In 1872, German mathematician Klein proposed a paper on the "Comparative Examination on Modern Geometry" in the University of Erranian University. He proposed that the so-called geometry is to study the study of the nature of a certain type of transformation remaining unchanged. According to this view, the "geometric nature" of the so-called graphic is the nature of the change in a transform group. In other words, how many different transformation groups have many different geometries. This view of Klein is later referred to as "Erlanggen".

Introduction SAFR group

Geometric transformation

gives a set M of any geometric object, and the agreed is called space. The process that changes each geometric object (or element) into another geometric object is referred to as a geometric transformation on M, referred to as transformation. A geometric object is indicated by alpha or a pattern configured by many objects, and α changes α to another object or graphic B below T, remember to T (α) = B, B The image of α, α is called the image source of B.

Take another transform S role to B, set S (b) = c, if the two transformation continuous effects, α change to с, so the process of changing to the с is also a transformation, Remember to p, ie p (α) = с. P The product of S and T is referred to as P = ST. The order of transforming product is generally unswerving, ie ST ≠ TS.


If there are three transformations T, S, R, prior action T, followed by S, and finally R, and the result is RST, the order of the symbol represents the right side to the left. The combination of transform product is established: (RS) T = R (ST) = RST.

If T is changed, each element B is the image of a unique element α, and then t is a one-to-one transformation. At this time, T has a determined inverse transform, remember T -1, t-1 product maintains each element, which is constant conversion, remembering E, TT-1 = T-1T = E.

The conversion group is a collection of limited or unlimited transformations on M, and satisfies the following two conditions: 1 The product of any two transformations in the collection G belong to G; 2 Each of the collateral g The transformation must have its inverse transformation, and this inverse transformation also belongs to G, which is called a conversion group on M.

If a part of the transformation is taken from a known transform group G, the whole constituting a transform group G1, and a transform subgroup of G1 is G1.

is defined by the definition: the integration of sports sets, affine conversion sets, reflection sets, etc. in planar or space constitutes a transform group, respectively, called a homage group, an affine group, a projectile group, and the like, respectively. Wait; the movement group is a subgroup of the affine group, the sport group and the affine group are the subgroup of the shooting group.

given a space M and a transform group G, if there is a transformation in G, the graphic α becomes graphic B, and the α and B are equivalent. Definition from the transformation group can be launched:

1 If the graphic α is equivalent to graphic B, the graphic B is also equivalent to graphic α equivalent. In fact, if the graphic α is equivalent to B, the group G must change T, so that T (α) = B; that is T-1 (b) = α, however T-1 belongs to G, which indicates that G There is a change to change B to α, so B is equivalent to α.

2 If two graphics α and b are equivalent to the third graphic с equivalent, α and b are equivalent to each other. In fact, if α is equivalent to с equivalence, the group g must change T, so that the T (α) = с; if B is equivalent to the с, the G must change S, so that S (b) = с Thus, S-1 (с) = B, therefore, S-1T (α) = B, so pattern α and B is equivalent.


Clein's equivalent nature of the graphics in space M is referred to as geometric nature or constant nature, and the geometric properties are any transform in the known group G. The unchanged amount is combined, which is obviously all equivalent graphics. All invariant properties under a group G are called the nature of G, and the geometry of the nature belonging to g is known as the geometry from G.

Klein's idea

Clein's theory of various geometries as the constant nature of various groups they have learned, making it in the 1980s The various geometries found have shown a more profound connection, and he proposed this group of view in the famous "Erranian Gang". Here, it takes out the idea of ​​geometric classification in accordance with the change group - the idea of ​​the Erlanggen. For example, the nature of the motion is the metrics, and the geometric geometric geometric geometry is called metrics (Ou's geometry); the nature of the affine transformation is the nature of the affine, and the geometric of the affine is called the affine geometry; The nature of the shooting transform is the material of the shooting, and the geometric geometry of the material is called the shooting geometry; Under the sport, the distance, angle, area, parallelism, single ratio, and cross-comparison; under affine transformation, distance, angle, area varies, but (in the same direction line segment) single, parallel Sexuality, a total linear, comparison, remain unchanged; for the shooting group, single ratio, parallelism changes, but co-linear, and the volume is maintained unchanged. This is because the motion group is a subgroup of the affine group, and the affine group is a subgroup of the shooting group.

Small junction

According to the above, the unchangeable properties under a certain change group must be the nature of its subgroup, but it is not necessary to establish it, that is, the group The bigger the difference, the less the geometry; the smaller the group, the more geometric content. For example, in the European geometry, the affine nature (single ratio, parallelism, etc.) can be discussed in the affine geometry (such as distance, angle, etc.).

The proposal of the Erlangen Ceremony is meant to deepen the geometric understanding. It puts all the geoments into a unified form, making people clarify the objects of classical geometry; showing how to establish a geometric method of abstract space, a guiding role in the future development, so far-reaching historical meaning.

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