ROTATIONAL SURFACES IN ISOTROPIC SPACES SATISFYING WEINGARTEN CONDITIONS
Alper Osman Öğrenmiş
Department of Mathematics, Faculty of Science
Firat University, Elazig, 23119, Turkey
Abstract. In this paper, we study the rotational surfaces in the isotropic 3-space satisfying Weingarten conditions in terms of the relative curvature (analogue of the Gaussian curvature) and the isotropic mean curvature . In particular, we classify such surfaces of linear Weingarten type in
Keywords: Isotropic space; rotational surface; Weingarten surface.
Math. Subject Classification 2010: , , .
The work of surfaces with special properties in the isotropic 3-space has important applications in several applied sciences, e.g., computer science, Image Processing, architectural design and microeconomics, see [3, 4, 6, 8], -.
Denote the Levi-Civita connection of and the normal vector field to Then the operator given by
is called the shape operator, where is a tangent vector field to It measures how bends in different directions. The eigenvalues of are called the principal curvatures donoted by and
The arithmetic mean of the principal curvatures are called the mean curvature, The Gaussian curvature is defined by
A surface in is called a Weingarten surface (W-surface) if it satisfies the following non-trivial functional relation
for a smooth function of two variables. The above relation implies the following
which is the equivalent to the vanishing of the corresponding Jacobian determinant, i.e. for a coordinate pair on
If fulfills the following condition
then it is called a linear Weingarten surface (LW-surface). In the particular case (resp. ) the LW-surfaces are indeed the surfaces with constant Gaussian curvature (resp. mean curvature). These phenomenal surfaces have been stuied by many geometers in various ambient spaces, see [14, 20], -, , .
The motivation of the present paper is to study Weingarten surfaces, in particular Weingarten rotational surfaces, in the isotropic 3-space which is one the Cayley–Klein spaces.
Most recently, M.E. Aydin () classified the helicoidal surfaces in , which are natural generalization of the rotational surfaces, with constant curvature and analyzed some special curves on such surfaces.
In the present paper, we provide that the rotational surfaces in are evidently Weingarten ones. Then we classified LW-rotational surfaces in satisfying the following relation
in which is the relative curvature and isotropic mean curvature.
The isotropic 3-space is obtained from the 3-dimensional projective space with the absolute figure which is an ordered triple , where is a plane in and are two complex-conjugate straight lines in (see ). The homogeneous coordinates in are introduced in such a way that the absolute plane is given by and the absolute lines by The intersection point of these two lines is called the absolute point. The group of motions of is a six-parameter group given in the normal form (in affine coordinates) by
Such affine transformations are called isotropic congruence transformations or i-motions.
Consider the points and The isotropic distance, so-called i-distance of two points and is defined by
The i-metric is degenerate along the lines in direction, and such lines are called isotropic lines.
(1) Non-isotropic planes are planes non-parallel to the direction. In these planes we basically have an Euclidean metric: This is not the one we are used to, since we have to make the usual Euclidean measurements in the top view. An i-circle (of elliptic type) in a non-isotropic plane is an ellipse, whose top view is an Euclidean circle. Such an i-circle with center and radius is the set of all points with
(2) Isotropic planes are planes parallel to the axis. There, induces an isotropic metric. An i-circle (of parabolic type) is a parabola with parallel axis and thus it lies in an isotropic plane
An i-circle of parabolic type is not the iso-distance set of a fixed point, but it may be seen as a curve with constant isotropic curvature: A curve in an isotropic plane P (without loss of generality we set ) which does not possess isotropic tangents can be written as graph . Then, the i-curvature of at is given by the second derivative . For an i-circle of parabolic type is quadratic and thus is constant.
There are also two types of isotropic spheres. An i-sphere of the cylindrical type is the set of all points with . Speaking in an Euclidean way, such a sphere is a right circular cylinder with parallel rulings; its top view is the Euclidean circle with center and radius . The more interesting and important type of spheres are the i-spheres of parabolic type,
From an Euclidean perspective, they are paraboloids of revolution with parallel axis. The intersections of these i-spheres with planes are i-circles. If is non-isotropic, then the intersection is an i-circle of elliptic type. If is isotropic, the intersection curve is an i-circle of parabolic type.
Curvature theory of surfaces. A surface immersed in is called admissible if it has no isotropic tangent planes. We restrict our framework to admissible regular surfaces. For such a surface , the coefficients of its first fundamental form are calculated with respect to the induced metric.
The normal field of is always the isotropic vector The coefficients of the second fundamental form of are calculated with respect to the normal field of (for details, see , p. 155).
The relative curvature (so called isotropic Gaussian curvature) and isotropic mean curvature are defined by
3 LW-rotational surfaces in
Let us consider the i-motions given by then the Euclidean rotations in the isotropic space is given by in affine coordinates
Definition 3.1. Let be a curve lying in the isotropic plane given by where . By rotating the curve around axis, we obtain that the rotational surface in is of the form
Similarly when the profile curve lies in the isotropic plane, then the rotational surface in is given by
Remark 3.1. The rotational surfaces given by and are locally isometric and thus we only consider the ones of the form .
Let be the rotational surface given by in Then the nonzero components of first fundamental form of are calculated by induced metric from as follows
The nonzero components of second fundamental form of are
where and From and we get
which yields that the curvatures and depend only on the variable , namely In the sequel, we have the following result.
Theorem 3.1. Rotational surfaces in are Weingarten surfaces.
We are also able to investigate the LW-rotational surfaces in with the relation
If in then those reduce to ones with constant relative curvature. Thus we aim to obtain the LW-rotational surfaces in with
The following result classifies the LW-rotational surfaces satisfying
Theorem 3.2. Let be a LW-rotational surface in . Then one of the following holds
(i) is of the form
(ii) is an elliptic paraboloid from the Euclidean perspective, i.e.
(iii) is given by
Proof. Assume is a LW-rotational surface in having the relation Then, from it follows
We have two cases:
Case a. Hence we can rewrite as
If in then and vanish which is not possible. Then we have
By solving we obtain
which gives the statement (i) of the theorem.
Case b. Then we have from
When then and This implies the statement (ii) of the theorem.
Otherwise, we conclude from that
After solving we derive
Therefore the proof is completed.
Example 3.1. Consider the elliptic paraboloid in from the Euclidean perspective given by
Then and We plot it as in Fig. 1.
4 Rotational surfaces in with
The authors in  introduced a new kind of curvature for the hypersurfaces of Euclidean spaces, called by amalgamatic curvature and explored its geometric meaning by proving an inequality related to the absolute mean curvature of the hypersurface. In the particular case , the amalgamatic curvature is indeed the harmonic ratio of the principal curvatures of any given surface, i.e., the ratio of the Gaussian curvature and the mean curvature.
By considering this argument, we can consider the rotational surfaces in satisfying Thus the statement (i) of Theorem 3.2 is indeed a classification of the rotational surfaces in satisfying
Therefore, we have the following trivial result.
Corollary 4.1. Let be a LW-rotational surface in satisfying Then it is of the form
Example 4.1. Take and in Then we obtain a rotational surface in with given by
where , Then it can be plotted as in Fig. 2.
-  M.E. Aydin, A generalization of translation surfaces with constant curvature in the isotropic space, J. Geom., 2015, DOI 10.1007/s00022-015-0292-0.
-  M.E. Aydin, Classification results on surfaces in the isotropic 3-space, arXiv:1601.03190v1 [math.DG], 2016.
-  M.E. Aydin and A. Mihai, Classification of quasi-sum production functions with Allen determinants, Filomat 29(6) (2015), 1351–1359.
-  M.E. Aydin and A. Mihai, Translation hypersurfaces and Tzitzeica translation hypersurfaces of the Euclidean space, Proc. Ro. Acad. Series A 16(4) (2015), 477-483.
-  C. Baikoussis and T. Koufogioros, Helicoidal surface with prescribed mean or Gauss curvature, J. Geom. 63 (1998), 25–29.
-  B. Y. Chen, S. Decu and L. Verstraelen, Notes on isotropic geometry of production models, Kragujevac J. Math. 37(2) (2013), 217–220.
-  C. T. R. Conley, R. Etnyre, B. Gardener, L. H. Odom and B. D. Suceava, New curvature inequalities for hypersurfaces in the Euclidean ambient space, Taiwanese J. Math. 17(3) (2013), 885–895.
-  S. Decu, L. Verstraelen, A note on the isotropical geometry of production surfaces, Kragujevac J. Math. 38(1) (2014), 23–33.
-  G. Delaunay, Sur la surface de revolution dont la courbure moyenne est constante, J. Math. Pures Appl. Series 6(1) (1841), 309-320.
-  F. Dillen and W. Kuhnel, Ruled Weingarten surfaces in Minkowski 3-space, Manuscripta Math., 98 (1999), 307-320.
-  M.P. Do Carmo and M. Dajczer, Helicoidal surfaces with constant mean curvature, Tohoku Math. J. 34 (1982), 425-435.
-  M.P. Do Carmo, Differential geometry of curves and surfaces, Prentice Hall: Englewood Cliffs, NJ, 1976.
-  Z. Erjavec, B. Divjak and D. Horvat, The general solutions of Frenet’s system in the equiform geometry of the Galilean, pseudo-Galilean, simple isotropic and double isotropic space, Int. Math. Forum 6(17) (2011), 837-856.
-  J. A. Galvez, A. Martinez and F. Milan, Linear Weingarten surfaces in , Monatsh. Math., 138 (2003), 133-144.
-  A. Gray, Modern differential geometry of curves and surfaces with mathematica. CRC Press LLC, 1998.
-  Z.H. Hou and F. Ji, Helicoidal surfaces with in Minkowski 3-space, J. Math. Anal. Appl. 325 (2007), 101–113.
-  I. Kamenarovic, On line complexes in the isotropic space Glasnik Matematicki 17(37) (1982), 321-329.
-  I. Kamenarovic, Associated curves on ruled surfaces in the isotropic space Glasnik Matematicki 29(49) (1994), 363-370.
-  K. Kenmotsu, Surfaces of revolution with prescribed mean curvature, Tohoku Math. J. 32 (1980), 147-153.
-  M. H Kim and D. W. Yoon, Weingarten quadric surfaces in a Euclidean 3-space, Turk. J. Math. 35 (2011), 479-485.
-  J. J. Koenderink and A. van Doorn, Image processing done right, Lecture Notes in Computer Science 2350 (2002), 158–172.
-  W. Kuhnel, Ruled W-surfaces, Arch. Math. 62 (1994), 475-480.
-  C.W. Lee, Linear Weingarten rotational surfaces in pseudo-Galilean 3-space, Int. J. Math. Anal. 9(50) (2015), 2469 - 2483.
-  H. Liu and G. Liu, Weingarten rotation surfaces in 3-dimensional de Sitter space, J. Geom. 79 (2004), 156 – 168.
-  R. Lopez and E. Demir, Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature, Cent. Eur. J. Math. 12(9) (2014), 1349-1361.
-  R. Lopez, Rotational linear Weingarten surfaces of hyperbolic type, Israel J. Math. 167 (2008), 283–301.
-  D. Palman, Spharische quartiken auf dem torus im einfach isotropen raum, Glasnik Matematicki 14(34) (1979), 345-357.
-  B. Pavkovic, An interpretation of the relative curvatures for surfaces in the isotropic space, Glasnik Matematicki 15(35) (1980), 149-152.
-  H. Pottmann and K. Opitz, Curvature analysis and visualization for functions defined on Euclidean spaces or surfaces, Comput. Aided Geom. Design 11 (1994), 655–674.
-  H. Pottmann and Y. Liu, Discrete surfaces of isotropic geometry with applications in architecture. In: Martin, R., Sabin, M., Winkler, J. (eds.) The Mathematics of Surfaces, pp. 341–363. Lecture Notes in Computer Science 4647. Springer (2007).
-  H. Pottmann, P. Grohs and N.J. Mitra, Laguerre minimal surfaces, isotropic geometry and linear elasticity, Adv. Comput. Math. 31 (2009), 391–419
-  H. Sachs, Ebene Isotrope Geometrie, Vieweg-Verlag, Braunschweig, Wiesbaden, 1990.
-  H. Sachs, Isotrope Geometrie des Raumes, Vieweg Verlag, Braunschweig, 1990.
-  H. Sachs, Zur Geometrie der Hyperspharen in n-dimensionalen einfach isotropen Raum, Jour. f. d. reine u. angew. Math. 298 (1978), 199-217.
-  Z. M. Sipus, Translation surfaces of constant curvatures in a simply isotropic space, Period. Math. Hung. 68 (2014), 160–175
-  Z. M. Sipus and B. Divjak, Curves in n-dimensional k-isotropic space, Glasnik Matematicki 33(53) (1998), 267-286.
-  K. Strubecker, Differentialgeometrie des isotropen Raumes III, Flachentheorie, Math. Zeitsch. 48 (1942), 369-427.
-  D. W. Yoon, Y. Tuncer and M. K. Karacan, Non-degenerate quadric surfaces of Weingarten type, Annales Polonici Math. 107 (2013), 59-69.