For a flat or a hyperbolic spatial geometry, the topology can be either compact or infinite: for example, Euclidean space is flat and infinite, but the torus is flat and compact.

The alternative two-dimensional spaces with a Euclidean metric are the cylinder and the Möbius strip, which are bounded in one direction but not the other, and the torus and Klein bottle, which are compact.

The compact manifolds with sol geometry are either the mapping torus of an Anosov map of the 2-torus ( an automorphism of the 2-torus given by an invertible 2 by 2 matrix whose eigenvalues are real and distinct, such as ), or quotients of these by groups of order at most 8.

In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.

For example in the surface case, an everywhere non-zero vector field can exist on an orientable compact surface only for the torus.

# A torus C < sup > n </ sup >/ Λ ( Λ a full lattice ) inherits a flat metric from the Euclidean metric on C < sup > n </ sup >, and is therefore a compact Kähler manifold.

This allows the vacuum chamber for the particles to be a large thin torus, rather than a disk as in previous, compact accelerator designs.

This high dimensionality is not necessarily a problem for string theory, because it can be formulated in such a way that along the 22 excess dimensions spacetime is folded up to form a small torus or other compact manifold.

that any compact hyperkähler 4-manifold is either a K3 surface or a compact torus.

The simplest viable compact spaces are those formed by modifying a torus.

where is a finite abelian group and is a product of a torus and a compact, connected, simply-connected Lie group K:

Hermann Weyl went on to give the detailed character theory of the compact connected Lie groups, based on maximal torus theory.

Now find a compact unknotted solid torus T < sub > 1 </ sub > inside the sphere.

The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary.

A Reebless foliation can fail to be taut but the only leaves of the foliation with no puncturing transverse circle must be compact, and in particular, homeomorphic to a torus.

This led to a re-awakening of compact torus research.

* The n-dimensional torus T < sup > n </ sup > ( the product of n circles ) is a compact n-manifold.

These groups were named by analogy with the theory of tori in Lie group theory ( see maximal torus ).

This is a real torus whose real points form the Lie group of nonzero complex numbers.

The kernel of this map is a nonsplit rank one torus called the norm torus of the extension C / R, and its real points form the Lie group U ( 1 ), which is topologically a circle.

Let G be a semisimple Lie group or algebraic group over, and fix a maximal torus T along with a Borel subgroup B which contains T. Let λ be an integral weight of T ; λ defines in a natural way a one-dimensional representation C < sub > λ </ sub > of B, by pulling back the representation on T = B / U, where U is the unipotent radical of B.

Examples of non-closed subgroups are plentiful ; for example take G to be a torus of dimension ≥ 2, and let H be a one-parameter subgroup of irrational slope, i. e. one that winds around in G. Then there is a Lie group homomorphism φ: R → G with H as its image.

While in the first two cases the surface X admits infinitely many conformal automorphisms ( in fact, the conformal automorphism group is a complex Lie group of dimension three for a sphere and of dimension one for a torus ), a hyperbolic Riemann surface only admits a discrete set of automorphisms.

For instance Dehn's algorithm does not solve the word problem for the fundamental group of the torus.

Notationally, this is written as T < sup > 2 </ sup >/ S < sub > 2 </ sub > – the 2-torus quotiented by the group action of the symmetric group on two letters ( switching coordinates ), and it can be thought of as the configuration space of two unordered points on the circle, possibly the same ( the edge corresponds to the points being the same ), with the torus corresponding to two ordered points on the circle.

The knot group of the unknot is an infinite cyclic group, and the knot complement is homeomorphic to a solid torus.

The Ricci-flat metric on a torus is actually a flat metric, so that the holonomy is the trivial group SU ( 1 ).

Compact manifolds with this geometry include the mapping torus of a Dehn twist of a 2-torus, or the quotient of the Heisenberg group by the " integral Heisenberg group ".

Sheaves endowed with nontrivial endomorphisms, such as the action of an algebraic torus or a Galois group, are of particular interest.

The quotient group is the symmetric group, and this construction is in fact the Weyl group of the general linear group: the diagonal matrices are a maximal torus in the general linear group ( and are their own centralizer ), the generalized permutation matrices are the normalizer of this torus, and the quotient, is the Weyl group.

As a torus, J carries a commutative group structure, and the image of C generates J as a group.

The next case is a Riemann surface of genus g = 1, such as a torus C / Λ, where Λ is a two-dimensional lattice ( a group isomorphic to Z < sup > 2 </ sup >).

As for the example: the first homotopy group of the torus T is

The dimension of a maximal torus in G is called the rank of G. The rank is well-defined since all maximal tori turn out to be conjugate.

* A maximal torus in G is a maximal abelian subgroup, but the converse need not hold.

* Given a maximal torus T in G, every element g ∈ G is conjugate to an element in T.

* Since the conjugate of a maximal torus is a maximal torus, every element of G lies in some maximal torus.

Given a torus T ( not necessarily maximal ), the Weyl group of G with respect to T can be defined as the normalizer of T modulo the centralizer of T. That is, Fix a maximal torus in G ; then the corresponding Weyl group is called the Weyl group of G ( it depends up to isomorphism on the choice of T ).

For example, when X is Euclidean space R < sup > n </ sup > of dimension n, and G is the lattice Z < sup > n </ sup > acting on it by translations, the quotient X / G is the n-dimensional torus.

In other words, there exists a faithfully flat map X → S such that any point in X has a quasi-compact open neighborhood U whose image is an open affine subscheme of S, such that base change to U yields a finite product of copies of GL < sub > 1, U </ sub > = G < sub > m </ sub >/ U. One particularly important case is when S is the spectrum of a field K, making a torus over S an algebraic group whose extension to some finite separable extension L is a finite product of copies of G < sub > m </ sub >/ L.

If a torus is isomorphic to a product of multiplicative groups G < sub > m </ sub >/ S, the torus is said to be split.

The weight lattice is the group of algebraic homomorphisms T → G < sub > m </ sub >, and the coweight lattice is the group of algebraic homomorphisms G < sub > m </ sub > → T. These are both free abelian groups whose rank is that of the torus, and they have a canonical nondegenerate pairing given by, where degree is the number n such that the composition is equal to the nth power map on the multiplicative group.