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William Thurston's **elliptization conjecture** states that a closed 3-manifold with finite fundamental group is spherical, i.e. has a Riemannian metric of constant positive sectional curvature. A 3-manifold with such a metric is covered by the 3-sphere, moreover the group of covering transformations are isometries of the 3-sphere. Note that this means that if the original 3-manifold had in fact a trivial fundamental group, then it is homeomorphic to the 3-sphere (via the covering map). Thus, proving the elliptization conjecture would prove the Poincaré conjecture as a corollary. In fact, the elliptization conjecture is logically equivalent to two simpler conjectures: the Poincaré conjecture and the spherical space form conjecture.

**William Paul Thurston** was an American mathematician. He was a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds. From 2003 until his death he was a professor of mathematics and computer science at Cornell University.

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an *n*-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension *n*. In this more precise terminology, a manifold is referred to as an ** n-manifold**.

In the mathematical field of algebraic topology, the **fundamental group** is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a topological invariant: homeomorphic topological spaces have the same fundamental group.

The elliptization conjecture is a special case of Thurston's geometrization conjecture, which was proved in 2003 by G. Perelman.

In mathematics, **Thurston's geometrization conjecture** states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries . In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.

**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 that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

**Grigori Yakovlevich Perelman** is a Russian mathematician. He has made contributions to Riemannian geometry and geometric topology. In 1994, Perelman proved the soul conjecture. In 2003, he proved Thurston's geometrization conjecture. The proof was confirmed in 2006. This consequently solved in the affirmative the Poincaré conjecture.

In differential geometry, the **Ricci curvature tensor**, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a narrow conical piece of a small geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. As such, it provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n-space. The Ricci tensor is defined on any pseudo-Riemannian manifold, as a trace of the Riemann curvature tensor. Like the metric itself, the Ricci tensor is a symmetric bilinear form on the tangent space of the manifold.

In Riemannian geometry, the **sectional curvature** is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature *K*(σ_{p}) depends on a two-dimensional plane σ_{p} in the tangent space at a point *p* of the manifold. It is the Gaussian curvature of the surface which has the plane σ_{p} as a tangent plane at *p*, obtained from geodesics which start at *p* in the directions of σ_{p}. The sectional curvature is a smooth real-valued function on the 2-Grassmannian bundle over the manifold.

In Riemannian geometry, the **scalar curvature** is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity.

In differential geometry, the **Ricci flow** is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric.

In mathematics and especially differential geometry, a **Kähler manifold** is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil.

In mathematics, **low-dimensional topology** is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. It can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.

In mathematics, a **3-manifold** is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

In mathematics, a **space form** is a complete Riemannian manifold *M* of constant sectional curvature *K*. The three obvious examples are Euclidean *n*-space, the *n*-dimensional sphere, and hyperbolic space, although a space form need not be simply connected.

In mathematics, a **spherical 3-manifold***M* is a 3-manifold of the form

In mathematics, more precisely in topology and differential geometry, a **hyperbolic 3–manifold** is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries.

In mathematics, **Hopf conjecture** may refer to one of several conjectural statements from differential geometry and topology attributed to either Eberhard Hopf or Heinz Hopf.

In the mathematical field of knot theory, the **hyperbolic volume** of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily a finite real number, and is a topological invariant of the link. As a link invariant, it was first studied by William Thurston in connection with his geometrization conjecture.

In mathematics, the **2π theorem** of Gromov and Thurston states a sufficient condition for Dehn filling on a cusped hyperbolic 3-manifold to result in a negatively curved 3-manifold.

The **Geometry Festival** is an annual mathematics conference held in the United States.

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

In mathematics, the **Poincaré conjecture** is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

- William Thurston.
*Three-dimensional geometry and topology. Vol. 1*. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN 0-691-08304-5. - William Thurston. The Geometry and Topology of Three-Manifolds, 1980 Princeton lecture notes on geometric structures on 3-manifolds, that states his elliptization conjecture near the beginning of section 3.

The **International Standard Book Number** (**ISBN**) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

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