  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.

Differential Geometry