Characterizations of Ruled Surfaces in $\mathbb{R}^3$ and of Hyperquadrics in $\mathbb{R}^{n+1}$ via Relative Geometric Invariants

TL;DR

This paper characterizes ruled surfaces in three-dimensional space and hyperquadrics in higher dimensions using relative geometric invariants, providing conditions for their classification based on curvature and normalization properties.

Contribution

It introduces necessary and sufficient conditions for identifying ruled surfaces and hyperquadrics via relative invariants and normalization proportionality.

Findings

Ruled surfaces in $ ext{R}^3$ with negative Gaussian curvature are characterized.

Hyperquadrics in $ ext{R}^{n+1}$ with positive Gaussian curvature are characterized.

Conditions for relative normalization to be proportional to equiaffine normalization are established.

Abstract

We consider hypersurfaces in the real Euclidean space $\mathbb{R}^{n+1}$ ($n\geq2$) which are relatively normalized. We give necessary and sufficient conditions a) for a surface of negative Gaussian curvature in $\mathbb{R}^3$ to be ruled, b) for a hypersurface of positive Gaussian curvature in $\mathbb{R}^{n+1}$ to be a hyperquadric and c) for a relative normalization to be constantly proportional to the equiaffine normalization.