Geometric Characterizations of C1 Manifold in Euclidean Spaces by Tangent Cones

TL;DR

This paper proves that a set in Euclidean space is a smooth manifold if and only if its lower and upper paratangent cones coincide at every point, providing a new geometric characterization and historical context.

Contribution

It offers a new elementary proof of the characterization of smooth manifolds via tangent cones and recovers classical and modern results in the field.

Findings

A set is a smooth manifold iff lower and upper paratangent cones coincide at each point.

Historical characterizations of manifolds by tangent cones are revisited.

Modern and classical results are unified under a geometric framework.

Abstract

A remarkable and elementary fact that a locally compact set F of Euclidean space is a smooth manifold if and only if the lower and upper paratangent cones to F coincide at every point, is proved. The celebrated von Neumann's result (1929) that a locally compact subgroup of the general linear group is a smooth manifold, is a straightforward application. A historical account on the subject is provided in order to enrich the mathematical panorama. Old characterizations of smooth manifold (by tangent cones), due to Valiron (1926, 1927) and Severi (1929, 1934) are recovered; modern characterizations, due to Gluck (1966, 1968), Tierno (1997), Shchepin and Repovs (2000) are restated.