On algebraic surfaces with an infinite set of planes of skew symmetry

Abstract

We consider (m-1)-dimensional noncylindrical algebraic surfaces in a real space R m with infinite symmetry groups G generated by skew reflections with respect to planes, along with the relative locations of linear hulls of four G-orbits of the directions of symmetry.Let F. be an (m-1)-dimensional noncylindrical surface of order n in a real space Era: assume that F is invariant with respect to an infinite group G that is generated by skew reflections relative to planes and does not admit extension. In proving the theorem (Sections l~ ~ we obtain equations for the surfaces F,, that are invariant under the corresponding groups G. t ~ We set dj = 1 and cr = v (there is no loss of generality). In the space E m we specify a Cartesian coordinate system Oyl ...y4zl ...z,zl ...zr (m = r + r + 4). We define a surface Fn (n > 2) with complete symmetry group G by means of the equation

Invariants of finite and infinite groups generated by reflections

“…Let us write the following quadratic forms: In [32], the following theorem was proved. A. I. Krivoruchko [43] gave a new proof of one of the auxiliary results of [21] (Sec.…”

“…The mutual arrangement of II € depends on the polynomial O0 [32]. In many aspects, its structure has been clarified in [34].…”

“…(1) j=0 the set B consists of diametrical planes of a quadic with the equation q0 = c [28], i.e., B = B(q0). The mutual arrangement of II € depends on the polynomial O0 [32].…”

“…The structure of a nondegenerated diametrical surface conjugate to the vector u and containing symmetry surface F, in the direction of symmetry n was considered in [27]. be defined in the space Era, if the contrary is not specified, F. is a noncylindrical surface of the order n invariant with respect to the infinite group G generated by oblique (in particular, orthogonal) reflections with respect to planes; /z i are planes of H ~j (j = 0~) spanned by G(u), the orbits of the symmetry directions u.…”