2018/09/21 by Naoyuki Koike, Koike, Naoyuki
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1809.07962
openalex publication_date 2018/09/21 · openalex created_date 2018/09/27 · openalex updated_date 2026/07/28
In this paper, we discuss how a Gromov-Hausdorff-like distance function over the space of all isometric classes of compact Ck-Riemannian manifolds should be defined in the aspect of the Riemannan submanifold theory, where k≥ 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→-∞. The key in the construction of the Gromov-Hausdorff-like distance function given in this paper is to define the distance of two Ck+1-isometric embeddings of distinct compact Ck-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.