Isotropic geometry of graph surfaces associated with product production functions in economics

Abstract

A production function is a mathematical formalization in economics which denotes the relations between the output generated by a firm, an industry or an economy and the inputs that have been used in obtaining it. In this paper, we study the product production functions of 2 variables in terms of the geometry of their associated graph surfaces in the isotropic $3-$space $\mathbb{I}^{3}$. In particular, we derive several classification results for the graph surfaces of product production functions in $\mathbb{I}^{3}$ with constant curvature.

Linear Weingarten factorable surfaces in isotropic spaces

“…The results of the present paper and [2,3] relating to the (affine) factorable surfaces in I 3 with K, H constants are summed up in Table 1 which x = |K 0 |x (y + ax) .…”

“…The results of the present paper and [2,3] relating to the (affine) factorable surfaces in I 3 with K, H constants are summed up in Table 1 which categorizes those surfaces. Notice also that, without emposing conditions, finding the affine factorable surfaces of type 2 with H = const.…”

The geometry of Gauss map and shape operator in simply isotropic and pseudo-isotropic spaces

“…The study of I 3 has been initiated by the Austrian geometer Karl Strubecker in the 1930's [17,18,19,20,21] (see also [14] and references therein), while that of I 3 p began only recently [3,7]. Besides its mathematical interest [1,3,9,23,24], see also the recent contributions by this Author [6,7], isotropic geometry finds applications in economics [4,5], image processing [10], and shape interrogation [12]. In addition, this theory may prove useful in understanding the geometry of surfaces with zero mean curvature in semi-Riemannian spaces [15,16].…”

Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spaces

“…The three dimensional (3d) simply isotropic I 3 and pseudo-isotropic I 3 p spaces are examples of 3d Cayley-Klein (CK) geometries [12,19,23,30], which is basically the study of those properties in projective space P 3 that preserves a certain configuration, the so-called absolute figure. Indeed, following Klein "Erlanger Program" [4,14], a CK geometry is the study of the geometry invariant by the action of the subgroup of projectivities that fix the absolute figure: e.g., Euclidean (Minkowski) space E 3 (E 3 1 ) can be modeled through an absolute figure given in homogeneous coordinates by a plane at infinity, usually identified with x 0 = 0, and a nondegenerate quadric of index zero (index one) usually identified with x 2 0 +• • •+x 2 3 = 0 (x 2 0 + x 2 1 + x 2 2 − x 2 3 = 0, respectively) [12]. In our case, i.e., isotropic geometries, the absolute figure is given by a plane at infinity and a degenerate quadric of index 0 or 1: x 2 0 + x 2 1 + δ x 2 2 = 0, with δ = 1 for the simply isotropic figure and δ = −1 for the pseudo-isotropic one.…”

“…Recently, isotropic geometry has been seen a renewed interest from both pure and applied viewpoints (a quite comprehensive and historical account before the 1990's can be found in [24]). We may mention investigations of special classes of curves [33] and surfaces [1,2,13,26], while applications may range from economics [3,8] and elasticity [21] to image processing and shape interrogation [15,22]. Another stimulus may come from the problem of characterizing curves on level set surfaces Σ = F −1 (c).…”