Geometric characterizations of manifolds in Euclidean spaces by tangent cones

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.