Limits of tangent spaces to definable sets

TL;DR

This paper investigates the behavior of tangent limits in definable sets within o-minimal structures, generalizing classical results from algebraic surfaces to higher dimensions and broader contexts.

Contribution

It characterizes exceptional tangent rays and extends O'Shea--Wilson's results to o-minimal structures and arbitrary dimensions.

Findings

Characterization of exceptional tangent rays

Criteria for identifying exceptional rays

Generalization of classical tangent limit results

Abstract

We study the set of tangent limits at a given point to a set definable in any o-minimal structure by characterizing the set of exceptional rays in the tangent cone to the set at that point and investigating the set of tangent limits along these rays. Several criteria for determining exceptional rays will be given. The main results of the paper generalize, to the o-minimal setting and to arbitrary dimension, the main results of O'Shea--Wilson which deals with algebraic surfaces in $\mathbb R^3$.