  A spacetime is a connected 4-dimensional semi-Riemannian manifold endowed
with a metric $g$ with signature $(- + + +)$. The geometry of a spacetime is
described by the metric tensor $g$ and the Ricci tensor $S$ of type $(0, 2)$
whereas the energy momentum tensor of type $(0,2)$ describes the physical
contents of the spacetime. Einstein's field equations relate $g$, $S$ and the
energy momentum tensor and describe the geometry and physical contents of the
spacetime. By solving Einstein's field equations for empty spacetime (i.e. $S =
0$) for a non-static spacetime metric, one can obtain the interior black hole
solution, known as the interior black hole spacetime which infers that a
remarkable change occurs in the nature of the spacetime, namely, the external
spatial radial and temporal coordinates exchange their characters to temporal
and spatial coordinates, respectively, and hence the interior black hole
spacetime is a non-static one as the metric coefficients are time dependent.
For the sake of mathematical generalizations, in the literature, there are many
rigorous geometric structures constructed by imposing the restrictions to the
curvature tensor of the space involving first order and second order covariant
differentials of the curvature tensor. Hence a natural question arises that
which geometric structures are admitted by the interior black hole metric. The
main aim of this paper is to provide the answer of this question so that the
geometric structures admitting by such a metric can be interpreted physically.
