u = cx + by, v = cx − by. This gives rise to Case 共23兲.

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083518-16

J. Math. Phys. 51, 083518 共2010兲

Bang-Yen Chen

Case (c.2.2): ␸ = 0. It follows from Eq. 共4.35兲 that b = 21 and ␥␦ = 0. Thus, we have either ␥ = 0 or ␦ = 0. After applying similar arguments as in Case 共c.2.1兲, we also obtain Cases 共19兲–共23兲 of the theorem. Case (c.2.3): ␤ , ␸ ⫽ 0. It follows from Eq. 共4.35兲 that ␥ and ␦ are nonzero real numbers and 2b ⫽ 1. So, Eqs. 共4.33兲 and 共4.34兲 yield h共e1,e1兲 = ␤e3 + ␥e4,

h共e1,e2兲 = e3 + be4,

冢

h共e2,e2兲 =

冣 冢

2b − 1 2␤ , A e3 = ␥ b b

2b − 1 共␤e3 + ␥e4兲, 2␤␥

共4.63兲

冣

共4.64兲

2b − 1 2␥ A e4 = . 1 ␤ 1

Thus, we have Lxx = ␤e3 + ␥e4,

Lxy = e3 + be4 + L,

Lyy =

2b − 1 共␤e3 + ␥e4兲. 2␤␥

共4.65兲

Moreover, since ␾ = 0, we obtain from Eq. 共4.64兲 that ˆ e = − 2b − 1 L − bL , ⵜ ⳵/⳵ y 3 x y 2␤

ˆ e = − bL − ␥L , ⵜ ⳵/⳵x 3 x y

ˆ e = − 2b − 1 L − L . ⵜ ⳵/⳵ y 4 x y 2␥

ˆ e = − L − ␤L , ⵜ ⳵/⳵x 4 x y

共4.66兲

From Eqs. 共4.65兲 and 共4.66兲, we get L共xv兲 + 2共b␤ + ␥兲Lx⵮ + 共b2␤2 + ␥2 + 2共1 − b兲␤␥兲Lx = 0. b ⫽ 21 ,

b ⬎ 21

共4.67兲

b ⬍ 21

Since we consider and separately. 1 Case (c.2.3.1): b ⬎ 2 . We divide this into several cases. Case (c.2.3.1.1): ␤ , ␥ ⬎ 0. If we put b = 共1 + a2兲 / 2 , ␤ = 2c2 and ␥ = k2 with a , c , k ⬎ 0, then after solving systems 共4.65兲 and 共4.66兲, we obtain L = c0 + c1 cos u + c2 sin u + c3 cos v + c4 sin v ,

冉

u = 冑c2 + 共ac + k兲2 x +

冊

ay , 2ck

冉

v = 冑c2 + 共ac − k兲2 x −

冊

ay . 2ck

Thus, after choosing suitable initial conditions, we obtain a special case of Case 共4兲. Case (c.2.3.1.2): ␤ , ␥ ⬍ 0. If we put b = 共1 + a2兲 / 2 , ␤ = −2c2 and ␥ = −k2 with a , c , k ⬎ 0, then after solving systems 共4.65兲 and 共4.66兲, we obtain L = c0 + c1 cosh u + c2 sinh u + c3 cosh v + c4 sinh v ,

冉

u = 冑c2 + 共ac − k兲2 x +

冊

ay , 2ck

冉

v = 冑c2 + 共ac + k兲2 x +

冊

ay . 2ck

This gives a special case of Case 共8兲. Case (c.2.3.1.3): ␤ ⬎ 0 and ␥ ⬍ 0. We put b = 共1 + a4兲 / 2 and ␥ = −k2 with a , k ⬎ 0. We divide this into three cases. Case (c.2.3.1.3.1): b␤ + ␥ = 0. In this case, Eq. 共4.67兲 reduces to

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083518-17

J. Math. Phys. 51, 083518 共2010兲

Parallel Lorentz surfaces in 4D pseudosphere

共1 + a4兲L共xv兲 + 4a4k4Lx = 0, which implies that L = eakx/

冑4 1 + a4

+ e−akx/

再 冉 冊 冉 冊冎 再 冉 冊 冉 冊冎 A共y兲cos

冑4 1 + a4

akx

冑1 + a 4

P共y兲cos

+ B共y兲cos

4

akx

akx

冑1 + a4 4

+ Q共y兲cos

冑4 1 + a4

akx

冑4 1 + a4

+ H共y兲.

Thus, after substituting this into systems 共4.65兲 and 共4.66兲, we obtain L = c0 + 兵c1 cos v + c2 sin v其cosh u + 兵c3 cos v + c4 sin v其sinh u,

u = ␮x − ␺y,

v = ␮x + ␺ y,

␮=

2ak2

2冑1 + a4k 4

,

␺=

a3冑1 + a4

2冑1 + a4k 4

.

Thus, after choosing suitable initial conditions, we have a special case of Case 共24兲. Case (c.2.3.1.3.2): b␤ + ␥ = c2 , c ⬎ 0. Equation 共4.67兲 reduces to L共xv兲 + 2c2Lx⵮ +

c4 + a4共c2 + 2k2兲2 Lx = 0. 1 + a4

共4.68兲

After solving this differential equation, we have L共x,y兲 = e␮x共A共y兲cos共␺x兲 + B共y兲sin共␺x兲兲 + e−␮x共P共y兲cos共␺x兲 + Q共y兲sin共␺x兲兲 + K共y兲, 共4.69兲 with

␮=

冑冑c4 + a4共c2 + 2k2兲2 − 冑1 + a4c2 冑2冑1 + a4 4

,

␺=

冑冑c4 + a4共c2 + 2k2兲2 + 冑1 + a4c2 冑2冑4 1 + a4

.

Thus, after substituting this into systems 共4.65兲 and 共4.66兲, we obtain L = c0 + 兵c1 cos v + c2 sin v其cosh u + 兵c3 cos v + c4 sin v其sinh u,

u = ␮x − sy,

t=

v = ␺x + ty,

s=

a2冑冑1 + a4冑c4 + a4共c2 + 2k2兲2 + c2共1 + a2兲 2冑2k冑c2 + k2

a2冑冑1 + a4冑c4 + a4共c2 + 2k2兲2 − c2共1 + a2兲 2冑2k冑c2 + k2

,

.

This yields a special case of Case 共24兲. Case (c.2.3.1.3.3): b␤ + ␥ = −c2 , c ⬎ 0. Since b , ␤ ⬎ 0, we get k2 ⬎ c2. Moreover, Eq. 共4.67兲 gives L共xv兲 − 2c2Lx⵮ +

c4 + a4共c2 − 2k2兲2 Lx = 0. 1 + a4

共4.70兲

After solving this differential equation, we have

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083518-18

J. Math. Phys. 51, 083518 共2010兲

Bang-Yen Chen

L共x,y兲 = e␺x共A共y兲cos共␮x兲 + B共y兲sin共␮x兲兲 + e−␺x共P共y兲cos共␮x兲 + Q共y兲sin共␮x兲兲 + K共y兲, 共4.71兲 with

␮=

冑冑c4 + a4共c2 + 2k2兲2 − 冑1 + a4c2 冑2冑4 1 + a4

␺=

,

冑冑c4 + a4共c2 + 2k2兲2 + 冑1 + a4c2 冑2冑4 1 + a4

.

Hence, after substituting Eq. 共4.71兲 into systems 共4.65兲 and 共4.66兲, we obtain L = c0 + 兵c1 cos v + c2 sin v其cosh u + 兵c3 cos v + c4 sin v其sinh u, u = ␺x − my,

v = ␮x + ny,

a2冑1 + a4冑冑c4 + a4共c2 − 2k2兲2 − c2冑1 + a4 4

m=

2冑2k冑k2 − c2

a2冑1 + a4冑冑c4 + a4共c2 − 2k2兲2 + c2冑1 + a4

,

4

n=

2冑2k冑k2 − c2

.

Thus, after choosing suitable initial conditions, we have a special case of Case 共24兲. Case (c.2.3.1.4): ␥ ⬎ 0 and ␤ ⬍ 0. If we put b = 共1 + a2兲 / 2 , ␤ = −2c2 and ␥ = k2 with a , c , k ⬎ 0, then Eq. 共4.67兲 becomes L共xv兲 + 2共k2 − c2 − a4c2兲Lx⵮ + 共k4 + 共1 + a4兲2c4 + 2共a4 − 1兲c2k2兲Lx = 0.

共4.72兲

Case (c.2.3.1.4.1): k = c冑1 + a4. After solving Eq. 共4.72兲, we find L共x,y兲 = e−ac

冑4 1 + a4x

+ e−ac

共A共y兲cos共ac冑1 + a4x兲 + B共y兲sin共ac冑1 + a4x兲兲, 4

冑4 1 + a4x

4

共P共y兲cos共ac冑1 + a4x兲 + Q共y兲sin共ac冑1 + a4x兲兲 + K共y兲. 4

4

Hence, after substituting this into systems 共4.65兲 and 共4.66兲, we obtain L = c0 + 兵c1 cos v + c2 sin v其cosh u + 兵c3 cos v + c4 sin v其sinh u, a4y , 2␸

u = ␸x −

v = ␸x +

a4y , 2␸

␸ = ac冑1 + a4 . 4

After choosing suitable initial conditions, we get a special case of Case 共24兲. Case (c.2.3.1.4.2): k ⬍ c冑1 + a4. After solving systems 共4.65兲 and 共4.66兲, we obtain L = c0 + 兵c1 cos v + c2 sin v其cosh u + 兵c3 cos v + c4 sin v其sinh u,

␺ 2 a cky, 2

u = ␮x −

␮=

冉 冑冑

v = ␺x +

␮ 2 a cky, 2

共1 + a4兲2c4 + 2共a4 − 1兲c2k2 + k4 + 共1 + a4兲c2 − k2

冑2

冊

,

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083518-19

J. Math. Phys. 51, 083518 共2010兲

Parallel Lorentz surfaces in 4D pseudosphere

␺=

冉 冑冑

共1 + a4兲2c4 + 2共a4 − 1兲c2k2 + k4 − 共1 + a4兲c2 + k2

冑2

冊

,

which gives rise a special case of Case 共24兲. Case (c.2.3.1.4.3): k ⬎ c冑1 + a4. After solving Eq. 共4.72兲, we find L = e␮x共A共y兲cos共␺x兲 + B共y兲sin共␺x兲兲 + e−␮x共P共y兲cos共␺x兲 + Q共y兲sin共␺x兲兲 + K共y兲, with a , c , k ⬎ 0 and

␮=

␺=

冑冑4a4c2k2 + 共k2 − 共1 + a4兲c2兲2 + 共1 + a4兲c2 − k2 冑2

冑冑4a4c2k2 + 共k2 − 共1 + a4兲c2兲2 − 共1 + a4兲c2 + k2 冑2

,

.

By substituting this into systems 共4.65兲 and 共4.66兲, we obtain L = c0 + 兵c1 cos v + c2 sin v其cosh u + 兵c3 cos v + c4 sin v其sinh u, u = ␮x −

a 2␺ y, 2ck

v = ␺x +

a 2␮ y. 2ck

Thus, after choosing suitable initial conditions, we get a special case of Case 共24兲. Case (c.2.3.2): b ⬍ 21 . We divide this case into several cases. Case (c.2.3.2.1): ␤ , ␥ ⬎ 0. If we put ␤ = 2c2 and ␥ = k2 with c , k ⬎ 0, then 共4.67兲 becomes L共xv兲 + 2共k2 + 2bc2兲Lx⵮ + 共共k2 − 2bc2兲2 + 4c2k2兲Lx = 0.

共4.73兲

Case (c.2.3.2.1.1): 2bc2 = −k2. In this case, we have b ⬍ 0. After solving systems 共4.65兲 and 共4.66兲, making suitable changes of variables, and choosing suitable initial conditions, we obtain a special case of Case 共24兲. Case (c.2.3.2.1.2): 2bc2 ⫽ −k2. By solving Eq. 共4.73兲 we get L = e px共A共y兲cos共qx兲 + B共y兲sin共qx兲兲 + e−px共P共y兲cos共qx兲 + Q共y兲sin共qx兲兲 + K共y兲,

p=

冑冑4c2k2 + 共k2 − 2bc2兲2 − 2bc2 − k2 冑2

,

q=

冑冑4c2k2 + 共k2 − 2bc2兲2 + 2bc2 + k2 冑2

.

After substituting this into systems 共4.65兲 and 共4.66兲, we obtain L = c0 + 兵c1 cos v + c2 sin v其cosh u + 兵c3 cos v + c4 sin v其sinh u,

u = px −

冑1 − 2bq 2ck

y,

v = qx +

冑1 − 2bq 2ck

y.

This gives a special case of Case 共24兲 after choosing suitable initial conditions. Case (c.2.3.2.2): ␤ , ␥ ⬍ 0. If we put ␤ = −2c2 and ␥ = −k2 with c , k ⬎ 0, then Eq. 共4.67兲 becomes L共xv兲 − 2共2bc2 + k2兲Lx⵮ + 共共2bc2 − k2兲2 + 4c2k2兲Lx = 0.

共4.74兲

Case (c.2.3.2.2.1): 2bc = −k . After solving systems 共4.65兲 and 共4.66兲, making suitable changes of variable, and choosing suitable initial conditions, we get a special case of Case 共24兲. 2

2

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083518-20

J. Math. Phys. 51, 083518 共2010兲

Bang-Yen Chen

Case (c.2.3.2.2.2): 2bc2 ⫽ −k2. After solving Eq. 共4.74兲 we get L = e px共A共y兲cos共qx兲 + B共y兲sin共qx兲兲 + e−px共P共y兲cos共qx兲 + Q共y兲sin共qx兲兲 + K共y兲,

p=

冑冑4c2k2 + 共2bc2 − k2兲2 + 2bc2 + k2 冑2

,

q=

冑冑4c2k2 + 共2bc2 − k2兲2 − 2bc2 − k2 冑2

.

Substituting this into systems 共4.65兲 and 共4.66兲 gives L = c0 + 兵c1 cos v + c2 sin v其cosh u + 兵c3 cos v + c4 sin v其sinh u,

u = px −

冑1 − 2b 2ck

qy,

v = qx +

冑1 − 2b 2ck

py.

This yields a special case of Case 共24兲. Case (c.2.3.2.3): ␤ ⬎ 0 and ␥ ⬍ 0. If we put ␤ = 2c2 and ␥ = −k2 with c , k ⬎ 0, then 共4.67兲 becomes L共xv兲 + 2共2bc2 − k2兲Lx⵮ + 共共2bc2 + k2兲2 − 4c2k2兲Lx = 0. 2

2

Case (c.2.3.2.3.1): 2bc = k . Since have

0 ⬍ b ⬍ 21 ,

共4.75兲

we get c ⬎ k ⬎ 0. Thus, after solving 共4.75兲, we

L = e pxA共y兲 + e−pxB共y兲 + cos共px兲P共y兲 + sin共px兲Q共y兲 + K共y兲, with p = 冑2k共c2 − k2兲1/4 for some vector-valued functions A , B , P , Q , K. Hence, after solving systems 共4.65兲 and 共4.66兲, we obtain L = c0 + c1 sinh u + c2 cosh u + c3 cos v + c4 sin v ,

u=

冑4 c2 − k2 2 冑2 2 冑2kc2 共2c kx + c − k y兲,

v=

冑4 c2 − k2 2 冑2 2 冑2kc2 共2c kx − c − k y兲,

with c ⬎ k ⬎ 0. Hence, we obtain a special case of Case 共2兲 after choosing suitable initial conditions. Case (c.2.3.2.3.2): 2bc2 ⬎ k2. We divide this into four cases. Case (c.2.3.2.3.2.1): b = k共2c − k兲 / 共2c2兲. Solving 共4.65兲 and 共4.66兲 gives L = c0 + c1u + c2u2 + c3 cos v + c4 sin v ,

u=

2c2kx + 共c − k兲y , 2c2k

v=

冑c − k 2 共2c kx − 共c − k兲y兲. c 2 冑k

So, after choosing suitable initial conditions, we get a special case of Case 共5兲. Case (c.2.3.2.3.2.2): 2bc2 = −共2c + k兲k. We get a special case of Case 共7兲. Case (c.2.3.2.2.2.3): b ⫽ 共2c − k兲k / 共2c2兲 and b ⫽ −k共2c + k兲 / 共2c2兲. In this case, we have 2bc2 2 ⫽ k + 2ck冑1 − 2b Case (c.2.3.2.3.2.3.1): 2bc2 ⬍ k2 + 2ck冑1 − 2b. Since 2bc2 ⬎ k2 and 1 ⬎ 2b holds, thus after solving Eq. 共4.75兲, we get L = A共y兲cos共px兲 + B共y兲sin共px兲 + P共y兲cosh共qx兲 + Q共y兲sinh共qx兲 + K共y兲,

共4.76兲

with p = 冑2bc2 − k2 + 2ck冑1 − 2b and q = 冑k2 − 2bc2 + 2ck冑1 − 2b. Hence, by substituting this into systems 共4.65兲 and 共4.66兲 and by choosing initial conditions, we obtain a special case of Case 共6兲. Case (c.2.3.2.3.2.3.2): 2bc2 ⬎ k2 + 2ck冑1 − 2b. Solving systems 共4.65兲 and 共4.66兲 gives

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083518-21

J. Math. Phys. 51, 083518 共2010兲

Parallel Lorentz surfaces in 4D pseudosphere

L = c0 + c1 cos u + c2 sin u + c3 cos v + c4 sin v ,

u = qx +

冑1 − 2b 2ck

v=

qy,

冑1 − 2b 2ck

p = 冑2bc2 − k2 + 2ck冑1 − 2b,

py − px,

q = 冑2bc2 − k2 − 2ck冑1 − 2b.

共4.77兲

Thus, we obtain a special case of Case 共4兲 after choosing suitable initial conditions. Case (c.2.3.2.3.3): 2bc2 ⬍ k2. We divide this into four cases. Case (c.2.3.2.3.3.a): 2bc2 = 共2c − k兲k and k ⬎ c. Solving 共4.65兲 and 共4.66兲 gives L = c0 + c1u + c2u2 + c3 cos v + c4 sin v , 2c2kx + cy − ky

u=

2c冑k

,

v=

冑k − c 2 共2c kx + ky − cy兲. c 2 冑k

共4.78兲

Thus, after choosing suitable initial conditions, we get a special case of Case 共7兲. Case (c.2.3.2.3.3.b): 2bc2 = 共2c − k兲k and k ⬍ c. After solving 共4.65兲 and 共4.66兲 and choosing suitable initial conditions, we obtain a special case of Case 共5兲. Case (c.2.3.2.3.3.c): 2bc2 ⫽ 共2c − k兲k. If k2 − 2bc2 = 2ck冑1 − 2b holds, we get 2bc2 = k共2c − k兲, which is a contraction. Therefore, either k2 − 2bc2 ⬎ 2ck冑1 − 2b holds or k2 − 2bc2 ⬍ 2ck冑1 − 2b holds. Case (c.2.3.2.3.3.c.1): k2 − 2bc2 ⬎ 2ck冑1 − 2b. After solving Eq. 共4.75兲, we get L = A共y兲cosh共px兲 + B共y兲sinh共px兲 + P共y兲cosh共qx兲 + Q共y兲sinh共qx兲 + K共y兲, with p = 冑k2 − 2bc2 − 2ck冑1 − 2b and q = 冑k2 − 2bc2 + 2ck冑1 − 2b. Thus, by solving 共4.65兲 and 共4.66兲, we find L = c0 + c1 cosh u + c2 sinh u + c3 cosh v + c4 sinh v ,

u=

冑1 − 2b 2ck

py − px,

v = qx +

p = 共冑k2 − 2bc2 − 2ck冑1 − 2b兲,

冑1 − 2b 2ck

qy,

q = 共冑k2 − 2bc2 + 2ck冑1 − 2b兲.

共4.79兲

Thus, after choosing suitable initial conditions, we obtain a special case of Case 共8兲. Case (c.2.3.2.3.3.c.2): k2 − 2bc2 ⬍ 2ck冑1 − 2b. After solving Eq. 共4.75兲, we have L = A共y兲cos共␮x兲 + B共y兲sin共␮x兲 + P共y兲cosh共qx兲 + Q共y兲sinh共qx兲 + K共y兲, with ␮ = 冑2bc2 − k2 + 2ck冑1 − 2b and q = 冑k2 − 2bc2 + 2ck冑1 − 2b. Thus, by solving 共4.65兲 and 共4.66兲, we find L = c0 + c1 cosh u + c2 sinh u + c3 cos v + c4 sin v ,

u = qx +

冑1 − 2b 2ck

qy,

v = ␮x +

冑1 − 2b 2ck

␮y.

共4.80兲

By after choosing suitable initial conditions, we obtain a special case of Case 共2兲. Case (c.2.3.2.4): ␥ ⬎ 0 and ␤ ⬍ 0. If we put ␤ = −2c2 and ␥ = k2 with c , k ⬎ 0, Eq. 共4.67兲 becomes

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083518-22

J. Math. Phys. 51, 083518 共2010兲

Bang-Yen Chen

L共xv兲 + 2共2bc2 + k2兲Lx⵮ + 共共2bc2 − k2兲2 + 4c2k2兲Lx = 0.

共4.81兲

After solving Eq. 共4.81兲, we have L = e−px共A共y兲cos共qx兲 + B共y兲sin共qx兲兲 + e px共P共y兲cos共qx兲 + Q共y兲sin共qx兲兲 + K共y兲 for some E53-valued functions A , B , P , Q , K, where p = 冑冑4c2k2 + 共2bc2 − k2兲2 − 2bc2 − k2/冑2, q = 冑冑4c2k2 + 共2bc2 − k2兲2 + 2bc2 + k2/冑2. Now, by substituting this into systems 共4.65兲 and 共4.66兲, we get L = c0 + cos u共c1 cosh v + c2 sinh v兲 + sin u共c3 cosh v + c4 sinh v兲,

u = qx −

1 − 2b y, 2q

v = px +

1 − 2b y. 2p

Thus, after choosing suitable initial conditions, we obtain a special case of Case 共3兲. This completes the proof of the theorem. 䊏 Remark 4.1: All of the 24 families of Lorentz parallel surfaces in E52 given in the Main Theorem are surfaces of finite type in the sense of Refs. 3 and 8. Remark 4.2: All of the flat parallel Lorentz surfaces given in the Main Theorem are geodesically complete; due to the fact that the flat parallel Lorentz surfaces are defined either for all x , y or for all u , v with nondegenerate flat metric g with constant coefficients gij and the universal coverings of those surfaces are E21. 共For a survey on completeness of Lorentz manifolds, cf. Ref. 15.兲 ACKNOWLEDGMENTS

The author thanks the referee for many valuable suggestions to improve the presentation of this paper. Blomstrom, C., Lect. Notes Math. 1156, 30 共1985兲. Chen, B. Y., Geometry of Submanifolds 共Dekker, New York, 1973兲. 3 Chen, B. Y., Total Mean Curvature and Submanifolds of Finite Type 共World Scientific, Singapore, 1984兲. 4 Chen, B. Y., Riemannian Submanifolds, Handbook of Differential Geometry 共North-Holland, Amsterdam, 2000兲, Vol. I, pp. 187–418. 5 Chen, B. Y., “Complete classification of parallel spatial surfaces in pseudo-Riemannian space forms with arbitrary index and dimension,” J. Geom. Phys. 60, 260 共2010兲. 6 Chen, B. Y., “Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space,” Cent. Eur. J. Math. 8, 706 共2010兲. 7 Chen, B. Y., Dillen, F., and Van der Veken, J., “Complete classification of parallel Lorentzian surfaces in Lorentzian complex space forms,” Int. J. Math. 21, 665 共2010兲. 8 Chen, B. Y., Dillen, F., Verstraelen, L., and Vrancken, L., “A variational minimal principle characterizes submanifolds of finite type,” C. R. Acad. Sci. Paris Sér. I Math. 317, 961 共1993兲. 9 Chen, B. Y. and Van der Veken, J., “Complete classification of parallel surfaces in 4-dimensional Lorentz space forms,” Tohoku Math. J. 61, 1 共2009兲. 10 Ferus, D., “Immersions with parallel second fundamental form,” Math. Z. 140, 87 共1974兲. 11 Graves, L. K., “On codimension one isometric immersions between indefinite space forms,” Tsukuba J. Math. 3, 17 共1979兲. 12 Graves, L. K., “Codimension one isometric immersions between Lorentz spaces,” Trans. Am. Math. Soc. 252, 367 共1979兲. 13 Magid, M. A., “Isometric immersions of Lorentz space with parallel second fundamental forms,” Tsukuba J. Math. 8, 31 共1984兲. 14 O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity 共Academic, New York, 1982兲. 15 Romero, A. and Sánchez, M., Geometry and Topology of Submanifolds 共Leuven, Brussels, 1993兲, Vol. IV, pp. 171–182. 16 Strübing, W., “Symmetric submanifolds of Riemannian manifolds,” Math. Ann. 245, 37 共1979兲. 1 2

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