In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

metric geometry In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

, Mikhael Gromov proved a fundamental compactness theorem for sequences of

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

s. In the special case of

Riemannian manifolds In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the n-sphere, hyperbolic space, and smooth surfaces in ...

, the key assumption of his compactness theorem is automatically satisfied under an assumption on

Ricci curvature In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...

. These theorems have been widely used in the fields of

geometric group theory Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these group ...

and

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

Metric compactness theorem

The Gromov–Hausdorff distance defines a notion of distance between any two

s, thereby setting up the concept of a sequence of metric spaces which converges to another metric space. This is known as

Gromov–Hausdorff convergence In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff distance. Gromov–Hausdorff distance The Gromov–Hausdorff dist ...

. Gromov found a condition on a sequence of

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

metric spaces which ensures that a

subsequence In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...

converges to some metric space relative to the Gromov–Hausdorff distance:

Let be a sequence of compact metric spaces with uniformly bounded diameter. Suppose that for every positive number there is a
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
and, for every , the set can be covered by metric balls of radius . Then the sequence has a subsequence which converges relative to the Gromov–Hausdorff distance.

The role of this theorem in the theory of Gromov–Hausdorff convergence may be considered as analogous to the role of the

Arzelà–Ascoli theorem The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded inte ...

in the theory of

uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E as the function domain i ...

. Gromov first formally introduced it in his 1981 resolution of the Milnor–Wolf conjecture in the field of

, where he applied it to define the

asymptotic cone In mathematics, an ultralimit is a geometric construction that assigns a limit metric space to a sequence of metric spaces X_n. The concept captures the limiting behavior of finite configurations in the X_n spaces employing an ultrafilter to bypass ...

of certain metric spaces. These techniques were later extended by Gromov and others, using the theory of

ultrafilter In the Mathematics, mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a Maximal element, maximal Filter (mathematics), filter on P; that is, a proper filter on P th ...

Riemannian compactness theorem

Specializing to the setting of geodesically complete

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

s with a fixed lower bound on the

, the crucial covering condition in Gromov's metric compactness theorem is automatically satisfied as a corollary of the Bishop–Gromov volume comparison theorem. As such, it follows that:

Consider a sequence of closed Riemannian manifolds with a uniform lower bound on the Ricci curvature and a uniform upper bound on the diameter. Then there is a subsequence which converges relative to the Gromov–Hausdorff distance.

The limit of a convergent subsequence may be a metric space without any smooth or Riemannian structure. This special case of the metric compactness theorem is significant in the field of

, as it isolates the purely metric consequences of lower Ricci curvature bounds.

References

Sources. * * * * * * * {{Metric spaces Differential geometry Theorems in Riemannian geometry