  This paper presents a rigorous treatment of the concept of extensivity in
equilibrium thermodynamics from a geometric point of view. This is achieved by
endowing the manifold of equilibrium states of a system with a smooth atlas
that is compatible with the pseudogroup of transformations on a vector space
that preserve the radial vector field. The resulting geometric structure allows
for accurate definitions of extensive differential forms and scaling, and the
well-known relationship between both is reproduced. This structure is
represented by a global vector field that is locally written as a radial one.
The submanifolds that are transversal to it are embedded, and locally defined
by extensive functions.
