2014/10/30 by Francisco Fontenele, Fontenele, Francisco, Frederico Xavier +1 · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1410.8465
openalex publication_date 2014/10/30 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
Although the Nash theorem solves the isometric embedding problem, matters are\ninherently more involved if one is further seeking an embedding that is\nwell-behaved from the standpoint of submanifold geometry. More generally,\nconsider a Lipschitz map F:Mm\→ mathbb Rn, where Mm is a Hadamard\nmanifold whose curvature lies between negative constants. The main result of\nthis paper is that F must perform a substantial compression: For every r>0\nand integer k\≥ 2 there exist k geodesic balls of radius r in Mm\nthat are arbitrarily far from each other, but whose images under F are\nbunched together arbitrarily close in the Hausdorff sense of mathbb Rn. In\nparticular, every isometric embedding mathbb Hm\→ mathbb Rn of hyperbolic\nspace must have a complex asymptotic behavior, regardless of how high the\ncodimension is. Hence, there is no truly simple way to realize mathbb Hm\nisometrically inside any Euclidean space.\n