In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

 is said to be flat if its curvature

Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

 is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°.

The universal cover of a complete

Complete space

In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....

 flat manifold is Euclidean space. This can be used to prove the theorem of
that all compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 flat manifolds are finitely covered by tori; the 3-dimensional case was proved earlier by .

Dimension 2

• The plane
• The cylinder
• The Moebius band
• The Klein bottle
  Klein bottle
  In mathematics, the Klein bottle is a non-orientable surface, informally, a surface in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a...
  
  
• The Real projective plane
  Real projective plane
  In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
  
  
• The 2-dimensional torus
  Torus
  In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
  
  . A flat torus can be isometrically embedded in three-dimensional Euclidean space with a C¹ map (by the Nash embedding theorem
  Nash embedding theorem
  The Nash embedding theorems , named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path...
  
  ) but not with a C² map, and the Clifford torus
  Clifford torus
  In geometric topology, the Clifford torus is a special kind of torus sitting inside R4. Alternatively, it can be seen as a torus sitting inside C2 since C2 is topologically the same space as R4...
  
   provides an isometric analytic embedding of a flat torus in four dimensions.

There are 17 compact 2-dimensional orbifolds with flat metric (including the torus and Klein bottle), listed in the article on orbifold

Orbifold

In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...

s, that correspond to the 17 wallpaper groups.

Dimension 3

For the complete list of the 6 orientable and 4 non-orientable compact examples see Seifert fiber space

Seifert fiber space

A Seifert fiber space is a 3-manifold together with a "nice" decomposition as a disjoint union of circles. In other words it is a S^1-bundle over a 2-dimensional orbifold...

.

Higher dimensions

• Euclidean space
• Tori
• Products of flat manifolds
• Quotients of flat manifolds by groups acting freely.

See also

• Space form
  Space form
  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.-Reduction to generalized crystallography:It is a...
  
  s
• Crystallographic groups
• Ricci-flat manifold
  Ricci-flat manifold
  In mathematics, Ricci-flat manifolds are Riemannian manifolds whose Ricci curvature vanishes. In physics, they represent vacuum solutions to the analogues of Einstein's equations for Riemannian manifolds of any dimension, with vanishing cosmological constant...
  
  
• Conformally flat manifold