Algebraic affine rotation surfaces of parabolic type

Abstract

Affine rotation surfaces, which appear in the context of affine differential geometry, are generalizations of surfaces of revolution. These affine rotation surfaces can be classified into three different families: elliptic, hyperbolic and parabolic. In this paper we investigate some properties of algebraic parabolic affine rotation surfaces, i.e. parabolic affine rotation surfaces that are algebraic, generalizing some previous results on algebraic affine rotation surfaces of elliptic type (classical surfaces of revolution) and hyperbolic type (hyperbolic surfaces of revolution). In particular, we characterize these surfaces in terms of the structure of their implicit equation, we describe the structure of the form of highest degree defining an algebraic parabolic affine rotation surface, and we prove that these surfaces can have either one, or two, or infinitely many axes of affine rotation. Additionally, we characterize the surfaces with more than one parabolic axis.

Appendix I: The cones \(y^2-2xz-wx^2=0\)

The matrix associated with the quadratic form $y^2-2xz-wx^2$ is

$$\begin{aligned} \mathbf{P}=\begin{bmatrix} -w&\quad 0&\quad -1\\ 0&\quad 1&\quad 0\\ -1&\quad 0&\quad 0 \end{bmatrix} \end{aligned}$$

The eigenvalues of the matrix $\mathbf{P}$ are

$$\begin{aligned} \lambda _1=1,\,\lambda _2=-\frac{1}{2}w+\frac{1}{2}\sqrt{w^2+4},\,\lambda _3=-\frac{1}{2}w-\frac{1}{2}\sqrt{w^2+4}. \end{aligned}$$

(9)

Observe that $\lambda _2\cdot \lambda _3=-1$, therefore $\lambda _2$ and $\lambda _3$ are never zero. Furthermore, if $w\ne 0$ then $\lambda _i\ne 1$ for $i=2,3$; also, $\lambda _2=\lambda _3$ iff $w^2+4=0$, i.e. $w=\pm 2i$, where $i^2=-1$. Now we distinguish the following three cases:

• (1) The matrix $\mathbf{P}$ has three distinct, eigenvalues, so $\mathbf{P}$ is diagonalizable and $y^2-2xz-wx^2=0$ defines a cone that is not a surface of revolution. Since $\lambda _2\cdot \lambda _3=-1$, $\lambda _2$ and $\lambda _3$ have opposite signs, so $y^2-2xz-wx^2=0$ defines a real cone iff w is real. Furthermore, in that case we get an elliptic cone. The eigenvector associated with $\lambda =1$ is (0, 1, 0), which defines one of the symmetry axes of the cone. The eigenvectors associated with the other two eigenvalues are parallel to the xz-plane.

• (2) The matrix $\mathbf{P}$ has only two distinct eigenvalues, $\lambda _1=1$ (simple) and $\lambda _2=\mp i$ (double), but $\mathbf{P}$ is not diagonalizable. The eigenvector associated with $\lambda =1$ is (0, 1, 0).

• (3) The matrix $\mathbf{P}$ has only two distinct eigenvalues, $\lambda _1=1$ (double) and $\lambda _2=-1$ (simple). The matrix $\mathbf{P}$ is diagonalizable, so $y^2-2xz=0$ defines a real cone of revolution about the line through the origin in the direction of the eigenvector associated with $\lambda _2=-1$, namely (1, 0, 1). Notice that the axis of revolution of this cone forms an angle of $\frac{\pi }{4}$ with both the x-axis and the z-axis.