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

TL;DR

This paper proposes a new Gromov-Hausdorff-like distance for compact Riemannian manifolds based on Riemannian submanifold theory, which aligns with Hamilton's convergence, enhancing geometric analysis tools.

Contribution

It introduces a novel distance function using Hausdorff distances in tangent bundles, bridging Riemannian submanifold theory with Gromov-Hausdorff convergence concepts.

Findings

The new distance converges to R. S. Hamilton's notion of convergence.

Hausdorff distance in tangent bundles effectively measures manifold similarity.

The approach generalizes classical Gromov-Hausdorff distance in Riemannian geometry.

Abstract

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.