Gromov's compactness theorem (geometry)
Encyclopedia
In Riemannian geometry
, Gromov's (pre)compactness theorem states that
the 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.
This theorem is a generalization of the Myers theorem.
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...
, Gromov's (pre)compactness theorem states that
the set of Riemannian manifolds of a given dimension, 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. It was proved by Mikhail Gromov.
This theorem is a generalization of the Myers theorem.