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

Gromov-Hausdorff-like distance function defined in the aspect of Riemannian submanifold theory