Gromov-Hausdorff metric

In mathematics , the Gromov-Hausdorff metric , named after the mathematicians Michail Leonidowitsch Gromow and Felix Hausdorff , is a metric on the class of isometric classes of compact metric spaces . Clearly, the Gromov-Hausdorff distance is smaller, the better the given rooms can be brought into congruence.

The convergence with respect to the Gromov-Hausdorff metric is called the Gromov-Hausdorff convergence .

definition

The Gromov-Hausdorff distance is the smallest possible Hausdorff distance that the given rooms can have when embedded in a metric room. So be compact metric spaces. Then the Gromov-Hausdorff distance is defined as: ${\ displaystyle X, Y}$ ${\ displaystyle d_ {GH} (X, Y)}$

{\ displaystyle d _ {\ mathrm {G} H} (X, Y): = \ inf \ {d _ {\ mathrm {H}} (f (X), g (Y)) \ mid f: X \ rightarrow Z , \; g: Y \ rightarrow Z \, {\ text {isometric embeddings}} \}}

in which

{\ displaystyle d_ {H} \, (f (X), g (Y))}

denotes the Hausdorff distance of f (X) and g (Y) in Z.

This is defined as:

{\ displaystyle d _ {\ mathrm {H}} (X, Y): = \ max \ {\, \ sup _ {x \ in X} \ inf _ {y \ in Y} d (x, y), \ , \ sup _ {y \ in Y} \ inf _ {x \ in X} d (x, y) \, \} {\ mbox {.}} \!}

The limit value of a sequence that is convergent in the sense of the Gromov-Hausdorff metric is referred to as the Gromov-Hausdorff limit value of the sequence; in this case one speaks of Gromov-Hausdorff convergence .

Dotted Gromov-Hausdorff convergence

The dotted Gromov-Hausdorff convergence is the appropriate analogue of the Gromov-Hausdorff convergence when considering non-compact metric spaces.

If a sequence of locally compact complete metric spaces, the metric of which is intrinsic, is called convergent if for each the closed -ball um in the Gromov-Hausdorff sense converge to the closed -ball um . ${\ displaystyle (X_ {n}, p_ {n})}$ ${\ displaystyle (Y, q)}$ ${\ displaystyle R> 0}$ ${\ displaystyle R}$ ${\ displaystyle p_ {n}}$ ${\ displaystyle R}$ ${\ displaystyle q}$

Gromov-Hausdorff convergence of manifolds

The limit of a Gromov-Hausdorff convergent sequence of -dimensional Riemannian manifolds in general does not have to be a manifold. ${\ displaystyle n}$ ${\ displaystyle (M_ {i}, g_ {i})}$

If the manifolds have uniformly restricted curvature and uniformly restricted diameters, it follows from Gromov's theorem that the limit is an Alexandrov space with the same curvature and diameter bounds and the dimension smaller or equal . ${\ displaystyle n}$

If (assuming that the curvature is uniformly bounded downwards) the limit value is a -dimensional manifold, then almost all of them must have been too homeomorphic - that is Perelman's theorem of stability . ${\ displaystyle M}$ ${\ displaystyle n}$ ${\ displaystyle M_ {i}}$ ${\ displaystyle M}$

More generally, if the limit value is a Riemannian manifold of arbitrary dimension (again under the assumption that the curvature is uniformly downward) , then almost all fiber bundles must have been over (Fukaya-Yamaguchi, V. Kapovitch-Wilking). ${\ displaystyle M}$ ${\ displaystyle M_ {i}}$ ${\ displaystyle M}$

literature

M. Gromov. Metric structures for Riemannian and non-Riemannian spaces , Birkhäuser (1999). ISBN 0-8176-3898-9 .