geometrization

In particular, the result of geometrization may be a geometry that is not isotropic.

Assuming the geometrization conjecture, the only open case was that of closed hyperbolic 3-manifolds.

The groups are related to the 8 geometries of Thurston's geometrization conjecture.

He later developed a program to prove the geometrization conjecture by Ricci flow with surgery.

Thurston was next led to formulate his geometrization conjecture.

Later came the development of a sculptural tradition known as Mezcala, characterized by its geometrization of the human form.

Perelman's work proves this claim and thus proves the geometrization conjecture.

According to the geometrization conjecture, these negatively curved 3-manifolds must actually admit a complete hyperbolic metric.

Yet again assuming the geometrization conjecture, these manifolds have a complete hyperbolic metric.

This later became a major ingredient in the development of geometrization, which can be seen as a partial-classification of 3-dimensional manifolds.

In 2003, he proved Thurston's geometrization conjecture.

For non-oriented manifolds the easiest way to state a geometrization conjecture is to first take the oriented double cover.

The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces.

To complete the picture, Thurston proved a geometrization theorem for Haken manifolds.

One form of Thurston's geometrization theorem states:

In 1981, he announced the orbifold theorem, an extension of his geometrization theorem to the setting of 3-orbifolds.

This result can be recovered from the combination of Mostow rigidity with Thurston's geometrization theorem.

Every closed 3-dimensional manifold can be cut into pieces that are geometrizable, by the geometrization conjecture, and there are 8 such geometries.

In dimension three, the conjecture had an uncertain reputation until the geometrization conjecture put it into a framework governing all 3-manifolds.

His early fears of the square marching battalions associated with dictatorships may have led him to oppose any "geometrization" of people and their architecture.

Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture.

For the proof of the conjectures, see the references in the articles on geometrization conjecture or Poincaré conjecture.

His papers say that he has proved what is known as the Geometrization Conjecture, a complete characterization of the geometry of three-dimensional spaces.

Grigori Perelman sketched a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery.

Thurston's geometrization conjecture proved by Perelman, a generalization of the hyperbolization theorem to all compact 3-manifolds.