What is geometry5.2. Automorphisms and invariants
The use of real numbers
Dimensions and semantic completeness
Multiple automorphisms and modelsThe below is under construction
The Galois connection (Aut, sInv)
The double meaning of "invariance"
Using morphisms to form geometries
Basis and algebraic structures
Subspaces, embeddings and sections
(For a short simplified version without reference to the
prerequisites, just read the text in green)
Some geometric spaces, such as vector spaces, Euclidean spaces,
and both space-times without gravitation: the classical one (the
Galilean space-time), and the one of Special Relativity (the
Minkowski space), are "richer" than affine geometry, i.e.
"contain an affine structure", in the following (roughly)
But other geometries do not contain an affine structure, either
because they have no notion of straight line, or because these
lines do not satisfy the axioms of affine geometry. For example
Like affine spaces, projective spaces are also based on the structure of alignment, but have no other affine structures from the above list. Parallels cannot be defined there as any pair of straight lines in a plane has one intersection point. Such spaces can be described as affine spaces together with extra points "at infinity" in the role of intersection points of parallel lines, while ignoring which points are at infinity and which are not. These points "at infinity" form the line of "horizon" that is a straight line like others. Projective transformations of the plane (automorphisms of the projective plane) are those involved for perspective representation. Among them, affine transformations are those keeping the horizon to itself, so that affine geometry is equivalent to projective geometry with a constant symbol named "horizon" with type "straight line".
It has a notion of circle (or sphere) but cannot distinguish
straight lines among them.
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Still, affine geometry is not far away from the above, as small regions of these spaces are approximately affine too: the smaller a region, the better its description can be approximated by affine geometry (with possibly more structures). Usual geometric spaces will have this property of being approximately affine in small regions : we will say they are smooth.
Differential geometry is the "geometry" whose only structure is
the notion of smoothness, and smooth curves.
In particular, smooth spaces have an approximation for ratios of small volumes as they become smaller and closer to each other.
The smoothness structure cannot be restricted to a structure of
"being approximately projective in small regions" because the
"approximately affine structure" of small regions would anyway be
definable from it, as the horizon relatively to a small region,
has its needed approximative definition as "what is not near this
This is even weaker than differential geometry, as it has a
notion of curve requesting their continuity, that is less
restrictive than smoothness. For example the Koch
snowflake is continuous but not smooth, so that it is
distinguished from curves by differential geometry, but not by
Introduction to topology
Let us first only present the structures, assumed to obey the
properties of the intended spaces. The axiomatic
specification of these properties will only be completed
Euclidean geometry (with any
dimension >1) consists in affine geometry together with one
structure, thus named the Euclidean structure, that can be
expressed (represented) in the following equivalent forms (any
one suffices to define the others) :
geometry may also admit one more structure from the previous
list : the unit of area (or unit of volume, in the case
of higher dimension), which finally (thanks to the above Euclidean
structure) can be called a choice
of unit of distance. The choice to include or not this
structure in the definition of "Euclidean geometry" is debatable,
with motivations for or against it, as follows
For any dimension, the operation
of distance d(A,B) between any two
points A and B (with values either among
real numbers or in a set of quantities, depending on the above
choice) can be seen as the
fundamental structure of Euclidean geometry, as it is completely
sufficient to define all other structures of this geometry in a
rather natural way.
Indeed, distance suffices to define the affine structures: the
betweenness relation is defined as
B∈[AC] ⇔ d(A,C) = d(A,B) + d(B,C)
and others can be
defined from it. Therefore, the transformations preserving
distance (even if its values are mere quantities), called isometries,
preserve all other structures of Euclidean geometry as well.
Other presentations of the Euclidean structure, assume a priori
As for the notions of circle (or sphere) and intersection angles,
they may suffice to define affine geometry but only
The details of these correspondences between different formulations of the Euclidean structure, will be explained in the introduction to inversive geometry.
The isometries of an Euclidean plane or space are called
moves (to be distinguished from the isometries of other
spaces such as a sphere, with also an operation of distance but
that does not satisfy all the same axioms). The space of
isometries of the Euclidean plane is 3-dimensional, and split in 2
pieces (each of which is also 3-dimensional):
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