Critical Curvature of Algebraic Surfaces in Three-Space

TL;DR

This paper investigates the algebraic and geometric properties of curvature on smooth algebraic surfaces in three-dimensional space, focusing on umbilical and critical curvature points, and establishes bounds on their quantities.

Contribution

It provides a new algebraic geometric analysis of curvature features on algebraic surfaces, including bounds and characterizations of critical points and umbilics.

Findings

Number of complex critical curvature points is of order d^3.

Full characterization of umbilics and critical points on general quadrics.

Establishment of bounds for the number of such points.

Abstract

We study the curvature of a smooth algebraic surface $X\subset \mathbb R^3$ of degree $d$ from the point of view of algebraic geometry. More precisely, we consider umbilical points and points of critical curvature. We prove that the number of complex critical curvature points is of order $d^3$. For general quadrics, we fully characterize the number of real and complex umbilics and critical curvature points.