Affine translation surfaces in the isotropic 3-space
AAFFINE TRANSLATION SURFACES IN THE ISOTROPIC3-SPACE
MUHITTIN EVREN AYDIN ∗ , MAHMUT ERGUT Abstract.
The isotropic 3-space I is a real affine 3-space endowed with themetric dx + dy . In this paper we describe Weingarten and linear Weingartenaffine translation surfaces in I . Further we classify the affine translationsurfaces in I that satisfy certain equations in terms of the position vector andthe Laplace operator. Introduction
It is well-known that a surface is called translation surface in a Euclidean 3-space R if it is the graph of a function z ( x, y ) = f ( x ) + g ( y ) for the standartcoordinate system of R . One of the famous minimal surfaces of R is Scherk’sminimal translation surface which is the graph of the function z ( x, y ) = 1 c log (cid:12)(cid:12)(cid:12)(cid:12) cos ( cx )cos ( cy ) (cid:12)(cid:12)(cid:12)(cid:12) , c ∈ R ∗ := R − { } . In order for more generalizations of the translation surfaces to see in various ambientspaces we refer to [4, 5, 7, 12, 16, 19, 20, 24, 26].In 2013, H. Liu and Y. Yu [14] defined the affine translation surfaces in R asthe graph of the function z ( x, y ) = f ( x ) + g ( y + ax ) , a ∈ R ∗ and described the minimal affine translation surfaces which are given by z ( x, y ) = 1 c log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos (cid:0) c √ a x (cid:1) cos ( c [ y + ax ]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , a, c ∈ R ∗ . These are called affine Scherk surface . Then H. Liu and S.D. Jung [15] classifiedthe affine translation surfaces in R of arbitrary constant mean curvature.In the isotropic 3-space I , there exist three different classes of translation sur-faces given by (see [18, 25]) z ( x, y ) = f ( x ) + g ( y ) ,y ( x, z ) = f ( x ) + g ( z ) ,x ( y, z ) = (cid:0) f (cid:0) y + z − π (cid:1) + g (cid:0) π − y + z (cid:1)(cid:1) , where x, y, z are the standart affine coordinates in I . These surfaces are respec-tively called translation surfaces of Type 1,2,3 in I . Such surfaces of constantisotropic Gaussian and mean curvature were obtained in [18] as well as Weingartenones. Mathematics Subject Classification.
Key words and phrases.
Isotropic space, affine translation surface, Weingarten surface. a r X i v : . [ m a t h . DG ] D ec MUHITTIN EVREN AYDIN ∗ , MAHMUT ERGUT The translation surfaces of Type 1 in I that satisfy the condition (cid:52) I,II r i = λ i r i , λ i ∈ R , i = 1 , , , were presented in [13], where r i is the coordinate function of the position vector and (cid:52) I,II the Laplace operator with respect to the first and second fundamental forms,respectively. This condition is natural, being related to the so-called submanifoldsof finite type , introduced by B.-Y. Chen in the late 1970’s (see [8, 9, 11]). Moredetails of translation surfaces in the isotropic spaces can be found in [2, 3, 6].In this paper, we investigate the affine translation surfaces of Type 1 in I , i.e.the graphs of the function z ( x, y ) = f ( ax + by ) + g ( cx + dy ) , ad − bc (cid:54) = 0and classify ones of Weinagarten type. Morever we describe the affine translationsurfaces of Type 1 that satisfy the condition (cid:52) I,II r i = λ i r i .2. Preliminaries
The isotropic 3-space I is a real affine space defined from the projective 3-space P (cid:0) R (cid:1) with an absolute figure consisting of a plane ω and two complex-conjugatestraight lines f , f in ω (see [1, 10, 17], [21]-[23]). Denote the projective coordinatesby ( X , X , X , X ) in P (cid:0) R (cid:1) . Then the absolute plane ω is given by X = 0 andthe absolute lines f , f by X = X + iX = 0 , X = X − iX = 0 . The intersectionpoint F (0 : 0 : 0 : 1) of these two lines is called the absolute point. The group ofmotions of I is a six-parameter group given in the affine coordinates x = X X ,y = X X , z = X X by( x, y, z ) (cid:55)−→ ( x (cid:48) , y (cid:48) , z (cid:48) ) : x (cid:48) = a + x cos φ − y sin φ,y (cid:48) = a + x sin φ + y cos φ,z (cid:48) = a + a x + a y + z, where a , ..., a , φ ∈ R . The metric of I is induced by the absolute figure, i.e. ds = dx + dy . Thelines in z − direction are called isotropic lines . The planes containing an isotropicline are called isotropic planes . Other planes are non-isotropic .Let M be a surface immersed in I . We call the surface M admissible if it hasno isotropic tangent planes. Such a surface can get the form r : D ⊆ R −→ I : ( x, y ) (cid:55)−→ ( r ( x, y ) , r ( x, y ) , r ( x, y )) . The components
E, F, G of the first fundamental form I of M can be calculatedvia the metric induced from I .Denote (cid:52) I the Laplace operator of M with respect to I . Then it is defined as(2.1) (cid:52) φ = 1 (cid:112) | W | (cid:40) ∂∂x (cid:32) Gφ x − F φ y (cid:112) | W | (cid:33) − ∂∂y (cid:32) F φ x − Eφ y (cid:112) | W | (cid:33)(cid:41) , where φ is a smooth function on M and W = EG − F .The unit normal vector field of M is completely isotropic, i.e. (0 , , II are(2.2) L = det ( r xx , r x , r y ) √ EG − F , M = det ( r xy , r x , r y ) √ EG − F , N = det ( r yy , r x , r y ) √ EG − F , where r xy = ∂ r∂x∂y , etc. FFINE TRANSLATION SURFACES IN THE ISOTROPIC 3-SPACE 3
The relative curvature (so-called the isotropic curvature or isotropic Gaussiancurvature ) and the isotropic mean curvature are respectively defined by(2.3) K = LN − M EG − F , H = EN − F M + LG EG − F ) . Assume that nowhere M has parabolic points, i.e. K (cid:54) = 0 . Then the Laplaceoperator with respect to II is given by(2.4) (cid:52) II φ = − (cid:112) | w | (cid:40) ∂∂x (cid:32) N φ x − M φ y (cid:112) | w | (cid:33) − ∂∂y (cid:32) M φ x − Lφ y (cid:112) | w | (cid:33)(cid:41) for a smooth function φ on M and w = LN − M .In particular, if M is a graph surface in I of a smooth function z ( x, y ) then themetric on M induced from I is given by dx + dy . Thus its Laplacian turns to(2.5) (cid:52) I = ∂ ∂x + ∂ ∂y . The matrix of second fundamental form II of M corresponds to the Hessianmatrix H ( z ), i.e., (cid:18) L MM N (cid:19) = (cid:18) z xx z xy z xy z yy (cid:19) . Accordingly, the formulas (2.3) reduce to(2.6) K = det ( H ( z )) , H = trace ( H ( z ))2 . Weingarten affine translation surfaces
Let M be the graph surface in I of the function z ( x, y ) = f ( u ) + g ( v ), where(3.1) u = ax + by, v = cx + dy. If ad − bc (cid:54) = 0, we call the surface M affine translation surface of Type 1 in I andthe pair ( u, v ) affine parameter coordinates .In the particular case a = d = 1 and b = c = 0 (or a = d = 0 and b = c = 1),such a surface reduces to the translation surface of Type 1 in I . Let us fix somenotations to use remaining part: ∂f∂x = a dfdu = af (cid:48) , ∂f∂y = bf (cid:48) , ∂g∂x = c dgdv = cg (cid:48) , ∂g∂y = dg (cid:48) , and so on. By (2 . , the relative curvature K and the isotropic mean curvature H of M turn to(3.2) K = ( ad − bc ) f (cid:48)(cid:48) g (cid:48)(cid:48) and 2 H = (cid:0) a + b (cid:1) f (cid:48)(cid:48) + (cid:0) c + d (cid:1) g (cid:48)(cid:48) . Now we can state the following result to describe the Weingarten affine transla-tion surfaces of Type 1 in I that satisfy the condition(3.3) K x H y − K y H x = 0 , where the subscript denotes the partial derivative. MUHITTIN EVREN AYDIN ∗ , MAHMUT ERGUT Theorem 3.1.
Let M be a Weingarten affine translation surface of Type 1 in I . Then one of the following occurs:(i) M is a quadric surface given by z ( x, y ) = c u + c (cid:0) a + b (cid:1) ( c + d ) v + c u + c v + c , c , ..., c ∈ R ; (ii) M is of the form either z ( x, y ) = f ( u ) + c v + c v + c , f (cid:48)(cid:48)(cid:48) (cid:54) = 0 , c , c , c ∈ R or z ( x, y ) = g ( v ) + c u + c u + c , g (cid:48)(cid:48)(cid:48) (cid:54) = 0 , c , c , c ∈ R , where ( u, v ) is the affine parameter coordinates given by (3.1).Remark . We point out that a quadric surface in I is the set of the pointssatisfying an equation of the second degree. Proof.
It follows from (3 .
2) and (3 .
3) that(3.4) (cid:2)(cid:0) a + b (cid:1) f (cid:48)(cid:48) − (cid:0) c + d (cid:1) g (cid:48)(cid:48) (cid:3) f (cid:48)(cid:48)(cid:48) g (cid:48)(cid:48)(cid:48) = 0 . To solve (3 . , we have several cases: Case (a). (cid:0) a + b (cid:1) f (cid:48)(cid:48) = (cid:0) c + d (cid:1) g (cid:48)(cid:48) . Then we derive z ( x, y ) = c u + c (cid:0) a + b (cid:1) ( c + d ) v + c u + c v + c , c , ..., c ∈ R , which gives the statement (i) of the theorem. Case (b). (cid:0) a + b (cid:1) f (cid:48)(cid:48) (cid:54) = (cid:0) c + d (cid:1) g (cid:48)(cid:48) . Then, by (3.4), the surface has the formeither z ( x, y ) = g ( v ) + c u + c u + c , g (cid:48)(cid:48)(cid:48) (cid:54) = 0or z ( x, y ) = f ( u ) + c v + c v + c , f (cid:48)(cid:48)(cid:48) (cid:54) = 0 , c , ..., c ∈ R . This implies the second statement of the theorem. Therefore the proof is completed. (cid:3)
Now we intend to find the linear Weingarten affine translation surfaces of Type1 in I that satisfy(3.5) αK + βH = γ, α, β, γ ∈ R , ( α, β, γ ) (cid:54) = (0 , , . Without lose of generality, we may assume α (cid:54) = 0 in (3 .
5) and thus it can berewritten as(3.6) K + 2 m H = n , m = βα , n = γα . Hence the following result can be given.
Theorem 3.2.
Let M be a linear Weingarten affine translation surface of Type 1in I that satisfies (3 . . Then we have:(i) M is a quadric surface given by z ( x, y ) = c u + c v + c u + c v + c , c , ..., c ∈ R ; FFINE TRANSLATION SURFACES IN THE ISOTROPIC 3-SPACE 5 (ii) M is of the form either z ( x, y ) = f ( u ) − m (cid:0) a + b (cid:1) ad − bc ) v + c v + c , f (cid:48)(cid:48)(cid:48) (cid:54) = 0 , c , c ∈ R or z ( x, y ) = g ( v ) − m (cid:0) c + d (cid:1) ad − bc ) u + c u + c , g (cid:48)(cid:48)(cid:48) (cid:54) = 0 , c , c ∈ R , where ( u, v ) is the affine parameter coordinates given by (3.1).Proof. Substituting (3 .
2) in (3 .
6) gives(3.7) ( ad − bc ) f (cid:48)(cid:48) g (cid:48)(cid:48) + m (cid:0) a + b (cid:1) f (cid:48)(cid:48) + m (cid:0) c + d (cid:1) g (cid:48)(cid:48) = n . After taking partial derivative of (3 .
7) with respect to u and v, we deduce f (cid:48)(cid:48)(cid:48) g (cid:48)(cid:48)(cid:48) =0. If both f (cid:48)(cid:48)(cid:48) and g (cid:48)(cid:48)(cid:48) are zero then we easily obtain the first statement of thetheorem. Otherwise, we have the second statement of the theorem. This provesthe theorem. (cid:3) Example 3.1.
Consider the affine translation surface of Type 1 in I with z ( x, y ) = cos ( x − y ) + ( x + y ) , − π ≤ x, y ≤ π . This surface plotted as in Fig. 1 satisfies the conditions to be Weingarten and linearWeingarten.4.
Affine translation surfaces satisfying (cid:52)
I,II r i = λ i r i This section is devoted to classify the affine translation surfaces of Type 1 in I that satisfy the conditions (cid:52) I,II r i = λ i r i , λ i ∈ R . For this, we get a localparameterization on such a surface as follows(4.1) r ( x, y ) = ( r ( x, y ) , r ( x, y ) , r ( x, y ))= ( x, y, f ( ax + by ) + g ( cx + dy )) . Thus we first give the following result.
Theorem 4.1.
Let M be an affine translation surface of Type 1 in I that satisfies (cid:52) I r i = λ i r i . Then it is congruent to one of the following surfaces:(i) ( λ , λ , λ ) = (0 , , z ( x, y ) = c u − c (cid:0) a + b (cid:1) ( c + d ) v + c u + c v + c ; (ii) ( λ , λ , λ ) = (0 , , λ > z ( x, y ) = c e (cid:113) λa b u + c e − (cid:113) λa b u + c e (cid:113) λc d v + c e − (cid:113) λc d v ; (iii) ( λ , λ , λ ) = (0 , , λ < z ( x, y ) = c cos (cid:18)(cid:113) − λa + b u (cid:19) + c sin (cid:18)(cid:113) − λa + b u (cid:19) + c cos (cid:18)(cid:113) − λc + d v (cid:19) + c sin (cid:18)(cid:113) − λc + d v (cid:19) , where ( u, v ) is the affine parameter coordinates given by (3.1) and c , ..., c ∈ R . MUHITTIN EVREN AYDIN ∗ , MAHMUT ERGUT Proof.
It is easy to compute from (2 .
5) and (4 .
1) that(4.2) (cid:52) I r = (cid:52) I r = 0and(4.3) (cid:52) I r = (cid:0) a + b (cid:1) f (cid:48)(cid:48) + (cid:0) c + d (cid:1) g (cid:48)(cid:48) . Assuming (cid:52) I r i = λ i r i , i = 1 , ,
3, in (4 .
2) and (4 .
3) yields λ = λ = 0 and(4.4) (cid:0) a + b (cid:1) f (cid:48)(cid:48) + (cid:0) c + d (cid:1) g (cid:48)(cid:48) = λ ( f + g ) , λ = λ. If λ = 0 in (4 . , then we derive f ( u ) = c u + c u + c and g ( v ) = − c (cid:0) a + b (cid:1) ( c + d ) v + c v + c , c , ..., c ∈ R , which proves the statement (i) of the theorem.If λ (cid:54) = 0 then (4 .
4) can be rewritten as(4.5) (cid:0) a + b (cid:1) f (cid:48)(cid:48) − λf = µ = − (cid:0) c + d (cid:1) g (cid:48)(cid:48) + λg, µ ∈ R . In the case λ > , by solving (4 .
5) we obtain f ( u ) = c exp (cid:16)(cid:113) λa + b u (cid:17) + c exp (cid:16) − (cid:113) λa + b u (cid:17) + µλ ,g ( v ) = c exp (cid:16)(cid:113) λc + d v (cid:17) + c exp (cid:16) − (cid:113) λc + d v (cid:17) − µλ , where c , ..., c ∈ R . This gives the statement (ii) of the theorem.Otherwise, i.e., λ < , then we derive f ( u ) = c cos (cid:16)(cid:113) − λa + b u (cid:17) + c sin (cid:16)(cid:113) − λa + b u (cid:17) + µλ ,g ( v ) = c cos (cid:16)(cid:113) − λc + d v (cid:17) + c sin (cid:16)(cid:113) − λc + d v (cid:17) − µλ for c , ..., c ∈ R . This completes the proof. (cid:3)
Example 4.1.
Take the affine translation surface of Type 1 in I with z ( x, y ) = cos ( x + y ) + sin ( x − y ) , − π ≤ x, y ≤ π. Then it satisfies (cid:52) I r i = λ i r i for λ = λ = 0, λ = − I that satisfies (cid:52) II r i = λ i r i , λ i ∈ R . Then its Laplace operator with respect to the second funda-mental form II has the form(4.6) (cid:52) II φ = ( f (cid:48)(cid:48) g (cid:48)(cid:48) ) − ad − bc ) (cid:104)(cid:0) − bφ x + aφ y (cid:1) ( f (cid:48)(cid:48) ) g (cid:48)(cid:48)(cid:48) + (cid:0) dφ x − cφ y (cid:1) f (cid:48)(cid:48)(cid:48) ( g (cid:48)(cid:48) ) (cid:105) + ( f (cid:48)(cid:48) g (cid:48)(cid:48) ) − ( ad − bc ) (cid:2)(cid:0) abφ xy − b φ xx − a φ yy (cid:1) f (cid:48)(cid:48) + (cid:0) cdφ xy − d φ xx − c φ yy (cid:1) g (cid:48)(cid:48) (cid:3) for a smooth function φ and f (cid:48)(cid:48) g (cid:48)(cid:48) (cid:54) = 0 . Hence we have the following result.
FFINE TRANSLATION SURFACES IN THE ISOTROPIC 3-SPACE 7
Theorem 4.2.
Let M be an affine translation surface of Type 1 in I that satisfies (cid:52) II r i = λ i r i . Then it is congruent to one of the following surfaces:(i) ( λ (cid:54) = 0 , λ (cid:54) = 0 , z ( x, y ) = ln (cid:16) x λ y λ (cid:17) + c , c ∈ R ; (ii) ( λ (cid:54) = 0 , λ, z ( x, y ) = ln (cid:16) ( uv ) λ (cid:17) + c , c ∈ R , where ( u, v ) is the affine parameter coordinates given by (3 . .Proof. Let us assume that (cid:52) II r i = λ i r i , λ i ∈ R . Then, from (4 .
1) and (4 . , westate the following system(4.7) d f (cid:48)(cid:48)(cid:48) ( f (cid:48)(cid:48) ) − b g (cid:48)(cid:48)(cid:48) ( g (cid:48)(cid:48) ) = 2 ( ad − bc ) λ x, (4.8) − c f (cid:48)(cid:48)(cid:48) ( f (cid:48)(cid:48) ) + a g (cid:48)(cid:48)(cid:48) ( g (cid:48)(cid:48) ) = 2 ( ad − bc ) λ y, (4.9) f (cid:48)(cid:48)(cid:48) f (cid:48) ( f (cid:48)(cid:48) ) + g (cid:48)(cid:48)(cid:48) g (cid:48) ( g (cid:48)(cid:48) ) − λ ( f + g ) . In order to solve above system we have to distinguish two cases depending on theconstants a, b, c, d for ad − bc (cid:54) = 0 . Case (a).
Two of a, b, c, d are zero. Without loss of generality we may assumethat b = c = 0 and a = d = 1. Then the equations (4.7) and (4.8) reduce to(4.10) f (cid:48)(cid:48)(cid:48) ( f (cid:48)(cid:48) ) = 2 λ x and(4.11) g (cid:48)(cid:48)(cid:48) ( g (cid:48)(cid:48) ) = 2 λ y. If λ = λ = 0 then we obtain a contradiction from (4.9) since f, g are non-constantfunctions. Thereby we need to consider the remaining cases: Case (a.1). λ = 0 , i.e. f (cid:48)(cid:48)(cid:48) = 0 . Then substituting (4 .
10) and (4 .
11) into (4 . λ = 0 and g ( y ) = 2 λ ln y + c , c ∈ R . Substituting it in (4.11) gives a contradiction.
Case (a.2). λ = 0 , i.e. g (cid:48)(cid:48)(cid:48) = 0 . Hence we can similarly obtain λ = 0 and f ( x ) = 2 λ ln x + c , c ∈ R , which gives a contradiction by considering it into (4.10). Case (a.3). λ λ (cid:54) = 0 . By substituting (4 .
10) and (4 .
11) into (4 .
9) we deduce(4.12) λ xf (cid:48) + λ yg (cid:48) − λ ( f + g ) . Case (a.3.1). If λ = 0, then (4.12) reduces to(4.13) λ xf (cid:48) + λ yg (cid:48) = 2 . MUHITTIN EVREN AYDIN ∗ , MAHMUT ERGUT By solving (4 .
13) we find(4.14) f ( x ) = ξλ ln x + c and g ( v ) = 2 − ξλ ln y + c , c , c ∈ R , ξ ∈ R ∗ . Substituting (4.14) into (4.10) and (4.11) yields ξ = 1. This proves the first state-ment of the theorem. Case (a.3.2). If λ (cid:54) = 0 in (4.12) then we can rewrite it as(4.15) λ xf (cid:48) − λ f − c = − λ yg (cid:48) + λ g, c ∈ R . After solving (4 . , we conclude(4.16) f ( x ) = − c λ + c x λ λ and(4.17) g ( y ) = c λ + c y λ λ , c , c ∈ R . By considering (4.16) and (4.17) into (4.10) and (4.11), respectively, we conclude λ = 0, which implies that this case is not possible. Case (b).
At most one of a, b, c, d is zero. Suppose that λ = 0 in (4.7). Itfollows from (4 .
7) that(4.18) f (cid:48)(cid:48)(cid:48) ( f (cid:48)(cid:48) ) = c d and g (cid:48)(cid:48)(cid:48) ( g (cid:48)(cid:48) ) = c b , c ∈ R , where we may assume that b (cid:54) = 0 (cid:54) = d since at most one of a, b, c, d can vanish. If c = 0 then we derive a contradiction from (4 .
9) since f (cid:48)(cid:48) g (cid:48)(cid:48) (cid:54) = 0 . Considering (4 . .
8) yields c bd = 2 λ y, which is no possible since y is an independent variable.This implies that λ is not zero and it can be similarly shown that λ is not zero.Hence from (4 .
7) and (4 .
8) we can write(4.19) f (cid:48)(cid:48)(cid:48) ( f (cid:48)(cid:48) ) = 2 ( λ ax + λ by )and(4.20) g (cid:48)(cid:48)(cid:48) ( g (cid:48)(cid:48) ) = 2 ( λ cx + λ dy ) . Compatibility condition in (4 .
19) or (4 .
20) gives λ = λ . Put λ = λ = λ. Bysubstituting (4 .
19) and (4 .
20) into (4 .
9) we deduce(4.21) λuf (cid:48) + λvg (cid:48) − λ ( f + g ) , where ( u, v ) is the affine parameter coordinates given by (3 . Case (b.1). If λ = 0, then (4.21) reduces to(4.22) λuf (cid:48) + λvg (cid:48) = 2 . By solving (4 .
22) we find(4.23) f ( u ) = ξλ ln u + c and g ( v ) = 2 − ξλ ln v + c , c , c ∈ R , ξ ∈ R ∗ . Substituting (4.23) into (4.19) and (4.20) yields ξ = 1. This proves the secondstatement of the theorem. Case (b.2). If λ (cid:54) = 0 in (4.11) then we can rewrite it as(4.24) λuf (cid:48) − λ f − c = − λvg (cid:48) + λ g, c ∈ R . FFINE TRANSLATION SURFACES IN THE ISOTROPIC 3-SPACE 9
After solving (4 . , we deduce(4.25) f ( u ) = − c λ + c u λ λ and(4.26) g ( v ) = c λ + c v λ λ , c , c ∈ R . Considering (4.25) and (4.26) into (4.19) and (4.20), respectively, we find λ = 0,however this is a contradiction. (cid:3) Example 4.2.
Given the affine translation surface of Type 1 in I as follows z ( x, y ) = ln (2 x + y ) + ln ( x − y ) , ( u, v ) ∈ [3 , × [1 , . Then it holds (cid:52) II r i = λ i r i for ( λ , λ , λ ) = (1 , ,
0) and we plot it as in Fig. 3.
Figure 1.
A (linear) Weingarten affine translation surface of Type 1.
Figure 2.
An affine translation surface of Type 1 with (cid:52) I r i = λ i r i , ( λ , λ , λ ) = (0 , , References [1] M.E. Aydin, I. Mihai,
On certain surfaces in the isotropic 4-space , Math. Commun., in press.[2] M.E. Aydin, A. Ogrenmis,
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