An elementary approach to tangent space variation on Riemannian submanifolds

TL;DR

This paper provides tight estimates for how tangent spaces vary on Riemannian submanifolds in Euclidean space, linking geometric properties to local feature size and deriving new structural insights.

Contribution

It introduces asymptotically tight bounds on tangent space variation based on local feature size, utilizing elementary Euclidean geometry and generalizing previous results.

Findings

Derived tight estimates for tangent space variation

Connected tangent variation to local feature size properties

Generalized a known structural property of local feature size

Abstract

We give asymptotically tight estimates of tangent space variation on Riemannian submanifolds of Euclidean space with respect to the local feature size of the submanifolds. We show that the result follows directly from structural properties of local feature size of the Riemannian submanifold and some elementary Euclidean geometry. We also show that using the tangent variation result one can prove a new structural property of local feature size function. This structural property is a generalization of a result of Giesen and Wagner [GW04, Lem. 7].