Geometric invariants of $5/2$-cuspidal edges

TL;DR

This paper introduces new geometric invariants for 5/2-cuspidal edges, explores their properties, and demonstrates their implications for isometric deformations and intrinsic/extrinsic distinctions.

Contribution

It defines the secondary cuspidal curvature and bias invariants, proves the secondary product curvature is intrinsic, and shows non-trivial isometric deformations exist for certain edges.

Findings

Secondary product curvature is an intrinsic invariant.

Non-trivial isometric deformations exist for edges with non-vanishing limiting normal curvature.

Bias invariant does not have an associated invariant counterpart.

Abstract

We introduce two invariants called the secondary cuspidal curvature and the bias on $5/2$-cuspidal edges, and investigate their basic properties. While the secondary cuspidal curvature is an analog of the cuspidal curvature of (ordinary) cuspidal edges, there are no invariants corresponding to the bias. We prove that the product (called the secondary product curvature) of the secondary cuspidal curvature and the limiting normal curvature is an intrinsic invariant. Using this intrinsity, we show that any real analytic $5/2$-cuspidal edges with non-vanishing limiting normal curvature admits non-trivial isometric deformations, which provide the extrinsity of various invariants.