  The Tolman~VII solution, an exact analytic solution to the spherically
symmetric, static Einstein equations with a perfect fluid source, has many
characteristics that make it interesting for modelling high density physical
astronomical objects. Here we supplement those characteristics with the
geometrical tensors that this solution possess, and find that the Weyl,
Riemann, and Ricci tensor components show unexpected mathematical behaviour
that change depending on physically motivated parameters, even though the mass
of the modelled objects is fixed. We show these features firstly through tensor
components, and then through the scalars in the null tetrad formalism of
Newmann and Penrose. The salient conclusion of this analysis is the intimate
relationship between the Tolman~VII solution and the constant density
Schwarzschild interior solution: the former being a straight forward
generalization of the latter while eschewing the unphysical constant density.
