3. LW-factorable surfaces of type 1

DOI: 10.24193/subbmath.2017.2.11

Linear Weingarten factorable surfaces in isotropic spaces

Muhittin Evren Aydin and Alper Osman Ogrenmis

Abstract. In this paper, we deal with a certain factorable surface in the isotropic 3-space satisfyingaK+bH=c, whereKis the relative curvature,Hthe isotropic mean curvature and a, b, c ∈ R. We obtain a complete classification for such surfaces. As a further study, we prove that a certain graph surface withK=H² is either a non-isotropic plane or a parabolic sphere.

Mathematics Subject Classification (2010):53A35, 53A40, 53B25.

Keywords:Isotropic space, factorable surface, Weingarten surface, Euler inequal- ity.

1. Introduction

Let M² be a regular surface of a Euclidean 3-space R³ and κ1, κ2 its princi- pal curvatures. Then M² is called a Weingarten surface if the following non-trivial functional relation occurs:

φ(κ₁, κ₂) = 0 (1.1)

for a smooth functionφof two variables. (1.1) immediately yields

δ(K, H) = 0, (1.2)

where K and H are respectively the Gaussian and mean curvatures of M². (1.2) is equivalent to the vanishing of the corresponding Jacobian determinant, i.e.

|∂(K, H)/∂(u, v)| = 0 for a coordinate pair (u, v) on M². If M² is a surface that satisfies

aH+bK=c, a, b, c∈R, (a, b, c)6= (0,0,0), (1.3) then it is called alinear Weingarten surface (LW-surface). Ifa= 0 orb= 0 in (1.3), then the LW-surfaces reduce to the surfaces with constant curvature. Such surfaces have been extensively studied, see [7, 8], [13]-[17], [30].

On the other hand, a surface in R³ that is the graph of the functionz(x, y) = f(x)g(y) is said to be factorable or homothetical. In various ambient spaces, these

surfaces have been desribed in terms of their curvatures and Laplace operator in [4, 9, 10, 12, 18, 19, 28, 29]. As distinct from the Euclidean case, a graph surface in the isotropic space I³ is said to be factorable if it is graph of either z(x, y) = f(x)g(y) orx(y, z) =f(y)g(z). We call them thefactorable surface of type 1 and type 2, respectively. Note that the factorable surface of one type cannot be carried into that of another type by the isometries of I³. These surfaces of both type in I³ withK, H=const.were obtained in [1]-[3].

The main purpose of this paper is to obtain LW-factorable surfaces of type 1 in I³.As a further study, we classify the graph surfaces of the functionz=z(x, y) inI³ withK=H².

2. Preliminaries

For general references of the isotropic geometry, see [5], [23]-[27]. The isotropic 3- spaceI³is a Cayley-Klein space defined from a 3-dimensional projective spaceP R³ with the absolute figure (ω, f₁, f₂), where ω is a plane in P R³

and f₁, f₂ are two complex-conjugate straight lines in ω. The homogeneous coordinates inP R³

are introduced in such a way that theabsolute planeωis given byX0= 0 and theabsolute lines f1, f2 by X0 = X1+iX2 = 0, X0 = X1−iX2 = 0. The intersection point F(0 : 0 : 0 : 1) of these two lines is called the absolute point. The affine coordinates in P R³

are given byx1= X1

X₀, x2= X2

X₀, x3= X3

X₀. The group of motions ofI³ is defined by

(x1, x2, x3)7−→(x⁰₁, x⁰₂, x⁰₃) :







x⁰₁=a1+x1cosφ−x2sinφ, x⁰₂=a2+x1sinφ+x2cosφ, x⁰₃=a₃+a₄x₁+a₅x₂+x₃, wherea1, ..., a5, φ∈R.

Consider the points x= (x₁, x₂, x₃) andy = (y₁, y₂, y₃). Theisotropic distance d_I(x, y) of two pointsxandy is defined as

d_I(x, y) = (y1−x1)²+ (y2−x2)².

The lines inx3−direction are calledisotropiclines. The plane containing an isotropic line is called an isotropic plane.Other planes are non-isotropic.

LetM²be a graph surface immersed inI³corresponding to a real-valued smooth functionz=z(x, y) on an open domainD⊆R². Then it is parameterized as follows:

r:D⊆R²−→I³, (x, y)7−→(x, y, z(x, y)). (2.1) It follows from (2.1) thatM²is an admissible (i.e. without isotropic tangent planes).

The metric on M² induced from I³ is given by g∗ = dx²+dy². This implies that M² is always flat with respect to the induced metric g∗ and thus its Laplacian is of the form4= ∂²

∂x² + ∂²

∂y².Therelative (orisotropic Gaussian)curvature K and the isotropic mean curvature H ofM² are defined by

K=zxxzyy−(zxy)² (2.2)

and

H= 4z

2 =zxx+zyy

2 . (2.3)

M² is calledisotropic minimal (resp.isotropic flat) ifH (resp.K) vanishes.

3. LW-factorable surfaces of type 1

Let M² be a factorable surface of type 1 in I³, i.e., the graph of z(x, y) =f(x)g(y).By (2.2) and (2.3), we get

K= (f⁰⁰f) (g⁰⁰g)−(f⁰)²(g⁰)² (3.1) and

2H =f⁰⁰g+f g⁰⁰, (3.2)

wheref⁰=_dx^df andg⁰=_dy^dg,etc. We mainly aim to classify the LW-factorable surfaces of type 1 inI³. For this, letM²satisfy the relation (1.3). Since at least one ofa, band c is nonzero in (1.3), without loss of generality, we may assumeb 6= 0. By dividing both sides of (1.3) withband putting ^a_b = 2m₀ and ^c_b =n₀, we write

2m0H+K=n0, m0, n0∈R. (3.3) If m0 = 0 , M² turns to be a factorable surface of type 1 in I³ with K = const.

however such surfaces were already provided in [1]. In our framework, it is meaningful to takem06= 0.By (3.1)−(3.3),we get

(f⁰⁰f) (g⁰⁰g)−(f⁰)²(g⁰)²+m0(f⁰⁰g+f g⁰⁰) =n0. (3.4) We have to distinguish several cases in order to solve (3.4).Remark that the roles of f andgare symmetric, so discussing on the cases based onf shall be sufficient. From now on, we use the notation c_i to denote nonzero constants and d_i to denote some constants,i= 1,2,3, ...

Case 1.f(x) =f0∈R− {0}.By (3.4), we find g(y) = n0

2f0m0

y²+d1y+d2. (3.5)

Thereby,M² is isotropic flat factorable surface of type 1 withH =_2mⁿ⁰

0. Case 2.f is a linear function, i.e.f(x) =c₁x+d₃.It follows from (3.4) that

m0d3g⁰⁰−c²₁(g⁰)²+ (m0c1g⁰⁰)x=n0. (3.6) (3.6) implies that g⁰⁰ = 0, namely g(y) = c₂y +d₄. In this case, M² is isotropic minimal factorable surface of type 1 withK=−(c1c2)².

Case 3.f is a non-linear function. From the symmetry,gis also a non-linear function.

By dividing (3.4) with the productf f⁰⁰,we get g⁰⁰g−(f⁰)²

f f⁰⁰ (g⁰)²+m0

g f +m0

g⁰⁰ f⁰⁰ = n0

f f⁰⁰. (3.7)

By taking partial derivative (3.7) with respect toy and then dividing with g⁰g⁰⁰, we deduce

1 + gg⁰⁰⁰

g⁰g⁰⁰ −2(f⁰)² f f⁰⁰ +

m0

f 1

g⁰⁰ + m0

f⁰⁰ g⁰⁰⁰

g⁰g⁰⁰ = 0. (3.8) We have two cases:

Case 3.1.g⁰⁰⁰= 0,i.e.

g(y) =c₃y²+d₅y+d₆. (3.9) Up to suitable translations ofy, we may assumed₅=d₆= 0. Then (3.8) reduces to

1−2(f⁰)² f f⁰⁰ +

m0

2c₃ 1

f = 0. (3.10)

(3.10) can be rewritten as m0

2c3

+f

f⁰⁰−2 (f⁰)²= 0. (3.11) After solving (3.11),we find

f(x) =− 1

c4x+d7

+m0

2c3

. (3.12)

Considering (3.9) and (3.12) into (3.4) gives the contradiction x=−1

c4

2m₀c₃ n0+m²₀ +d7

due to the fact thatxis an independent variable.

Case 3.2.g⁰⁰⁰ 6= 0.By taking partial derivatives of (3.8) with respect to xandy, we conclude

f⁰ f²

g⁰⁰⁰

(g⁰⁰)²− f⁰⁰⁰ (f⁰⁰)²

g⁰⁰⁰ g⁰g⁰⁰

0

= 0. (3.13)

Due tof⁰g⁰⁰⁰ 6= 0, neither f⁰⁰⁰ nor g⁰⁰⁰

g⁰g⁰⁰ 0

can vanish in (3.13). Then (3.13) can be rewritten as

f⁰(f⁰⁰)²

f²f⁰⁰⁰ = (g⁰⁰)² g⁰⁰⁰

g⁰⁰⁰ g⁰g⁰⁰

0

. (3.14)

Since the left side of (3.14) is a function ofx,however the right side is a function of y.Then both sides have to be equal a nonzero constant, namely

f⁰(f⁰⁰)²

f²f⁰⁰⁰ =c₅=(g⁰⁰)² g⁰⁰⁰

g⁰⁰⁰ g⁰g⁰⁰

0

. (3.15)

From the left side of (3.15),we write f⁰⁰⁰ (f⁰⁰)² = 1

c5

f⁰

f² (3.16)

or, by taking once integral with respect tox, f⁰⁰= c5f

c5d8f+ 1. (3.17)

Likewise, by the right side of (3.15), we deduce g⁰⁰⁰

g⁰g⁰⁰ = −c₅

g⁰⁰ +d₉. (3.18)

Substituting (3.17) and (3.18) into (3.8) yields 1 + (m0d8+g)d9−c5(m0d8+g)

g⁰⁰ +m0d9

c₅f −

−2 (c5d8f + 1) (f⁰)² c5f² = 0.

(3.19)

Taking partial derivative of (3.19) with respect toy and considering (3.18) leads to g⁰⁰=−c5(g+m0d8). (3.20) After substituting (3.20) into (3.19), we conclude

2 +m0d8d9+d9g+m₀d₉

c5f −2 (f⁰)² f f⁰⁰ = 0,

which yieldsd₉= 0 andf f⁰⁰= (f⁰)².Solving this one givesf(x) =c₆exp (c₇x).By putting this in (3.4) we derive the polynomial equation on (f):

c²₇h

gg⁰⁰−(g⁰)²i

f²+m₀ c²₇g+g⁰⁰

f −n₀= 0, which implies that the coefficients must be zero; namelyn₀= 0,

gg⁰⁰−(g⁰)²= 0 andc²₇g+g⁰⁰= 0. (3.21) (3.21) leads to the contradiction c²₇g²+ (g⁰)² = 0 and therefore we have proved the following:

Theorem 3.1. Let M² be a LW-factorable surface of type 1 which is the graph of z(x, y) =f(x)g(y)inI³. Then we have either

(A) f(x) =f0∈R− {0}, g(y) =c6y²+d10y+d11; (B) orz(x, y) = (c7x+d12) (c8y+d13).

4. Graph surfaces with K = H

LetM² be a surface of the Euclidean 3-space R³. The Euler inequality for M² including the Gaussian and mean curvature follows

K≤H². (4.1)

The equality sign of (4.1) holds onM²if and only if it is totally umbilical, i.e. a part of a plane or a two sphere inE³. For more generalizations, see [6, 11], [20]-[22]. Now we are interested in the factorable surfaces of type 1 in I³ satisfying K =H². For this, let us reconsider (3.1) and (3.2). IfK=H², then

(f⁰⁰g−f g⁰⁰)²+ 4 (f⁰g⁰)²= 0. (4.2) (4.2) immediately implies that

f⁰⁰g−f g⁰⁰= 0 andf⁰g⁰= 0. (4.3)

By (4.3) we conclude that either f =const. andg(y) =c1y+d1 or g=const. and f(x) =c2x+d2, which yields the following result:

Proposition 4.1. The factorable surfaces of type 1 inI³ satisfying K =H² are only non-isotropic planes.

As a generalization, we are able to investigate the graph surfaces of type 1 inI³ satisfyingK=H².More precisely, letM² be a graph surface ofz=z(x, y) inI³.If K=H² onM²,then we get

(zxx−zyy)²+ 4 (zxy)²= 0, (4.4) which yields that

zxy= 0 (4.5)

and

z_xx=z_yy. (4.6)

By (4.5),we derive

z(x, y) =α(x) +β(y) (4.7)

and considering (4.7) into (4.6) gives d²α dx² =d²β

dy² =d3, d3∈R. (4.8)

By solving (4.8),we find α(x) =d₃

2x²+d₄x+d₅, β(y) =d₃

2 y²+d₆y+d₇. (4.9) (4.9) implies that M² is either a non-isotropic plane (d₃= 0) or a parabolic sphere (d₃6= 0). Consequently, we have

Theorem 4.2. A graph surface of a functionz=z(x, y)inI³ with K=H² is either (a piece of ) a non-isotropic plane or (a piece of ) a parabolic sphere given by

z(x, y) =c3 x²+y²

+d8x+d9y+d10.

References

[1] Aydin, M.E., Ergut, M., Isotropic geometry of graph surfaces associated with product production functions in economics, Tamkang J. Math.,47(4)(2016), 433-443.

[2] Aydin, M.E., Constant curvature factorable surfaces in 3-dimensional isotropic space, arXiv:1612.02458 [math.DG].

[3] Aydin, M.E., Ogrenmis, A.O.,Homothetical and translation hypersurfaces with constant curvature in the isotropic space, BSG Proceedings,23(2016), 1-10.

[4] Bekkar, M., Senoussi, B., Factorable surfaces in the three-dimensional Euclidean and Lorentzian spaces satisfying4ri=λiri,J. Geom.,103(2012), 17-29.

[5] Chen, B.Y., Decu, S., Verstraelen, L.,Notes on isotropic geometry of production models, Kragujevac J. Math.,38(2014), no. 1, 23-33.

[6] Chen, B.Y.,Mean curvature and shape operator of isometric immersions in real-space- forms, Glasg. Math. J.,38(1996), 87-97.

[7] Dillen, F., Kuhnel, W.,Ruled Weingarten surfaces in Minkowski 3-space, Manuscripta Math.,98(1999), 307-320.

[8] Galvez, J.A., Martinez, A., Milan, F., Linear Weingarten surfaces in R³, Monatsh.

Math.,138(2003), 133-144.

[9] Goemans, W., Van de Woestyne, I.,Translation and homothetical lightlike hypersurfaces of semi-Euclidean space, Kuwait J. Sci. Eng.,38(2011), no. 2A, 35-42.

[10] Gray, A., Abbena, E., Salamon, S.,Modern differential geometry of curves and surfaces with mathematica, Third Edition, Chapman and Hall/CRC Press, New York, 2006.

[11] Hou, Z.H., Ji, F.,Helicoidal surfaces withH²=Kin Minkowski 3-space, J. Math. Anal.

Appl.,325(2007), 101-113.

[12] Jiu, L., Sun, H.,On minimal homothetical hypersurfaces, Colloq. Math.,109(2007), 239- 249.

[13] Kim, M.H., Yoon, D.W.,Weingarten quadric surfaces in a Euclidean 3-space, Turk. J.

Math.,35(2011), 479-485.

[14] Kuhnel, W.,Ruled W-surfaces, Arch. Math.,62(1994), 475-480.

[15] Lee, C.W., Linear Weingarten rotational surfaces in pseudo-Galilean 3-space, Int. J.

Math. Anal.,9(50)(2015), 2469-2483.

[16] Liu, H., Liu, G.,Weingarten rotation surfaces in 3-dimensional de Sitter space, J. Geom., 79(2004), 156-168.

[17] Lopez, R., Rotational linear Weingarten surfaces of hyperbolic type, Israel J. Math., 167(2008), 283-301.

[18] Lopez, R., Moruz, M., Translation and homothetical surfaces in Euclidean space with constant curvature, J. Korean Math. Soc.,52(2015), no. 3, 523-535.

[19] Meng, H., Liu, H., Factorable surfaces in Minkowski space, Bull. Korean Math. Soc., 46(2009), no. 1, 155-169.

[20] Mihai, I.,On the generalized Wintgen inequality for Lagrangian submanifolds in complex space forms, Nonlinear Analysis,95(2014), 714-720.

[21] Mihai, I.,On the generalized Wintgen inequality for Legendrian submanifolds in Sasakian space forms, Tohoku J. Math, to appear.

[22] Mihai, A.,Geometric inequalities for purely real submanifolds in complex space forms, Results Math.,55(2009), 457-468.

[23] Milin-Sipus, Z.,Translation surfaces of constant curvatures in a simply isotropic space, Period. Math. Hung.,68(2014), 160-175.

[24] Pottmann, H., Opitz, K.,Curvature analysis and visualization for functions defined on Euclidean spaces or surfaces, Comput. Aided Geom. Design,11(1994), 655-674.

[25] Pottmann, H., Grohs, P., Mitra, N.J., Laguerre minimal surfaces, isotropic geometry and linear elasticity, Adv. Comput. Math.,31(2009), 391-419.

[26] Sachs, H., Isotrope Geometrie des Raumes, Vieweg-Verlag, Braunschweig, Wiesbaden, 1990.

[27] Strubecker, K., Differentialgeometrie des isotropen Raumes III, Flachentheorie, Math.

Zeitsch.,48(1942), 369-427.

[28] Van de Woestyne, I., Minimal homothetical hypersurfaces of a semi-Euclidean space, Results Math.,27(1995), 333-342.

[29] Yu, Y., Liu, H.,The factorable minimal surfaces, Proceedings of The Eleventh Interna- tional Workshop on Diff. Geom.,11(2007), 33-39.

[30] Yoon, D.W., Tuncer, Y., Karacan, M.K.,Non-degenerate quadric surfaces of Weingarten type, Annales Polonici Math.,107(2013), 59-69.

Muhittin Evren Aydin

Firat University, Faculty of Science Department of Mathematics 23200 Elazig, Turkey

e-mail:[email protected] Alper Osman Ogrenmis

Firat University, Faculty of Science Department of Mathematics 23200 Elazig, Turkey

e-mail:[email protected]