Plücker interpretation

A model that realizes the geometry of the three-dimensional projective space $ P _ {3} $ in the hyperbolic space $ {} ^ {3} S _ {5} $. The Plücker interpretation is based on a special interpretation of the Plücker coordinates of a straight line, which are defined for any straight line in $ P _ {3} $.

Under projective transformations of $ P _ {3} $ the Plücker coordinates transform linearly; the Plücker coordinates of straight lines in $ P _ {3} $ give a one-to-one correspondence between the straight lines of $ P _ {3} $ and the points in the projective space $ P _ {5} $ whose coordinates are numerically equal to the Plücker coordinates in $ P _ {3} $.

Straight lines in $ P _ {3} $ are represented by the points of a non-singular quadric in $ P _ {5} $ of index three.

If one takes this quadric as the absolute and defines a projective (non-Euclidean) metric in $ P _ {5} $, one gets the five-dimensional hyperbolic space $ {} ^ {3} S _ {5} $. Under each collineation and correlation of $ P _ {3} $ the Plücker coordinates transform linearly, i.e. each collineation and correlation is represented by a collineation of $ P _ {5} $ that maps the absolute into itself. These collineations are thus displacements of $ {} ^ {3} S _ {5} $. The displacements of $ {} ^ {3} S _ {5} $ represent either collineations or correlations in $ P _ {3} $.

Each line complex in $ P _ {3} $ is put into correspondence with a point in $ {} ^ {3} S _ {5} $. The projective geometry of $ P _ {3} $ can be considered as a non-Euclidean geometry of $ {} ^ {3} S _ {5} $. This interpretation of the geometry of $ P _ {3} $ in $ {} ^ {3} S _ {5} $ is called the Plücker interpretation, in connection with the role of the Plücker coordinates.

If one takes a straight line as the basic object in $ P _ {3} $, the geometry of this space can be considered as the geometry on the absolute of $ {} ^ {3} S _ {5} $.

The group of projective transformations of $ P _ {3} $ is isomorphic to the group of displacements of $ {} ^ {3} S _ {5} $, and any involutory projective transformation of $ P _ {3} $ corresponds to an involutory displacement in $ {} ^ {3} S _ {5} $. For example, a null system in $ P _ {3} $ corresponds to a reflection in a point and its polar hyperplane in $ {} ^ {3} S _ {5} $; an involutory homology in $ P _ {3} $ corresponds to a hyperbolic paratactic displacement by a half-line in $ {} ^ {3} S _ {5} $, etc. Each connected component of the group of projective transformations for $ P _ {3} $ corresponds to a connected component of the group of displacements for $ {} ^ {3} S _ {5} $.

1) A collineation of $ P _ {3} $ with positive determinant, including the identity transformation, corresponds to a displacement in $ {} ^ {3} S _ {5} $ with determinant $ + 1 $( identity transformations are included here).

2) Any correlation in $ P _ {3} $ with positive determinant (including the null system) corresponds to a displacement in $ {} ^ {3} S _ {5} $ with determinant equal to $ - 1 $ that transforms the proper and ideal domains, respectively, into themselves (including reflections in a point).

3) Any collineation of $ P _ {3} $ having negative determinant corresponds to a displacement in $ {} ^ {3} S _ {5} $ with determinant $ + 1 $ that transforms the proper domain into the ideal domain and vice versa, and this component contains a hyperbolic displacement by a half-line.

4) Any correlation in $ P _ {3} $ having negative determinant corresponds to a displacement in $ {} ^ {3} S _ {5} $ with determinant $ - 1 $ that transforms the proper domain into the ideal domain and vice versa.

The images under symmetries that correspond to one another in $ P _ {3} $ and $ {} ^ {3} S _ {5} $ are put into correspondence with numerical invariants, between which there are certain relationships.

The Plücker interpretation is used in research on the displacement groups for the three-dimensional non-Euclidean spaces $ S _ {3} $, $ {} ^ {1} S _ {3} $ and $ {} ^ {2} S _ {3} $, which are isomorphic to certain subgroups of the displacement group for $ {} ^ {3} S _ {5} $. There is also a relation between the groups of motions for these three-dimensional spaces (elliptic and hyperbolic) and groups of displacements of spaces of lower dimensions (see Fubini model; Kotel'nikov interpretation). The Plücker interpretation is also used in examining the interpretation of the three-dimensional symplectic space $ \mathop{\rm Sp} _ {3} $ in $ {} ^ {3} S _ {5} $.

The Plücker interpretation was proposed by J. Plücker [1].

References

[1]	J. Plücker, "Neue Geometrie des Raumes gegründet auf die Betrachtung der geraden Linie als Raumelement" , Teubner (1868)
[2]	B.A. Rozenfel'd, "Non-Euclidean geometry" , Moscow (1955) (In Russian)
[3]	F. Klein, "Vorlesungen über höhere Geometrie" , Springer (1926)

Comments

The quadric in $ P _ {5} $ whose points represent the lines in $ P _ {3} $ is often referred to as the Plücker quadric.

References

[a1]	J.W.P. Hirschfeld, "Finite projective spaces of three dimensions" , Oxford Univ. Press (1985)
[a2]	B.L. van der Waerden, "Einführung in die algebraische Geometrie" , Springer (1939) pp. Chapt. 1
[a3]	D.M.Y. Sommerville, "Analytical geometry of three dimensions" , Cambridge Univ. Press (1934)