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In mathematics, a Riemannian manifold is said to be flat if its
Riemann curvature tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
is everywhere zero. Intuitively, a flat manifold is one that "locally looks like"
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°. The
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
of a complete flat manifold is Euclidean space. This can be used to prove the theorem of that all compact flat manifolds are finitely covered by tori; the 3-dimensional case was proved earlier by .


Examples

The following manifolds can be endowed with a flat metric. Note that this may not be their 'standard' metric (for example, the flat metric on the 2-dimensional torus is not the metric induced by its usual embedding into \mathbb^3).


Dimension 1

Every one-dimensional Riemannian manifold is flat. Conversely, given that every connected one-dimensional smooth manifold is diffeomorphic to either \mathbb or S^1, it is straightforward to see that every connected one-dimensional Riemannian manifold is isometric to one of the following (each with their standard Riemannian structure): * the real line * the open interval (0,x) for some number x>0 * the open interval (0,\infty) * the circle \ of radius r, for some number r>0. Only the first and last are complete. If one includes Riemannian manifolds-with-boundary, then the half-open and closed intervals must also be included. The simplicity of a complete description in this case could be ascribed to the fact that every one-dimensional Riemannian manifold has a smooth unit-length vector field, and that an isometry from one of the above model examples is provided by considering an integral curve.


Dimension 2


The five possibilities, up to diffeomorphism

If (M,g) is a smooth two-dimensional connected complete flat Riemannian manifold, then M must be diffeomorphic to \mathbb^2, S^1\times\mathbb, S^1\times S^1, the Möbius strip, or the Klein bottle. Note that the only compact possibilities are S^1\times S^1 and the Klein bottle, while the only orientable possibilities are \mathbb^2, S^1\times \mathbb, and S^1\times S^1. It takes more effort to describe the distinct complete flat Riemannian metrics on these spaces. For instance, the two factors of S^1\times S^1 can have any two real numbers as their radii. These metrics are distinguished from each other by the ratio of their two radii, so this space has infinitely many different flat product metrics which are not isometric up to a scale factor. In order to talk uniformly about the five possibilities, and in particular to work concretely with the Möbius strip and the Klein bottle as abstract manifolds, it is useful to use the language of group actions.


The five possibilities, up to isometry

Given (x_0,y_0)\in\mathbb^2, let T_ denote the translation \mathbb^2\to\mathbb^2 given by (x,y)\mapsto(x+x_0,y+y_0). Let R denote the reflection \mathbb^2\to\mathbb^2 given by (x,y)\mapsto(x,-y). Given two positive numbers a,b, consider the following subgroups of \operatorname(\mathbb^2), the group of isometries of \mathbb^2 with its standard metric. * G_=\ * G_(a)=\ * G_(a,b)=\ provided a * G_(a)=\\cup\ * G_(b)=\\cup\ These are all groups acting freely and properly discontinuously on \mathbb^2, and so the various coset spaces \mathbb^2/G all naturally have the structure of two-dimensional complete flat Riemannian manifolds. None of them are isometric to one another, and any smooth two-dimensional complete flat connected Riemannian manifold is isometric to one of them.


Orbifolds

There are 17 compact 2-dimensional orbifolds with flat metric (including the torus and Klein bottle), listed in the article on
orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...
s, that correspond to the 17
wallpaper groups A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformati ...
.


Remarks

Note that the standard 'picture' of the torus as a doughnut does not present it with a flat metric, since the points furthest from the center have positive curvature while the points closest to the center have negative curvature. According to Kuiper's formulation of the
Nash embedding theorem The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedding, embedded into some Euclidean space. Isometry, Isometric means preserving the length of every ...
, there is a C^1 embedding S^1\times S^1\to\mathbb^3 which induces any of the flat product metrics which exist on S^1\times S^1, but these are not easily visualizable. Since S^1 is presented as an embedded submanifold of \mathbb^2, any of the (flat) product structures on S^1\times S^1 are naturally presented as submanifolds of \mathbb^2\times\mathbb^2=\mathbb^4. Likewise, the standard three-dimensional visualizations of the Klein bottle do not present a flat metric. The standard construction of a Möbius strip, by gluing ends of a strip of paper together, does indeed give it a flat metric, but it is not complete.


Dimension 3

There are 6 orientable and 4 non-orientable compact flat 3-manifolds, which are all
Seifert fiber space A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle ( circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
s; they are the quotient groups of \mathbb^3 by the 10 torsion-free crystallographic groups. There are also 4 orientable and 4 non-orientable non-compact spaces.


Orientable

The 10 orientable flat 3-manifolds are: #
Euclidean 3-space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
, \mathbb^3. # The 3-torus T^3, made by gluing opposite faces of a cube. # The manifold made by gluing opposite faces of a cube with a 1/2 twist on one pair. # The manifold made by gluing opposite faces of a cube with a 1/4 twist on one pair. # The manifold made by gluing opposite faces of a
hexagonal prism In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices.. Since it has 8 faces, it is an octahedron. However, the term ''octahedron'' is primarily used ...
with a 1/3 twist on the hexagonal faces. # The manifold made by gluing opposite faces of a hexagonal prism with a 1/6 twist on the hexagonal faces. # The Hantzsche–Wendt manifold. # The manifold S^1 \times \mathbb^2 made as the space between two parallel planes that are glued together. # The manifold T^2 \times \mathbb made by gluing opposite walls of an infinite square chimney. # The manifold made by gluing opposite walls of an infinite square chimney with a 1/2 twist on one pair.


Non-orientable

The 8 non-orientable 3-manifolds are: # The Cartesian product of a circle and a Klein bottle, S^1 \times K. # A manifold similar to the aforementioned, but translationally offset in one direction parallel to the
glide plane In geometry and crystallography, a glide plane (or transflection) is a symmetry operation describing how a reflection in a plane, followed by a translation parallel with that plane, may leave the crystal unchanged. Glide planes are noted by ''a'' ...
; moving in this direction returns to the opposite side of the manifold. # The manifold made by reflecting a point across two perendicular glide planes and translating along the third direction. # A manifold similar to the aforementioned, but translationally offset in one direction parallel to one glide plane; moving in this direction returns to the opposite side of the manifold. # The Cartesian product of a circle and an (unbounded) Möbius strip. # The manifold K \times \mathbb made by translating a point along one axis and reflecting it across a perpendicular glide plane. # The manifold made by translating a point along one axis and reflecting it across a parallel glide plane. # The manifold made by reflecting a point across two perpendicular glide planes.


Higher dimensions

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


Relation to amenability

Among all closed manifolds with non-positive sectional curvature, flat manifolds are characterized as precisely those with an amenable fundamental group. This is a consequence of the Adams- Ballmann theorem (1998), which establishes this characterization in the much more general setting of discrete cocompact groups of isometries of
Hadamard space In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space. In the literature they are also equivalently defined as complete CAT(0) spaces. A Hadamard space is defined to be a nonempty complete ...
s. This provides a far-reaching generalisation of Bieberbach's theorem. The discreteness assumption is essential in the Adams-Ballmann theorem: otherwise, the classification must include
symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...
s, Bruhat-Tits buildings and Bass-Serre trees in view of the "indiscrete" Bieberbach theorem of Caprace- Monod.


See also

* Space forms * Crystallographic groups * Ricci-flat manifold * Conformally flat manifold * Affine manifold


References

*. *. * *. *


External links

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References

{{Manifolds Riemannian manifolds