On a Certain Class of Translation Surfaces in a Pseudo-Galilean Space

Transcription

1 International Mathematical Forum, Vol. 6,, no. 3, 3-5 On a Certain Class of Translation Surfaces in a Pseudo-Galilean Space Željka Milin Šipuš Department of Mathematics Universit of Zagreb Bijenička cesta 3 Zagreb, Croatia Abstract In this paper we describe a special class of translation surfaces in a pseudo-galilean space. We anale translation surfaces having constant Gaussian and mean curvatures, as well as translation Weingarten surfaces. Mathematics Subject Classification: 53A35, 53A4 Kewords: pseudo-galilean space, translation surface Introduction In this paper we describe a special class of translation surfaces in a pseudo- Galilean space. We are speciall interested in the analogues of the results from the Euclidean space concerning translation surfaces having constant Gaussian K and mean curvature H, and translation surfaces that are Weingarten surfaces as well. A translation surface is a surface that can locall be written as the sum of two curves α, β (u, v) =α(u)+β(v). Translation surfaces in the Euclidean and Minkowski space having constant Gaussian and mean curvature are described in [8]: Theorem. Let S be a translation surface with constant Gaussian curvature K in 3-dimensional Euclidean space or 3-dimensional Minkowski space. Then S is congruent to a clindrical surface (i.e. generalied clinder), so K =.

2 4 Ž. M. Šipuš Theorem. Let S be a translation surface with constant mean curvature H in 3-dimensional Euclidean space. Then S is congruent to the following surface or a part of it (class of clindrical surfaces) = +a 4H H a, a R, and for H =to a plane or to the Scherk minimal surface = a log(cos(a)) log(cos(a)), a R \{}. a For the translation Weingarten surfaces in Euclidean space the following theorem holds ([]): Theorem.3 A translation surface in R 3 is a Weingarten surface if and onl if it is either (a part of). a plane,. a clindrical surface, 3. the minimal surface of Scherk, 4. an orthogonal elliptic paraboloid parametried b (s, t) =(s, t, a(s + t )). Counterparts of these results for surfaces in Minkowski space can be found in []. In [9] the same problems were treated in a Galilean space. Preliminar Notes The pseudo-galilean space G 3 is a Cale-Klein space with absolute figure consisting of the ordered triple {ω, f, I}, where ω is the ideal (absolute) plane in the real three-dimensional projective space P 3 (R), f the line (absolute line) in ω and I the fied hperbolic involution of points of f. Homogeneous coordinates in G 3 are introduced in such a wa that the absolute plane ω is given b =, the absolute line f b = = and the hperbolic involution b ( : : : 3 ) ( : : 3 : ). The last condition is equivalent to the requirement that the conic 3 = is the absolute conic. Metric relations are introduced with respect to the absolute figure. In affine coordinates defined b ( : : : 3 ) = ( : : : ), distance between points P i =( i, i, i ), i =,, is defined b d(p,p )= {, if, ( ) ( ), if =. ()

3 Translation surfaces 5 The group of motions of G 3 is a si-parameter group given (in affine coordinates) b = a + ȳ = b + c + cosh ϕ + sinh ϕ () = d + e + sinh ϕ + cosh ϕ. It leaves invariant the absolute figure as well the pseudo-galilean distance () of points. The pseudo-galilean space G 3 can be also treated as a Cale-Klein space equipped with the projective metric of signature (,, +, ), as eplained in []. According to the description of the Cale-Klein spaces in [7], it is denoted b P 3 and also called the Galilean space of inde. In the pseudo-galilean space, a vector is called isotropic if it is of the form (,,). Among these vectors, there are also vectors with supplementar norm equal to ero, the are called lightlike vectors (vectors parallel to the planes = ±). Isotropic vectors satisfing > are said to be spacelike vectors, and vectors satisfing < timelike vectors. In Figure planes = ± and hperbolic clinders = ± are presented. A plane of the form = const. is called a pseudo-euclidean plane (since its induced geometr is pseudo-euclidean, i.e. Minkowski plane geometr), otherwise it is called isotropic (since its induced geometr is isotropic, i.e. Galilean plane geometr). A C r -surface, r, is a subset Φ G 3 for which there eists an open subset D of R and C r -mapping : D G 3 satisfing Φ = (D). A Cr - surface Φ G 3 is called regular if is an immersion, and simple if is an embedding. It is admissible if it does not have pseudo-euclidean tangent planes. If we denote = ((u,u ),(u,u ),(u,u )),,i =,,i =, u i u i,i =, i =,, then a surface is admissible if and onl if,i, for some u i i =,. Let Φ G 3 be a regular admissible surface. Then the unit normal vector field of a surface (u, v) is equal to N(u, v) = W (u, v) = W (u, v) (,,,,,,,,,, ), (,,,, ) (,,,, ) (3) The function W is equal to the pseudo-galilean norm the vector,,,,. Vector defined b σ = ( W,,,, ) is called a side tangential vector. We will not consider surfaces with W =, i.e. surfaces having lightlike side tangential vector (lightlike surfaces).

4 6 Ž. M. Šipuš Since the normal vector field satisfies N N = ɛ = ±, we distinguish two basic tpes of admissible surfaces: spacelike surfaces having timelike surface normals (ɛ = ) and timelike surfaces having spacelike normals (ɛ = ). A surface is spacelike if (,,,, )=(,,,, ) (,,,, ) > in all of its points, timelike otherwise. The first fundamental form of a surface is induced from the metric of the ambient space G 3 where δ = ds =(, du +, du ) + δ(, du +, du ), {, if direction du :du is non isotropic,, if direction du :du is isotropic. B above a vector, the projection of a vector on the pseudo-euclidean plane is denoted. The square on the second summand is the scalar square of the pseudo-euclidean scalar product (, ) (, )=. We introduce the coefficients of the first fundamental form g i =,i, h ij =,i,j, i,j =,. Now we can write the first fundamental form as ds =(g du + g du ) + δ(h du +h du du + h du )=ds +ds. Notice that a surface is spacelike if g h g g h + g h >, timelike otherwise. Furthermore, the function W satisfies W = ɛ(g, g, ) = ɛ(g h g g h + g h ) >. Therefore, notice that a spacelike surface has both parts of the first fundamental form, ds and ds = ɛw du g, g, positive definite, while a timelike surfaces has the form ds is positive definite and ds negative definite. We have assumed here, without loss of generalit, g. The Gaussian curvature of a regular admissible surface is defined b K = ɛ L L L W

5 Translation surfaces 7 Figure : Lightlike planes, a spacelike and a timelike hperbolic sphere where L ij, i, j =,, are the coefficients of the second fundamental form L ij = ɛ g (g,i,j g i,j, ) N = ɛ g (g,i,j g i,j, ) N, The mean curvature of a surface is defined b H = ɛ W (g L g g L + gl ). As in the Galilean space G 3 ([]), the following theorem holds: Theorem. Minimal surfaces in a pseudo-galilean space G 3 are ruled conoidal surfaces, i.e. the are cones with vertices on the absolute line or ruled surfaces with the absolute line as a director curve in infinit. The geometr of the pseudo-galilean space G 3 is developed in []. Some more results on surfaces in G 3 can be found in [3], [4], [5], [6]. 3 Translation surfaces in the pseudo-galilean space We will consider translation surfaces in the pseudo-galilean space G 3 that are obtained b translating two planar curves. In order to obtain an admissible surface, translated curves can be, with respect to the absolute figure, either an isotropic and a non-isotropic curve (obtained surfaces will be called translation surfaces of Tpe ) or two non-isotropic curves (translation surfaces of Tpe ). In this paper we shall treat the first ones, i.e. translation surfaces of Tpe.

6 8 Ž. M. Šipuš These surfaces can be given b = f()+g(). (4) The are obtained b translating a non-isotropic curve α() = (,, f()) along an isotropic curve β() =(,,g()) (or vice-versa). The curve β is a spacelike (resp. timelike) curve if and onl if g () > (resp. g () < ). The side-tangential vector of a translation surface (4) is given b and the unit normal field σ(, ) = W (,,g ()), W = g () N(, ) = W (,g (), ). Since N = g (), the surface is a spacelike (resp. timelike) if and onl if N = g () < (resp. g () > ). Therefore, we have: Proposition 3. A translation surface of Tpe in G 3 is spacelike (resp. timelike) if and onl if its generating isotropic curve is spacelike (resp. timelike). The Gaussian curvature of a surface (4) is given b K = ɛ f ()g () ( g ()), and the mean curvature for a spacelike surface b and for a timelike surface b H = H = g () ( g ()) 3/, g () (g () ) 3/. B we have denoted derivatives with respect to corresponding variables. First we eamine surfaces of Tpe having constant Gaussian curvature. Contrar to the Euclidean case, since variables, in the function K can be separated, K is constant if and onl if either f () =const. and f () = or g () = const., (5) ( g ()) g () =. (6) ( g ())

7 Translation surfaces 9 Therefore, b solving (5) we obtain f() =a + b + c, a, b, c R with g() which is the solution of the ordinar differential equation g () g () + log +g () = A + B, A, B R. g () The previous ordinar differential equation can not be solved in elementar functions. Therefore, we will reparametrie the generating curve β b the arclength u, β(u) =(,h(u),k(u)), h (u) k (u) =±. For this parametriation we get Therefore σ(u, v) =(,h (u),k (u)), N(u, v) =(,k (u),h (u)), W(u, v) =. L = ɛf ()h (u), M =, N = ɛ(k (u)h (u) h (u)k (u)). Since h (u) k (u) =±, then h (u)h (u) k (u)k (u) =, and therefore N = k (u) f (u). Now we get K = f ()k (u), regardless whether a surface is spacelike or timelike. B solving the equation K = const., we get k(u) = Au + Bu + C and h(u) = Au + B +(Au + B) A + A arcsinh(au + B)+C, A,B,C,C R, (7) in the case when a surface (i.e. generating curve β is spacelike) and h(u) = Au + B (Au + B) A + ( A log (Au + B)+ (Au + B) ) +C, (8) in the case when a surface timelike (i.e. generating curve β is timelike), see Figures 3, 5. We can notice that parametriations with (7), (8) are parametriations with principal curves of the surface. Furthermore, from (6) we have that the Gaussian curvature is constant the obtained surface is clindrical (, ) =a + b + g(), or (, ) =f()+c + d. Therefore, we have proved in the pseudo-galilean space the following theorem: Theorem 3. If S is a translation surface of Tpe of constant Gaussian curvature in the pseudo-galilean space, then S is congruent to a special surface with f() =a +b+c, a, b, c R and k, h given b (7) for a spacelike surface or (8) for a timelike surface (K ), or to a clindrical surface (K =) having either non-isotropic or isotropic rulings.

8 Z. M. S ipus Figure : Spacelike surfaces with K = Figure 3: Generating curves for spacelike surfaces with K = a parabola and the isotropic curve (7) Figure 4: Timelike surfaces with K = Now we eamine surfaces of Tpe of constant mean curvature. Theorem 3.3 If S is a translation surface of Tpe of constant mean

9 Translation surfaces Figure 5: Generating curves for timelike surfaces with K a parabola and the isotropic curve (8) curvature H in the pseudo-galilean space, then S is congruent to a surface = f()+ (H + c ) H ± +c, c,c = const. (9) Speciall, if f() =a+b, a, b, R, then S is congruent to a hperbolic sphere (an equilateral hperbolic clinder). Proof. If we put h() =g (), one should solve the ordinar differential equation h () =H( h ()) 3, H = const in the case when a surface is spacelike, and h () =H(h () ) 3, H = const in the case when a surface is timelike. We get H + c h() =± (H + c ) + () and H + c h() =± (H + c ), c R. () Furthermore, notice that curves β)) =(,,h()) satisf the equation = const. and therefore the are hperbolas. If the curve α is a line, then S is a hperbolic sphere (an equilateral hperbolic clinder), see Figure. A unit sphere = (resp. = ) is a spacelike (resp. timelike) surface. Notice that contrar to the Euclidean situation, a surface having constant mean curvature need not be ruled, due to the fact that H is a function of a variable onl.

10 Ž. M. Šipuš Figure 6: A surface with H = const. and a parabolic (spacelike) sphere Furtheremore, notice that among surfaces with K = const. or H = const. there is also a parabolic sphere (a parabolic clinder). A unit parabolic spacelike (resp. timelike) sphere in a pseudo-galilean space is a surfaces given b (its normal form) = (resp. = ). It is an admissible surface, a clindrical surface with isotropic rulings, having K = H =, see Figure 6. Finall, for minimal translation surfaces in G 3 the following statement is obvious. Theorem 3.4 If S is a translation surface of Tpe of ero mean curvature in the pseudo-galilean space, then S is congruent to a clindrical surface with isotropic rulings (and therefore K =) = f()+a + b, a, b R. In other words, the obtained surface is a ruled surface with rulings having the constant isotropic direction (,,a). It is spacelike (resp. timelike) if and onl if a > (resp. a < ). The obtained results agree with the statement of Theorem.. 4 Translation Weingarten surfaces Weingarten surfaces are surfaces whose Gaussian and mean curvature satisf a functional relationship (of class C at least). The class of Weingarten surfaces contains alread mentioned surfaces of constant curvatures K or H. Furthermore, a C r -surface, r>3, is Weingarten if and onl if K H K H =. In the case of translation surfaces of Tpe the following theorem holds: Theorem 4. A translation surface of Tpe in the pseudo-galilean space is a Weingarten surface if and onl if it is either (a part of)

11 Translation surfaces 3. an isotropic plane,. a clindrical surface with isotropic or non-isotropic rulings, 3. a translation surface of constant Gaussian curvature of Theorem 3., 4. a translation surface of constant mean curvature of Theorem 3.3, 5. a surface = a + b + c + g(), a, b, c R. Proof. Since H is a function of -variable onl, a C r - surface, r>3, is Weingarten if and onl if K H =. This condition is satisfied either if K =orh =. The latter condition gives surfaces of constant mean curvature (Theorem 3.3, Theorem 3.4). The first condition K = f ()g () ( g ()) 4 is satisfied b surfaces of constant Gaussian curvature (Theorem 3.) or surfaces which have f () =. These are the surfaces = a + b + c + g(), a,b,c R. Among them, depending on a curve β, there are spacelike ( g () > ) as well as timelike g () < surfaces. Notice, that the functional relationship between K and H of these surfaces is given b ( g ()) / K H = Figure 7: Translation Weingarten surface of tpe In Euclidean space, if the curvatures K, H of a surface satisf ak +bh =, with at least one of the constant a, b different from, then a surface is the Scherk s minimal surface or a surface is flat ([]).

12 4 Ž. M. Šipuš Proposition 4. The onl translation Weingarten surfaces of Tpe in the pseudo-galilean space that satisf linear condition ak + bh =, with at least one of the constant a, b different from, are clindrical surfaces with non-isotropic or isotropic rulings (speciall parabolic spheres). Proof. If a =,b, then S is a minimal surface of Theorem 3.4. If a,b = then S is flat surface of Theorem 3.. If ab, then the condition ak+bh = implies either g () =orf () =W () =const. = A. Therefore we get either g() =a+b, which generates clindrical surfaces with isotropic rulings, or f() = A + B+ C and g() =a + b, A, B, C, a, b R. These surfaces are parabolic spheres. References [] F. Dillen, W. Goemans, I. Van de Woestne, Translation surfaces of Weingarten tpe in 3-space, Bulletin of Transilvania Universit of Brasov, (5)(8), Series III: Mathematics, Informatics, Phsics, 9. [] B. Divjak, Geometrija pseudogalilejevih prostora, PH. D. thesis, Universit of Zagreb, 997. [3] B. Divjak, Curves in pseudo-galilean geometr, Annales Univ. Sci. Budapest, 4(998), 7 8. [4] B. Divjak, Ž. Milin Šipuš, Special curves on ruled surfaces in Galilean and pseudo-galilean Space, Acta Math. Hungar. 98(3)(3), 3 5. [5] B. Divjak, Ž. Milin Šipuš, Minding isometries of ruled surfaces in pseudo- Galilean space, J. geom. 77(3), [6] B. Divjak, Ž. Milin Šipuš, Transversal surfaces of ruled surfaces in pseudo- Galilean space, Situngsber. Abt. II 3(4), 3 3. [7] O. Giering, Vorlesungen über höhere Geometrie, Friedr. Vieweg & Sohn, Brauschweig, Wiesbaden, 98. [8] H. Liu, Translation surfaces with constant mean curvature in 3- dimensioanl spaces, J. Geom. 64(999), [9] Ž. Milin Šipuš, B. Divjak, Translation surfaces in the Galilean space, accepted for publication in Glas.Mat.Ser.III. [] E. Molnar, The projective interpretation of the eight 3-dimensional homogeneous geometries, Beitr. Algebra Geom. 38(997), 6-88.

13 Translation surfaces 5 [] M. I. Munteanu, A. I. Nistor, Polnomial translation Weingarten surfaces in 3-dimensional Euclidean space, Proceedings of the VIII International Colloquuim on Differential Geometr, (E. Vidal Abascal Centennial Congress), World Scientific 9, 36 3, Eds. J.A.Alvare Lope and E. Garcia Rio. [] O. Röschel, Die Geometrie des Galileischen Raumes, Habilitationsschrift, Leoben, 984. Received: October,