Curvature loci of 3-manifolds

TL;DR

This paper refines the classification of curvature loci for 3-manifolds in Euclidean spaces, analyzing their singularities and solution structures to better understand their geometric properties.

Contribution

It introduces a refined affine classification of real nets of quadrics to characterize curvature loci of 3-manifolds and studies their singularities and solution behaviors.

Findings

Characterization of curvature loci via solutions of systems of three cubics

Analysis of how singularities of curvature loci behave under orthogonal projection

Refinement of affine classification for real nets of quadrics

Abstract

We refine the affine classification of real nets of quadrics in order to obtain generic curvature loci of regular $3$-manifolds in $\mathbb{R}^6$ and singular corank $1$ $3$-manifolds in $\mathbb{R}^5$. For this, we characterize the type of the curvature locus by the number and type of solutions of a system of equations given by 4 ternary cubics (which is a determinantal variety in some cases). We also study how singularities of the curvature locus of a regular 3-manifold can go to infinity when the manifold is projected orthogonally in a tangent direction.