Real projective line
In
An example of a real projective line is the projectively extended real line, which is often called the projective line.
Formally, a real projective line P(R) is defined as the set of all one-dimensional linear subspaces of a two-dimensional vector space over the reals. The
Topologically, real projective lines are
Definition
The points of the real projective line are usually defined as
v ~ w to hold when there exists a nonzero real number t such that v = tw. The definition of a vector space implies almost immediately that this is an equivalence relation. The equivalence classes are the vector lines from which the zero vector has been removed. The real projective line P(V) is the set of all equivalence classes. Each equivalence class is considered as a single point, or, in other words, a point is defined as being an equivalence class.If one chooses a basis of V, this amounts (by identifying a vector with its coordinate vector) to identify V with the direct product R × R = R2, and the equivalence relation becomes (x, y) ~ (w, z) if there exists a nonzero real number t such that (x, y) = (tw, tz). In this case, the projective line P(R2) is preferably denoted P1(R) or . The equivalence class of the pair (x, y) is traditionally denoted [x: y], the colon in the notation recalling that, if y ≠ 0, the
As P(V) is defined through an equivalence relation, the
Charts
The projective line is a
- Chart #1:
- Chart #2:
The equivalence relation provides that all representatives of an equivalence class are sent to the same real number by a chart.
Either of x or y may be zero, but not both, so both charts are needed to cover the projective line. The
The inverse function of chart #1 is the map
It defines an
The same construction may be done with the other chart. In this case, the point at infinity is [0: 1]. This shows that the notion of point at infinity is not intrinsic to the real projective line, but is relative to the choice of an embedding of the real line into the projective line.
Structure
The real projective line is a
The real projective line has a cyclic order that extends the usual order of the real numbers.
Automorphisms
The projective linear group and its action
Matrix-vector multiplication defines a left action of GL2(R) on the space R2 of column vectors: explicitly,
Since each matrix in GL2(R) fixes the zero vector and maps proportional vectors to proportional vectors, there is an induced action of GL2(R) on P1(R): explicitly,[2]
(Here and below, the notation for homogeneous coordinates denotes the equivalence class of the
The elements of GL2(R) that act trivially on P1(R) are the nonzero scalar multiples of the identity matrix; these form a subgroup denoted R×. The projective linear group is defined to be the quotient group PGL2(R) = GL2(R)/R×. By the above, there is an induced faithful action of PGL2(R) on P1(R). For this reason, the group PGL2(R) may also be called the group of linear automorphisms of P1(R).
Linear fractional transformations
Using the identification R ∪ ∞ → P1(R) sending x to [x:1] and ∞ to [1:0], one obtains a corresponding action of PGL2(R) on R ∪ ∞ , which is by linear fractional transformations: explicitly, since
the class of in PGL2(R) acts as [3][4][5] and ,[6] with the understanding that each fraction with denominator 0 should be interpreted as ∞.[7]
Properties
- Given two ordered triples of distinct points in P1(R), there exists a unique element of PGL2(R) mapping the first triple to the second; that is, the action is sharply 3-transitive. For example, the linear fractional transformation mapping (0, 1, ∞) to (−1, 0, 1) is the Cayley transform .
- The of the real line, consisting of the transformations for all a ∈ R× and b ∈ R.
See also
Notes
- ^ The argument used to construct P1(R) can also be used with any field K and any dimension to construct the projective space Pn(K).
- ^ Miyake, Modular forms, Springer, 2006, §1.1. This reference and some of the others below work with P1(C) instead of P1(R), but the principle is the same.
- ^ Lang, Elliptic functions, Springer, 1987, 3.§1.
- ^ Serre, A course in arithmetic, Springer, 1973, VII.1.1.
- ^ Stillwell, Mathematics and its history, Springer, 2010, §8.6
- ^ Lang, Complex analysis, Springer, 1999, VII, §5.
- ^ Koblitz, Introduction to elliptic curves and modular forms, Springer, 1993, III.§1.
References
- Juan Carlos Alvarez Paiva (2000) The Real Projective Line, course content from New York University.
- Santiago Cañez (2014) Notes on Projective Geometry from Northwestern University.