Gromov–Hausdorff convergence
In
Gromov–Hausdorff distance
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Gromov-Hausdorff_distance.png/220px-Gromov-Hausdorff_distance.png)
The Gromov–Hausdorff distance was introduced by David Edwards in 1975,
The Gromov–Hausdorff distance turns the set of all isometry classes of compact metric spaces into a metric space, called Gromov–Hausdorff space, and it therefore defines a notion of convergence for sequences of compact metric spaces, called Gromov–Hausdorff convergence. A metric space to which such a sequence converges is called the Gromov–Hausdorff limit of the sequence.
Some properties of Gromov–Hausdorff space
The Gromov–Hausdorff space is
Pointed Gromov–Hausdorff convergence
The pointed Gromov–Hausdorff convergence is an analog of Gromov–Hausdorff convergence appropriate for non-compact spaces. A pointed metric space is a pair (X,p) consisting of a metric space X and point p in X. A sequence (Xn, pn) of pointed metric spaces converges to a pointed metric space (Y, p) if, for each R > 0, the sequence of closed R-balls around pn in Xn converges to the closed R-ball around p in Y in the usual Gromov–Hausdorff sense.[10]
Applications
The notion of Gromov–Hausdorff convergence was used by Gromov to prove that any discrete group with polynomial growth is virtually nilpotent (i.e. it contains a nilpotent subgroup of finite index). See Gromov's theorem on groups of polynomial growth. (Also see D. Edwards for an earlier work.) The key ingredient in the proof was the observation that for the Cayley graph of a group with polynomial growth a sequence of rescalings converges in the pointed Gromov–Hausdorff sense.
Another simple and very useful result in Riemannian geometry is Gromov's compactness theorem, which states that the set of Riemannian manifolds with
The Gromov–Hausdorff distance metric has been applied in the field of computer graphics and computational geometry to find correspondences between different shapes.[12] It also has been applied in the problem of motion planning in robotics.[13]
The Gromov–Hausdorff distance has been used by Sormani to prove the stability of the Friedmann model in Cosmology. This model of cosmology is not stable with respect to smooth variations of the metric.[14]
In a special case, the concept of Gromov–Hausdorff limits is closely related to large-deviations theory.[15]
The Gromov–Hausdorff distance metric has been used in neuroscience to compare brain networks.[16]
See also
References
- ^ David A. Edwards, "The Structure of Superspace", in "Studies in Topology", Academic Press, 1975, pdf Archived 2016-03-04 at the Wayback Machine
- arXiv:1612.00728 [math.MG].
- ^ M. Gromov. "Structures métriques pour les variétés riemanniennes", edited by Lafontaine and Pierre Pansu, 1981.
- Zbl 0474.20018.
- ^ D. Burago, Yu. Burago, S. Ivanov, A Course in Metric Geometry, AMS GSM 33, 2001.
- S2CID 39754495.
- arXiv:1603.02385 [math.MG].
- arXiv:1806.02100 [math.MG].
- arXiv:1611.04484 [math.MG].
- ISBN 978-3-0348-9946-8.
- .
- S2CID 207156533.
- arXiv:2209.04800 [cs.RO].
- S2CID 53312009.
- S2CID 122531716.
- PMID 21995042.
- M. Gromov. Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser (1999). ISBN 0-8176-3898-9(translation with additional content).