  In this paper, we discuss how a Gromov-Hausdorff-like distance function over
the space of all isometric classes of compact $C^k$-Riemannian manifolds should
be defined in the aspect of the Riemannan submanifold theory, where $k\geq 1$.
The most important fact in this discussion is as follows. The Hausdorff
distance function between two spheres of mutually distinct radii isometrically
embedded into the hypebolic space of curvature $c$ converges to zero as
$c\to-\infty$. The key in the construction of the Gromov-Hausdorff-like
distance function given in this paper is to define the distance of two
$C^{k+1}$-isometric embeddings of distinct compact $C^k$-Riemannian manifolds
into a higher dimensional Riemannian manifold by using the Hausdorff distance
function in the tangent bundle of order $k+1$ equipped with the Sasaki metric.
Furthermore, we show that the convergence of a sequence of compact Riemannian
manifolds with respect to this distance function coincides with the convergence
in the sense of R. S. Hamilton.
