Gromov-Hausdorff convergence
Encyclopedia
In mathematics
, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff
, is a notion for convergence of metric space
s which is a generalization of Hausdorff convergence
.
metric spaces are from being isometric
.
If X and Y are two compact metric spaces, then dGH (X,Y )
is defined to be the infimum of all numbers dH(f (X ), g (Y )) for all metric spaces M and all isometric embeddings
f :X→M and g :Y→M. Here dH denotes Hausdorff distance
between subsets in M and the isometric embedding is understood in the global sense, i.e it must preserve all distances, not only infinitesimally small ones; for example no compact Riemannian manifold
of negative sectional curvature
admits such an embedding into Euclidean space
.
The Gromov–Hausdorff distance turns the set of all isometry classes of compact metric spaces into a metric space, and it therefore defines a notion of convergence for sequence
s of compact metric spaces, called Gromov–Hausdorff convergence. A metric space to which such a sequence converges is called the Hausdorff limit of the sequence.
Given a sequence (Xn, pn) of locally compact complete
length metric spaces
with distinguished points, it converges to (Y, p) if for any R > 0 the closed R-balls around pn in Xn converge to the closed R-ball around p in Y in the usual Gromov–Hausdorff sense.
any discrete group
with polynomial growth
is almost 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 Ricci curvature
≥ c and diameter
≤ D is relatively compact in the Gromov–Hausdorff metric.
The Gromov–Hausdorff distance metric has been applied in the field of computer graphics and computational geometry to find correspondences between different shapes.
The Gromov–Hausdorff distance has been used 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff
Felix Hausdorff
Felix Hausdorff was a Jewish German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.-Life:Hausdorff studied at the University of Leipzig,...
, is a notion for convergence of metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s which is a generalization of Hausdorff convergence
Hausdorff distance
In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right...
.
Gromov–Hausdorff distance
Gromov–Hausdorff distance measures how far two compactCompact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
metric spaces are from being isometric
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
.
If X and Y are two compact metric spaces, then dGH (X,Y )
is defined to be the infimum of all numbers dH(f (X ), g (Y )) for all metric spaces M and all isometric embeddings
f :X→M and g :Y→M. Here dH denotes Hausdorff distance
Hausdorff distance
In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right...
between subsets in M and the isometric embedding is understood in the global sense, i.e it must preserve all distances, not only infinitesimally small ones; for example no compact Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
of negative sectional curvature
Sectional curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...
admits such an embedding into Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
.
The Gromov–Hausdorff distance turns the set of all isometry classes of compact metric spaces into a metric space, and it therefore defines a notion of convergence for sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
s of compact metric spaces, called Gromov–Hausdorff convergence. A metric space to which such a sequence converges is called the Hausdorff limit of the sequence.
Pointed Gromov–Hausdorff convergence
Pointed Gromov–Hausdorff convergence is an appropriate analog of Gromov–Hausdorff convergence for non-compact spaces.Given a sequence (Xn, pn) of locally compact complete
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
length metric spaces
Intrinsic metric
In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are a given distance from each other, it is natural to expect that one should be able to get from one point to another along a path whose arclength is equal to that distance...
with distinguished points, it converges to (Y, p) if for any R > 0 the closed R-balls around pn in Xn converge to the closed R-ball around p in Y in the usual Gromov–Hausdorff sense.
Applications
The notion of Gromov–Hausdorff convergence was first used by Gromov to prove thatany discrete group
Discrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
with polynomial growth
Growth rate (group theory)
In group theory, the growth rate of a group with respect to a symmetric generating set describes the size of balls in the group. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length...
is almost nilpotent (i.e. it contains a nilpotent subgroup
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
of finite index
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
). See Gromov's theorem on groups of polynomial growth
Gromov's theorem on groups of polynomial growth
In geometric group theory, Gromov's theorem on groups of polynomial growth, named for Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups which have nilpotent subgroups of finite index....
.(Also see D. Edwards for an earlier work.)
The key ingredient in the proof was the observation that for the
Cayley graph
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem and uses a specified, usually finite, set of generators for the group...
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
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
is Gromov's compactness theorem
Gromov's compactness theorem (geometry)
In Riemannian geometry, Gromov's compactness theorem states thatthe set of Riemannian manifolds of a given dimension, with Ricci curvature ≥ c and diameter ≤ D is relatively compact in the Gromov-Hausdorff metric. It was proved by Mikhail Gromov....
, which states that
the set of Riemannian manifolds with Ricci curvature
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space...
≥ c and diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
≤ D is relatively compact in the Gromov–Hausdorff metric.
The Gromov–Hausdorff distance metric has been applied in the field of computer graphics and computational geometry to find correspondences between different shapes.
The Gromov–Hausdorff distance has been used 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.