Projective space

The projective space is in mathematics, a fundamental concept of both differential geometry and algebraic geometry. Geometrically, one can understand the projective space as a projective extension of the affine space. With this extension you added to the points of the space for each family of parallel straight lines in the affine space each have their direction as a new point ( farthest point ). Thus in sentences and proofs omitted many case distinctions that are necessary in the affine formulation due to the parallelism. In the coordinate representation happens that extension if homogeneous coordinates are introduced in affine space and the restriction that the additionally introduced coordinate must not be 0, it is dropped. This brings us to a projective coordinate system of the projective space.

For projective spaces a duality principle.

Definition

The real projective space is the set of all lines through the origin in. Formally, we define this as follows.

On is the equivalence relation

Defined. In words, this means that if and is equivalent to, if there is a, so that the following applies. All points on a straight line through the origin - the origin is not included - are so identified with each other and no longer distinguished. The quotient space is real, dimensional projective space called and recorded.

In case one speaks of the projective line (also: projective line) and in the case of a projective plane.

If one chooses instead the complex vector space, one obtains the analogue definition with the complex projective space of (complex) DIMENSION as the space of complex 1- dimensional subspaces of.

The coordinates of the points of the projective space, which are indeed equivalence classes of points are recorded by and called homogeneous coordinates. ( According to the complex projective space. ) For defines the mapping is a bijection between and.

General also projective space on any other articles can be designed ( in place of or ).

A more general term of the projective space is used in the synthetic geometry, especially in the case of the projective plane. The axioms of this general concept is represented in the main article projective geometry.

Projective linear group ( collineations )

Projective linear group PGL (n 1, r) is the group of invertible projective images is defined as the ratio of GL ( n 1, R) of the equivalence relation

The effects of the GL (n 1, R) is a well-defined action of PGL (n 1, r). The corresponding elements of the pictures are projective, which here means double ratio loyal collineations. In other words:

Analogously to define a PGL activity of (n 1, C).

In the case of projective straight acting PGL (2, R) by fractional- linear transformations. Having identified with (or with ) acts PGL ( 2, R ) or PGL (2, C).

Example: Riemann number sphere

The complex projective line is as defined above is just the set of complex lines in which pass through the origin.

The complex projective line can be also known as the real - two-dimensional sphere or Riemann number sphere

Interpret. Compliance with these terms is as follows: Denote by the " North Pole ". Consider the stereographic projection

Which is given by. Clearly puts you through the North Pole and a (real) line and selects the intersection of this line with the equatorial plane as a pixel of the image, with the north pole is identified with. The correspondence between, and in homogeneous coordinates is then.

Properties

• The real and complex projective spaces are compact manifolds.
• The projective space is an example of a non-affine algebraic variety or a non- affine scheme. In algebraic- geometric context, one can use any body instead of the real or complex numbers.
• Subvarieties of projective space are called projective varieties (obsolete as projective manifolds ) refers.
• Local projective space modeled after the locally - homogeneous manifolds are called projective manifolds.

Topology

The projective line is homeomorphic to the circle. For the fundamental group of the projective space is the group Z/2Z, the 2- fold covering of the sphere.

For odd n is orientable, n is just he is not orientable.

The projective plane is a non- orientable surface, which can not be embedded in the. But there are immersions of the, for example, the so-called Boy's surface.

The complex projective line is homeomorphic to the sphere, the quaternionisch - projective line is homeomorphic to the Cayley projective line homeomorphic to.

All complex or quaternionisch projective spaces are simply connected.

The Hopf fibrations form ( for ) each of the unit sphere into on that fiber is the unit sphere in. Obtained in this way fibrations

These fibrations have Hopf invariant 1

Projective subspaces and derived spaces

In this section the above general definition of a dimensional projective space is considered over an arbitrary field in the sense that the points of space can therefore be regarded as one-dimensional subspaces of.

• Each -dimensional subspace of a - dimensional projective subspace of assigned. It is also called a (generalized projective ) plane for hyperplane for straight into. The empty set is here considered as a projective subspace that is assigned to and the dimension of the null space.
• The intersection of two projective subspaces is again a projective subspace.
• If one forms the subspaces that are associated with two projective spaces and the linear hull of their union in so belongs to this subspace again a projective subspace of the connection space ( also listed as a sum ) of and.
• For cutting and joining of projective subspaces projective dimension formula holds:

• The set of all subspaces of the projective space forms with respect to the links " section " and " connection" a finite length, modular, complementary association.
• Each projective point can be assigned to its coordinates a homogeneous coordinate equation whose solution set describes a hyperplane. Due to the hyper -plane coordinates so defined are the hyperplanes in turn points of a projective space, the dual space. (→ see projective coordinate system # coordinates equations and hyper -plane coordinates ).
• Generally, the set of hyperplanes, containing a solid -dimensional sub-space, a projective space, which is referred to as a bundle, in the special case as tufts of hyperplanes. ie carrier of the bundle or tuft.