In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
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, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
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
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
flat manifold is Euclidean space. This can be used to prove the theorem of
that all
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
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
).
Dimension 1
Every one-dimensional Riemannian manifold is flat. Conversely, given that every connected one-dimensional smooth manifold is diffeomorphic to either
or
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
for some number
* the open interval
* the circle
of radius
for some number
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
is a smooth two-dimensional connected complete flat Riemannian manifold, then
must be diffeomorphic to
the
Möbius strip
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Augu ...
, or the
Klein bottle
In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...
. Note that the only compact possibilities are
and the Klein bottle, while the only orientable possibilities are
and
It takes more effort to describe the distinct complete flat Riemannian metrics on these spaces. For instance, the two factors of
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
let
denote the translation
given by
Let
denote the reflection
given by
Given two positive numbers
consider the following subgroups of
the group of isometries of
with its standard metric.
*
*
*
provided