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Projective Spaces

  • Igor R. Shafarevich
  • Alexey O. Remizov

Abstract

The theory of projective spaces and their transformations is presented. The notions of a projective space, projective subspace, homogeneous and inhomogeneous coordinates, projective algebraic variety, projective transformation, cross ratio, etc., are introduced and discussed. The principle of projective duality is presented. At the end of the chapter, the topological properties of real and complex projective spaces are investigated. In particular, compactness of projective spaces and projective algebraic varieties is established. Also, it is proved that a real projective space of arbitrary odd dimension has two orientations, and a real projective space of arbitrary even dimension has one orientation.

Keywords

Projective Space Projective Plane Projective Line Projective Transformation Cross Ratio 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

9.1 Definition of a Projective Space

In plane geometry, points and lines in the plane play very similar roles. In order to emphasize this symmetry, the fundamental property that connects points and lines in the plane is called incidence, and the fact that a point A lies on a line l or that a line l passes through a point A expresses in a symmetric form that A and l are incident. Then one might hope that to each assertion of geometry about incidence of points and lines there would correspond another assertion obtained from the first by everywhere interchanging the words “point” and “line.” And such is indeed the case, with some exceptions. For example, to every pair of distinct points, there is incident one and only one line. But it is not true that to every pair of distinct lines, there is incident one and only one point: the exception is the case that the lines are parallel. Then not a single point is incident to the two lines.

Projective geometry gives us the possibility of eliminating such exceptions by adding to the plane certain points called points at infinity. For example, if we do this, then two parallel lines will be incident at some point at infinity. And indeed, with a naive perception of the external world, we “see” that parallel lines moving away from us converge and intersect at a point on the “horizon.” Strictly speaking, the “horizon” is the totality of all points at infinity by which we extend the plane.

In analyzing this example, we may say that a point p of the plane seen by us corresponds to the point where the line passing through p and the center of our eye meets the retina. Mathematically, this situation is described using the notion of central projection.

Let us assume that the plane Π that we are investigating is contained in three-dimensional space. Let us choose in this same space some point O not contained in the plane Π. Every point A of the plane Π can be joined to O by the line OA. Conversely, a line passing through the point O intersects the plane Π in a certain point, provided that the line is not parallel to Π. Thus most straight lines passing through the point O correspond to points AΠ. But lines parallel to Π intuitively correspond precisely to points at infinity of the plane Π, or “points on the horizon.” See Fig. 9.1.
Fig. 9.1

Central projection

We shall make this notion the basis of the definition of projective space and shall develop it in more detail in the sequel.

Definition 9.1

Let L be a vector space of finite dimension. The collection of all lines 〈x〉, where x is a nonnull vector of the space L, is called a projectivization of L or projective space ℙ(L). Here the lines 〈x〉 themselves are called points of the projective space ℙ(L). The dimension of the space ℙ(L) is defined as the number dimℙ(L)=dimL−1.

As we saw in Chap.  3, all vector spaces of a given dimension n are isomorphic. This fact is expressed by saying that there exists only one theory of n-dimensional vector spaces. In the same sense, there exists only one theory of n-dimensional projective space.

We shall frequently denote the projective space of dimension n by ℙ n if we have no need of indicating the (n+1)-dimensional vector space on the basis of which it was constructed.

If dimℙ(L)=1, then ℙ(L) is called the projective line, and if dimℙ(L)=2, then it called the projective plane. Lines in an ordinary plane are points on the projective line, while lines in three-dimensional space are points in the projective plane.

And as earlier, we give the reader the choice whether to consider L a real or complex space, or even to consider it as a space over an arbitrary field \({\mathbb{K}}\) (with the exception of certain questions related specifically to real spaces). In accordance with the definition given above, we shall say that dimℙ(L)=−1 if dimL=0. In this case, the set ℙ(L) is empty.

In order to introduce coordinates in a space ℙ(L) of dimension n, we choose a basis e 0,e 1,…,e n in the space L. A point A∈ℙ(L) is by definition a line 〈x〉, where x is some nonnull vector in L. Thus we have the representation
$$ {\boldsymbol{x} }= \alpha _0{\boldsymbol{e} }_0 + \alpha _1{\boldsymbol{e} }_1 + \cdots+ \alpha _n{\boldsymbol{e} }_n. $$
(9.1)
The numbers (α 0,α 1,…,α n ) are called homogeneous coordinates of the point A. But the point A is the entire line 〈x〉. It can also be obtained in the form 〈y〉 if y=λ x and λ≠0. Then
$$ {\boldsymbol{y} }= \lambda \alpha _0{\boldsymbol{e} }_0 + \lambda \alpha _1 {\boldsymbol{e} }_1 + \cdots+ \lambda \alpha _n{\boldsymbol{e} }_n. $$
From this it follows that the numbers (λα 0,λα 1,…,λα n ) are also homogeneous coordinates of the point A. That is, homogeneous coordinates are defined only up to a common nonzero factor. Since by definition, A=〈x〉 and x0, they cannot all be simultaneously equal to zero. In order to emphasize that homogeneous coordinates are defined only up to a nonzero common factor, they are written in the form
$$ (\alpha _0 : \alpha _1 : \alpha _2 : \cdots: \alpha _n). $$
(9.2)
Thus if we wish to express some property of the point A in terms of its homogeneous coordinates, then that assertion must continue to hold if all the homogeneous coordinates (α 0,α 1,…,α n ) are simultaneously multiplied by the same nonzero number.
Let us assume, for example, that we are considering the points of projective space whose homogeneous coordinates satisfy the relationship
$$ F(\alpha _0, \alpha _1,\ldots, \alpha _n) = 0, $$
(9.3)
where F is a polynomial in n+1 variables. In order for this requirement actually to be related to the points and not depend on the factor λ by which we can multiply their homogeneous coordinates, it is necessary that along with the numbers (α 0,α 1,…,α n ), the relationship (9.3) be satisfied as well by the numbers (λα 0,λα 1,…,λα n ) for an arbitrary nonzero factor λ.
Let us elucidate when this requirement is satisfied. To this end, in the polynomial F(x 0,x 1,…,x n ) let us collect all terms of the form \(a x_{0}^{k_{0}} x_{1}^{k_{1}} \cdots x_{n}^{k_{n}}\) with k 0+k 1+⋯+k n =m and denote their sum by F m . We thereby obtain the representation
$$F(x_0, x_1,\ldots, x_n) = \sum _{m=0}^N F_m (x_0, x_1,\ldots, x_n). $$
It follows at once from the definition of F m that
$$F_m (\lambda x_0, \lambda x_1,\ldots, \lambda x_n) = \lambda ^m F_m (x_0, x_1,\ldots, x_n). $$
From this, we obtain
$$F (\lambda x_0, \lambda x_1,\ldots, \lambda x_n) = \sum _{m=0}^N \lambda ^m F_m (x_0, x_1,\ldots, x_n). $$
Our condition means that the equality \(\sum_{m=0}^{N} \lambda ^{m} F_{m} = 0\) is satisfied for the coordinates of the points in question and simultaneously for all nonzero values of λ. Let us denote by c m the value F m (α 0,α 1,…,α n ) for some concrete choice of homogeneous coordinates (α 0,α 1,…,α n ). Then we arrive at the condition \(\sum_{m=0}^{N} c_{m} \lambda ^{m} = 0\) for all nonzero values λ. This means that the polynomial \(\sum_{m=0}^{N} c_{m} \lambda ^{m}\) in the variable λ has an infinite number of roots (for simplicity, we are now assuming that the field \({\mathbb{K}}\) over which the vector space L is being considered is infinite; however, it would be possible to eliminate this restriction). Then, by a well-known theorem on polynomials, all the coefficients c m are equal to zero. In other words, our equality (9.3) is reduced to the satisfaction of the relationship
$$ F_m (\alpha _0, \alpha _1,\ldots, \alpha _n) = 0,\quad m = 0,1,\ldots, N. $$
(9.4)
The polynomial F m contains only monomials of the same degree m, that is, it is homogeneous. We see that the property of the point A expressed by an algebraic relationship between its homogeneous coordinates does not depend on the permissible selection of coordinates but only on the point A itself if it is expressed by setting the homogeneous polynomials in its coordinates equal to zero.

If L′⊂L is a vector subspace, then ℙ(L′)⊂ℙ(L), since every line 〈x〉 contained in L′ is also contained in L. Such subsets ℙ(L′)⊂ℙ(L) are called projective subspaces of the space ℙ(L). Every ℙ(L′) is by definition itself a projective space. Its dimension is thus defined by dimℙ(L′)=dimL′−1. By analogy with vector spaces, a projective subspace ℙ(L′)⊂ℙ(L) is called a hyperplane if dimℙ(L′)=dimℙ(L)−1, that is, if dimL′=dimL−1, and consequently, L′ is a hyperplane in L.

A set of points of the space ℙ(L) defined by the relationships
$$ \begin{cases} F_1 (\alpha _0,\alpha _1,\ldots, \alpha _n) = 0, \cr F_2 (\alpha _0, \alpha _1,\ldots,\alpha _n) = 0, \cr \ldots\ldots\ldots\ldots\ldots\ldots\ldots \cr F_m (\alpha _0, \alpha _1,\ldots,\alpha _n) = 0, \end{cases} $$
(9.5)
where F 1,F 2,…,F m are homogeneous polynomials of differing (in general) degrees, is called a projective algebraic variety.

Example 9.2

The simplest example of a projective algebraic variety is a projective subspace. Indeed, as we saw in Sect.  3.7, every vector subspace L′⊂L can be defined with the aid of a system of linear homogeneous equations, and consequently, a projective subspace ℙ(L′)⊂ℙ(L) can be defined by formula (9.5), in which m=dimℙ(L)−dimℙ(L′) and the degree of each of the homogeneous polynomials F 1,…,F m is equal to 1. Here in the case m=1, we obtain a hyperplane.

Example 9.3

Another important example of a projective algebraic variety is what are called projective quadrics. They are given by formula (9.5), where m=1 and the degree of the sole homogeneous polynomial F 1 is equal to 2. We shall consider quadrics in detail in Chap.  11. The simplest examples of projective quadrics appear in a course in analytic geometry, namely curves of degree 2 in the projective plane.

Example 9.4

Let us consider the set of points of the projective space ℙ(L) whose ith homogeneous coordinate (in some basis e 0,e 1,…,e n of the space L) is equal to zero, and let us denote by L i the set of vectors of the space L associated with these points. The subset L i L is defined in L by a single linear equation α i =0, and therefore is a hyperplane. This means that ℙ(L i ) is a hyperplane in the projective space ℙ(L). We shall denote the set of points of the projective space ℙ(L) whose ith homogeneous coordinate is nonzero by V i . It is obvious that V i is already not a projective subspace in ℙ(L).

The following construction is a natural generalization of Example 9.4. In the space L let an arbitrary basis e 0,e 1,…,e n be chosen. Let us consider some linear function φ on the space L not identically equal to zero. Vectors xL for which φ(x)=0 form a hyperplane L φ L. It is a subspace of the solutions of the “system” consisting of a single linear homogeneous equation. To it is associated the projective hyperplane ℙ(L φ )⊂ℙ(L). It is obvious that L φ coincides with the hyperplane L i from Example 9.4 if the linear function φ maps each vector xL onto its ith coordinate, that is, φ is the ith vector of the basis of the space L , the dual of the basis e 0,e 1,…,e n of the space L.

Let us now denote by W φ the set of vectors xL for which φ(x)=1. This is again the set of solutions of the “system” consisting of a single linear equation, but now inhomogeneous. It can be viewed naturally as an affine space with space of vectors L φ . Let us denote the set ℙ(L)∖ℙ(L φ ) by V φ . Then for every point AV φ there exists a unique vector xW φ for which A=〈x〉.

In this way, we may identify the set V φ with the set W φ , and with the aid of this identification, consider V φ an affine space. By definition, its space of vectors is L φ , and if A and B are two points in V φ , then there exist two vectors x and y for which φ(x)=1 and φ(y)=1 such that A=〈x〉 and B=〈y〉, and then \(\overrightarrow {AB} = {\boldsymbol{y} }-{\boldsymbol{x} }\). Thus the n-dimensional projective space ℙ(L) can be represented as the union of the n-dimensional affine space V φ and the projective hyperplane ℙ(L φ )⊂ℙ(L); see Fig. 9.2. In the sequel, we shall call V φ an affine subset of the space ℙ(L).
Fig. 9.2

Affine subset of a projective space

Let us choose in the space L a basis e 0,…,e n such that φ(e 0)=1 and φ(e i )=0 for all i=1,…,n. Then the vector e 0 is associated with the point O=〈e 0〉 belonging to the affine subset V φ , while all the remaining vectors e 1,…,e n are in L φ , and they are associated with the points 〈e 1〉,…,〈e n 〉 lying in the hyperplane ℙ(L φ ). We have thus constructed in the affine space (V φ ,L φ ) a frame of reference (O;e 1,…,e n ). The coordinates (ξ 1,…,ξ n ) of the point AV φ with respect to this frame of reference are called inhomogeneous coordinates of the point A in our projective space. We wish to emphasize that they are defined only for points in the affine subset V φ . If we return to the definitions, then we see that the inhomogeneous coordinates (ξ 1,…,ξ n ) are obtained from the homogeneous coordinates (9.2) through the formula
$$ \xi_i = \frac{\alpha _i}{\alpha _0},\quad i=1, \ldots, n. $$
(9.6)
It is obvious here that for x from formula (9.1), the function φ that we have chosen assumes the value φ(x)=α 0.

In order to extend the concept of inhomogeneous coordinates to all points of a projective space ℙ(L)=V φ ∪ℙ(L φ ), it remains also to consider the points of the projective hyperplane ℙ(L φ ). For such points it is natural to assign the value α 0=0. Sometimes this is expressed by saying that the inhomogeneous coordinates (ξ 1,…,ξ n ) of the point A∈ℙ(L φ ) assume infinite values, which justifies thinking of ℙ(L φ ) as a set of “points at infinity” (horizon) for the affine subset V φ .

Of course, one could also choose a linear function φ such that φ(e i )=1 for some number i∈{0,…,n}, not necessarily equal to 0, as was done above, and φ(e j )=0 for all ji. We will denote the associated spaces V φ and L φ by V i and L i . In this case, the projective space ℙ(L) can be represented in the analogous form V i ∪ℙ(L i ), that is, as the union of an affine part V i and a hyperplane ℙ(L i ) for the corresponding value i∈{0,…,n}. Sometimes this fact is expressed by saying that in the projective space ℙ(L), one may introduce various affine charts. It is not difficult to see that every point A of a projective space ℙ(L) is “finite” for some value i∈{0,…,n}, that is, it belongs to the subset V i for the corresponding value i. This follows from the fact that by definition, homogeneous coordinates (9.2) of the point A are not simultaneously equal to zero. If α i ≠0 for some i∈{0,…,n}, then A is contained in the associated affine subset V i .

If L′ and L″ are two subspaces of a space L, then it is obvious that
$$ {\mathbb{P}}\bigl({\mathsf{L}}'\bigr) \cap {\mathbb{P}}\bigl({\mathsf{L}}'' \bigr) = {\mathbb{P}}\bigl({\mathsf{L}}' \cap {\mathsf{L}}''\bigr). $$
(9.7)

It is somewhat more complicated to interpret the set ℙ(L′+L″). It is obvious that it does not coincide with ℙ(L′)∪ℙ(L″). For example, if L′ and L″ are two distinct lines in the plane L, then the set ℙ(L′)∪ℙ(L″) consisting of two points is in general not a projective subspace of the space ℙ(L).

To give a geometric interpretation to the sets ℙ(L′+L″), we shall introduce the following notion. Let P=〈e〉 and P′=〈e′〉 be two distinct points of the projective space ℙ(L). Let us set L 1=〈e,e′〉 and consider the one-dimensional projective subspace ℙ(L 1). It obviously contains both points P and P′, and moreover, it is contained in every projective subspace containing the points P and P′. Indeed, if L 2L is a vector subspace such that ℙ(L 2) contains the points P and P′, then this means that L 2 contains the vectors e and e′, which implies that it also contains the entire subspace L 1=〈e,e′〉. Therefore, by the definition of a projective subspace, we have that ℙ(L 1)⊂ℙ(L 2).

Definition 9.5

The one-dimensional projective subspace ℙ(L 1) constructed from two given points PP′ is called the line connecting the points P and P′.

Theorem 9.6

Let Land Lbe two subspaces of a vector space L. Then the union of lines connecting all possible points of ℙ(L′) with all possible points of ℙ(L″) coincides with the projective subspace ℙ(L′+L″).

Proof

We shall denote by Σ the union of lines described in the statement of the theorem. Every such line has the form ℙ(L 1), where L 1=〈e′,e″〉, for vectors e′∈L′ and e″∈L″. Since e′+e″∈L′+L″, it follows from the preceding discussion that every such line ℙ(L 1) belongs to ℙ(L′+L″). Thus we have proved the set inclusion Σ⊂ℙ(L′+L″).

Conversely, suppose now that the point S∈ℙ(L) belongs to the projective subspace ℙ(L′+L″). This means that S=〈e〉, where the vector e is in L′+L″. And this implies that the vector e can be represented in the form e=e′+e″, where e′∈L′ and e″∈L″. This means that S=〈e〉 and the vector e belongs to the plane 〈e′,e″〉, that is, S lies on the line connecting the point 〈e′〉 in ℙ(L′) to the point 〈e″〉 in ℙ(L″). In other words, we have SΣ, and thus the subspace ℙ(L′+L″) is contained in Σ. Taking into account the reverse inclusion proved above, we obtain the required equality Σ=ℙ(L′+L″). □

Definition 9.7

The set ℙ(L′+L″) is called a projective cover of the set ℙ(L′)∪ℙ(L″) and is denoted by
$$ {\mathbb{P}}\bigl({\mathsf{L}}' + {\mathsf{L}}'' \bigr) = \overline{{\mathbb{P}}\bigl({\mathsf{L}}'\bigr) \cup {\mathbb{P}}\bigl( {\mathsf{L}}''\bigr)}. $$
(9.8)

Recalling Theorem 3.41, we obtain the following result.

Theorem 9.8

If ℙ′ and ℙ″ are two projective subspaces of a projective space ℙ(L), then
$$ \dim \bigl({\mathbb{P}}' \cap {\mathbb{P}}''\bigr) + \dim \bigl(\overline{{\mathbb{P}}' \cup {\mathbb{P}}''}\bigr) = \dim {\mathbb{P}}' + \dim {\mathbb{P}}''. $$
(9.9)

Example 9.9

If ℙ′ and ℙ″ are two lines in the projective plane ℙ(L), dimL=3, then dimℙ′=dimℙ″=1 and \(\dim (\overline{{\mathbb{P}}' \cup {\mathbb{P}}''}) \le2\), and from relationship (9.9), we obtain that dim(ℙ′∩ℙ″)≥0, that is, every pair of lines in the projective plane intersect.

The theory of projective spaces exhibits a beautiful symmetry, which goes under the name duality (we have already encountered an analogous phenomenon in the theory of vector spaces; see Sect.  3.7).

Let L be the dual space to L. The projective space ℙ(L ) is called the dual of ℙ(L). Every point of the dual space ℙ(L ) is by definition a line 〈f〉, where f is a linear function on the space L not identically zero. Such a function determines a hyperplane L f L, given by the linear homogeneous equation f(x)=0 in the vector space L, which means that the hyperplane ℙ f is equal to ℙ(L f ) in the projective space ℙ(L).

Let us prove that the correspondence constructed above between points 〈f〉 of the dual space ℙ(L ) and hyperplanes ℙ f of the space ℙ(L) is a bijection. To do so, we must prove that the equations f=0 and α f=0 are equivalent, defining one and the same hyperplane, that is, ℙ f =ℙ α f . As was shown in Sect.  3.7, every hyperplane L′⊂L is determined by a single nonzero linear equation. Two different equations f=0 and f 1=0 can define one and the same hyperplane only if f 1=α f, where α is some nonzero number. Indeed, in the contrary case, the system of the two equations f=0 and f 1=0 has rank 2, and therefore, it defines a subspace L″ of dimension n−2 in L and a subspace ℙ(L″)⊂ℙ(L) of dimension n−3, which is obviously not a hyperplane. Thus the dual space ℙ(L ) can be interpreted as the space of hyperplanes in ℙ(L). This is the simplest example of the fact that certain geometric objects cannot be described by numbers (such as, for example, vector spaces can be described by their dimension), but constitute a set having a geometric character. We shall encounter more complex examples in Chap.  10.

There is also a much more general fact, namely that there is a bijection between m-dimensional projective subspaces of the space ℙ(L) (dimension n) and subspaces of dimension nm−1 of the space ℙ(L ). We shall now describe this correspondence, and the reader will easily verify that for m=n−1, this coincides with the above-described correspondence between hyperplanes in ℙ(L) and points in ℙ(L ).

Let L′⊂L be a subspace of dimension m+1, so that dimℙ(L′)=m. Let us consider in the dual space L , the annihilator (L′) a of the subspace L′. Let us recall that the annihilator is the subspace (L′) a L consisting of all linear functions fL such that f(x)=0 for all vectors xL′. As we established in Sect.  3.7 (formula ( 3.54)), the dimension of the annihilator is equal to
$$ \dim\bigl({\mathsf{L}}'\bigr)^a = \dim {\mathsf{L}}- \dim {\mathsf{L}}' = n - m. $$
(9.10)
The projective subspace ℙ((L′) a )⊂ℙ(L ) is called the dual to the subspace ℙ(L′)⊂ℙ(L). By (9.10), its dimension is nm−1. What we have here is a variant of a concept that is well known to us. If a nonsingular symmetric bilinear form (x,y) is defined on the space L, then we can identify (L′) a with the orthogonal complement to L′, which was denoted by (L′); see p. 198. If we write the bilinear form (x,y) in some orthonormal basis of the space L, then it takes the form \(\sum_{i=0}^{n} x_{i} y_{i}\), and the point with coordinates (y 0,y 1,…,y n ) will correspond to the hyperplane defined by the equation
$$\sum_{i=0}^n x_i y_i = 0, $$
in which y 0,…,y n are taken as fixed, and x 0,…,x n are variables.

The assertions we have proved together with the duality principle established in Sect.  3.7 leads automatically to the following result, called the principle of projective duality.

Proposition 9.10

(Principle of projective duality)

If a theorem is proved for all projective spaces of a given finite dimension n over a given field \({\mathbb{K}}\) in a formulation that uses only the concepts of projective subspace, dimension, projective cover, and intersection, then for all such spaces, one has also the dual theorem obtained from the original one by the following substitutions:
$$\begin{array}{r@{\quad}||@{\quad}l} \textit{dimension $m$} & \textit{dimension $n-m-1$} \\ \textit{intersection ${\mathbb{P}}_{1} \cap {\mathbb{P}}_{2}$} & \textit{projective cover $\overline{{\mathbb{P}}_{1} \cup {\mathbb{P}}_{2}}$ } \\ \textit{projective cover $\overline{{\mathbb{P}}_{1} \cup {\mathbb{P}}_{2}}$} & \textit{intersection ${\mathbb{P}}_{1} \cap {\mathbb{P}}_{2}$.} \end{array} $$

For example, the assertion “through two distinct points of the projective plane there passes one line” has as its dual assertion “every pair of distinct lines in the projective plane intersect in one point.”

One may try to extend this principle in such a way that it will cover not only projective spaces, but also the projective algebraic varieties described by equation (9.5). However, in this regard there appear some new difficulties, which we shall only mention here without going into detail.

Assume, for example, that a projective algebraic variety X⊂ℙ(L) is given by the single equation
$$F (x_0, x_1,\ldots, x_n) = 0, $$
where F is a homogeneous polynomial. To every point AX there corresponds a hyperplane given by the equation
$$ \sum_{i=0}^n \frac{\partial F}{\partial x_i} (A) x_i = 0, $$
(9.11)
called the tangent hyperplane to X at the point A (this notion will be discussed later in greater detail). By the above considerations, we can assign to this hyperplane the point B of the dual space ℙ(L ).

It is natural to suppose that as A runs through all points X, then the point B also runs through some projective algebraic variety in the space ℙ(L), called the dual to the original variety X. This is indeed the case, except for certain unpleasant exceptions. Namely, for some point A, it could be the case that all partial derivatives \(\frac{\partial F}{\partial x_{i}} (A)\) are equal to 0 for i=0,1,…,n, and equation (9.11) takes the form of the identity 0=0. Such points are called singular points of the projective algebraic variety X. In this case, we do not obtain any hyperplane, and therefore, we cannot use the indicated method to assign to the point A a given point of the space ℙ(L ). It is possible to prove that singular points are in some sense exceptional. Moreover, many very interesting varieties have no singular points at all, so that for them, the dual variety exists. But then in the dual variety, there appear singular points, so that the beautiful symmetry nevertheless disappears. Overcoming all these difficulties is the task of algebraic geometry. We shall not go deeply into this, and we have mentioned it only in connection to the fact that in Chap.  11, devoted to quadrics, we shall consider precisely the special case in which these difficulties do not appear.

9.2 Projective Transformations

Let Open image in new window be a linear transformation of a vector space L into itself. It is natural to entertain the idea of extending it to the projective space ℙ(L). It would seem to be something easy to do: one has only to associate with each point P∈ℙ(L) corresponding to the line 〈e〉 in L, the line Open image in new window , which is some point of the projective space ℙ(L). However, here we encounter the following difficulty: If Open image in new window , then we cannot construct the line Open image in new window , since all vectors proportional to Open image in new window are the null vector. Thus the transformation that we wish to construct is not defined in general for all points of the projective space ℙ(L). However, if we wished to define it for all points, then we must require that the kernel of the transformation Open image in new window be (0). As we know, this condition is equivalent to the transformation Open image in new window being nonsingular. Thus to all nonsingular transformations Open image in new window of the space L into itself (and only these) there correspond mappings of the projective space ℙ(L) into itself. We shall denote them by Open image in new window .

We have seen that a nonsingular transformation Open image in new window defines a bijective mapping of the space L into itself. Let us prove that in this case, the corresponding mapping Open image in new window is also a bijection. First, let us verify that its image coincides with all ℙ(L). Let P be a point of the space ℙ(L). It corresponds to some line 〈e〉 in L. Since the transformation Open image in new window is nonsingular, it follows that Open image in new window for some vector e′∈L, and moreover, e′≠0, since e0. If P′ is a point of the space ℙ(L) corresponding to the line 〈e′〉, then Open image in new window . It remains to show that Open image in new window cannot map two distinct points into one. Let us suppose that PP′ and where the points P, P′, and \({\overline{P}}\) correspond to the lines 〈e〉, 〈e′〉, and \(\langle {\overline{{\boldsymbol{e} }}}\rangle\) respectively.

The condition PP′ is equivalent to the vectors e and e′ being linearly independent, while from equality (9.12) it follows that Open image in new window , which means that the vectors Open image in new window and Open image in new window are linearly dependent. But if Open image in new window , where α≠0 or β≠0, then Open image in new window , and since the transformation Open image in new window is nonsingular, we have α e+β e′≠0, which contradicts the condition PP′. Thus we have proved that the mapping Open image in new window is a bijection. Consequently, the inverse mapping Open image in new window is also defined.

Definition 9.11

A mapping Open image in new window of the projective space ℙ(L) corresponding to the nonsingular transformation Open image in new window of a vector space L into itself is called a projective transformation of the space ℙ(L).

Theorem 9.12

We have the following assertions:
  1. (1)

    Open image in new window if and only if Open image in new window , where λ is some nonzero scalar.

     
  2. (2)

    If Open image in new window and Open image in new window are two nonsingular transformations of a vector space L, then Open image in new window .

     
  3. (3)

    If Open image in new window is a nonsingular transformation, then Open image in new window .

     
  4. (4)

    A projective transformation Open image in new window carries every projective subspace of the space ℙ(L) into a subspace of the same dimension.

     

Proof

All the assertions of the proof follow directly from the definitions.

(1) If Open image in new window , then it is obvious that Open image in new window and Open image in new window map lines of the vector space L in exactly the same way, that is, Open image in new window . Now suppose, conversely, that Open image in new window for an arbitrary point A∈ℙ(L). If the point A corresponds to the line 〈e〉, then we have Open image in new window , that is, where λ is some scalar. However, in theory, the number λ in relationship (9.13) could have had its own value for each vector e. Let us consider two linearly independent vectors x and y and for the vectors x, y, and x+y, let us write down condition (9.13): In view of the linearity of Open image in new window and Open image in new window , we have Having substituted expressions (9.15) into the third equality of (9.14), we then subtract from it the first and second inequalities. We then obtain Since the transformation Open image in new window is nonsingular (by the definition of a projective transformation), it follows that (νλ)x+(νμ)y=0, and in view of the linear independence of the vectors x and y, it follows from this that λ=ν and μ=ν, that is, all the scalars λ,μ,ν in (9.14) are the same, and therefore the scalar λ in relationship (9.13) is one and the same for all vectors eL.

(2) We must prove that for every point P of the corresponding line 〈e〉, we have the equality Open image in new window , and this, by the definition of a projective transformation, follows from the fact that Open image in new window . The last equality follows from the definition of the product of linear transformations.

(3) By what we have proven, we have the equality Open image in new window . It is obvious that Open image in new window is the identity transformation of the space ℙ(L) into itself. From this, it follows that Open image in new window .

(4) Finally, let L′ be an m-dimensional subspace of the vector space L and let ℙ(L′) be the associated (m−1)-dimensional projective subspace. The mapping Open image in new window takes ℙ(L′) into a collection of points of the form Open image in new window , where P′=〈(e′)〉 runs through all points of ℙ(L′). This holds because e′ runs through all vectors of the space L′. Let us prove that here, all vectors Open image in new window coincide with the nonnull vectors of some vector subspace L″ having the same dimension as L′. This will give us the required assertion.

In the subspace L′, let us choose a basis e 1,…,e m . Then every vector e′∈L′ can be represented in the form
$${\boldsymbol{e} }' = \alpha _1{\boldsymbol{e} }_1 + \cdots+ \alpha _m{\boldsymbol{e} }_m, $$
while the condition e′≠0 is equivalent to not all the coefficients α i being equal to zero. From this, we obtain The vectors Open image in new window are linearly independent, since the transformation Open image in new window is nonsingular. Let us consider the m-dimensional subspace Open image in new window . From the relationship (9.16), it follows that the transformation Open image in new window takes the points of the subspace ℙ(L′) precisely into the points of the subspace ℙ(L″). From the equality dimL′=dimL″=m, we obtain dimℙ(L′)=dimℙ(L″)=m−1. □
By analogy with linear and affine transformations, there is a hope that we can describe a projective transformation unambiguously by how it maps a certain number of “sufficiently independent” points. As a first attempt, we may consider the points P i =〈e i 〉 for i=0,1,…,n, where e 0,e 1,…,e n is a basis of the space L. But this path does not lead to our goal, for there exist too many distinct transformations taking each point P i into itself. Indeed, such are all the transformations of the form Open image in new window if Open image in new window with arbitrary λ i ≠0, that is, in other words, if Open image in new window has, in the basis e 0,e 1,…,e n , the matrix In this case, Open image in new window for all i=0,1,…,n. However, the image of an arbitrary vector
$${\boldsymbol{e} }= \alpha _0{\boldsymbol{e} }_0 + \alpha _1{\boldsymbol{e} }_1 + \cdots+ \alpha _n{\boldsymbol{e} }_n $$
is equal to and this vector is already not proportional to e unless all λ i are identical. Thus even knowing how the transformation Open image in new window maps the points P 0,P 1,…,P n , we are not yet able to determine it uniquely. But it turns out that the addition of one more point (under some weak assumptions) describes the transformation uniquely. For this, we need to introduce a new concept.

Definition 9.13

In the n-dimensional projective space ℙ(L), n+2 points
$$ P_0, P_1, \ldots, P_n, P_{n+1} $$
(9.17)
are said to be independent if no n+1 of them lie in a subspace of dimension less than n.

For example, four points in the projective plane are independent if no three of them are collinear.

Let us explore what the condition of independence means if to the point P i there corresponds the line 〈e i 〉, i=0,…,n+1. Since by definition, the points P 0,P 1,…,P n do not lie in a subspace of dimension less than n, it follows that the vectors e 0,e 1,…,e n do not lie in a subspace of dimension less than n+1, that is, they are linearly independent, and this means that they constitute a basis of the space L. Thus the vector e n+1 is a linear combination of these vectors:
$$ {\boldsymbol{e} }_{n+1} = \alpha _0 {\boldsymbol{e} }_0 + \alpha _1 {\boldsymbol{e} }_1 + \cdots+ \alpha _n {\boldsymbol{e} }_n. $$
(9.18)
If some scalar α i is equal to 0, then from (9.18), it follows that the vector e n+1 lies in the subspace \({\mathsf{L}}' = \langle {\boldsymbol{e} }_{0}, \ldots, \breve{{\boldsymbol{e} }}_{i},\ldots, {\boldsymbol{e} }_{n} \rangle\), where the sign \(\breve{\phantom{o}}\) indicates the omission of the corresponding vector. Consequently, the vectors \({\boldsymbol{e} }_{0},\ldots, \breve{{\boldsymbol{e} }}_{i}, \ldots, {\boldsymbol{e} }_{n}, {\boldsymbol{e} }_{n+1}\) lie in a subspace L′ whose dimension does not exceed n. But this means that the points \(P_{0},\ldots, \breve{P}_{i},\ldots, P_{n}, P_{n+1}\) lie in the projective space ℙ(L′), and moreover, dimℙ(L′)≤n−1, that is, they are dependent.
Let us show that for the independence of points (9.17), it suffices that in the decomposition (9.18), all coefficients α i be nonzero. Let the vectors e 0,e 1,…,e n form a basis of the space L, while the vector e n+1 is a linear combination (9.18) of them such that all the α i are nonzero. Let us show that then, the points (9.17) are independent. If this were not the case, then some n+1 vectors from among e 0,e 1,…,e n+1 of the space L would lie in a subspace of dimension not greater than n. This cannot be the vectors e 0,e 1,…,e n , since by assumption, they constitute a basis of L. So let it be the vectors \({\boldsymbol{e} }_{0}, \ldots, \breve{{\boldsymbol{e} }}_{i},\ldots, {\boldsymbol{e} }_{n}, {\boldsymbol{e} }_{n+1}\) for some i<n+1, and their linear dependence is expressed by the equality
$$\lambda _0 {\boldsymbol{e} }_0 + \cdots+ \lambda _{i-1} {\boldsymbol{e} }_{i-1} + \lambda _{i+1} {\boldsymbol{e} }_{i+1} + \cdots+ \lambda _{n+1} {\boldsymbol{e} }_{n+1} = {\boldsymbol{0} }, $$
where λ n+1≠0, since the vectors e 0,e 1,…,e n are linearly independent. From this, it follows that the vector e n+1 is a linear combination of the vectors \({\boldsymbol{e} }_{0},\ldots, \breve{{\boldsymbol{e} }}_{i},\ldots, {\boldsymbol{e} }_{n}\). But this contradicts the condition that in the expression (9.18), all the α i are nonzero, since the vectors e 0,e 1,…,e n form a basis of the space L, and the decomposition (9.18) for an arbitrary vector e n+1 uniquely determines its coordinates α i .

Thus, n+2 independent points (9.17) are always obtained from n+1 points P i =〈e i 〉 whose corresponding vectors e i form a basis of the space L by the addition of one more point P=〈e〉 for which the vector e is a linear combination of the vectors e i with all nonzero coefficients.

We can now formulate our main result.

Theorem 9.14

Let
$$ P_0, P_1, \ldots, P_n, P_{n+1};\qquad P'_0, P'_1, \ldots, P'_n, P'_{n+1} $$
(9.19)
be two systems of independent points of the projective space ℙ(L) of dimension n. Then there exists a projective transformation taking the point P i to \(P'_{i}\) for all i=0,1,…,n+1, and moreover, it is unique.

Proof

We shall use the interpretation of the property of independence of points obtained above. Let points P i correspond to the lines 〈e i 〉, and let the points \(P'_{i}\) correspond to the lines \(\langle {\boldsymbol{e} }'_{i} \rangle\). We may assume that the vectors e 0,…,e n and the vectors \({\boldsymbol{e} }'_{0},\ldots, {\boldsymbol{e} }'_{n}\) are bases of an (n+1)-dimensional subspace of L. Then as we know, for every collection of nonzero scalars λ 0,…,λ n , there exists (and it is unique) a nonsingular linear transformation Open image in new window mapping e i to \(\lambda _{i} {\boldsymbol{e} }_{i}'\) for all i=0,1,…,n.

By definition, for such a transformation Open image in new window , we have Open image in new window for all i=0,1,…,n. Since dimL=n+1, we have the relationships
$$ {\boldsymbol{e} }_{n+1} = \alpha _0 {\boldsymbol{e} }_0 + \alpha _1 {\boldsymbol{e} }_1 + \cdots+ \alpha _n {\boldsymbol{e} }_n,\qquad {\boldsymbol{e} }'_{n+1} = \alpha '_0 {\boldsymbol{e} }'_0 + \alpha '_1 {\boldsymbol{e} }'_1 + \cdots+ \alpha '_n {\boldsymbol{e} }'_n. $$
(9.20)
From the condition of independence of both collections of points (9.19), it follows that in the representations (9.20), all the coefficients α i and \(\alpha '_{i}\) are nonzero. Applying the transformation Open image in new window to both sides of the first relationship in (9.20), taking into account the equalities Open image in new window , we obtain After setting the scalars λ i equal to \(\alpha '_{i} \alpha _{i}^{-1}\) for all i=0,1,…,n and substituting them into the relationship (9.21), taking into account the second equality of formula (9.20), we obtain that Open image in new window , that is, Open image in new window .

The uniqueness of the projective transformation Open image in new window that we have obtained follows from its construction. □

For example, for n=1, the space ℙ(L) is the projective line. Three points P 0,P 1,P 2 are independent if and only if they are distinct. We see that any three distinct points on the projective line can be mapped into three other distinct points by a unique projective transformation.

Let us now consider how a projective transformation can be given in coordinate form. In homogeneous coordinates (9.2), the stipulation of a projective transformation Open image in new window in fact coincides with that of a nonsingular linear transformation Open image in new window , and indeed, the homogeneous coordinates of a point A∈ℙ(L) coincide with the coordinates of the vector x from (9.1) that determines the line 〈x〉 corresponding to the point A. Using formula ( 3.25), we obtain for the homogeneous coordinates β i of the point Open image in new window the following expressions in homogeneous coordinates α i of the point A:
$$ \begin{cases} \beta_0 = a_{00} \alpha _0 + a_{01} \alpha _1 + a_{02} \alpha _2 + \cdots+ a_{0n} \alpha _n, \cr \beta_1 = a_{10} \alpha _0 + a_{11} \alpha _1 + a_{12} \alpha _2 + \cdots+ a_{1n} \alpha _n, \cr \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \ldots\ldots \ldots \cr \beta_n = a_{n0} \alpha _0 + a_{n1} \alpha _1 + a_{n2} \alpha _2 + \cdots+ a_{nn} \alpha _n. \end{cases} $$
(9.22)
Here we must recall that the homogeneous coordinates are defined only up to a common factor, and both collections (α 0:α 1:⋯:α n ) and (β 0:β 1:⋯:β n ) are not identically zero. Clearly, in multiplying all the α i by the common factor λ, all β i in formula (9.22) are also multiplied by this factor. All the β i cannot become zero if all the α i cannot become zero (this follows from the fact that the transformation Open image in new window is nonsingular). The condition of nonsingularity of the transformation Open image in new window is expressed as the determinant of its matrix being nonzero:

Another way of writing a projective transformation is in inhomogeneous coordinates of affine spaces. Let us recall that a projective space ℙ(L) contains affine subsets V i , i=0,1,…,n, and it can be obtained from any of the V i by the addition of the corresponding projective hyperplane ℙ(L i ) consisting of “points at infinity,” that is, in the form ℙ(L)=V i ∪ℙ(L i ). For simplicity of notation, we shall limit ourselves to the case i=0; all the remaining V i are considered analogously.

To an affine subset V 0 there corresponds (as its subspace of vectors) the vector subspace L 0L defined by the condition α 0=0. For assigning coordinates in the affine space V 0, we must fix in the space some frame of reference consisting of a point OV 0 and a basis in the space L 0. In the (n+1)-dimensional space L, let us choose a basis e 0,e 1,…,e n . For the point OV 0, let us choose the point associated with the line 〈e 0〉, and for the basis in L 0, let us take the vectors e 1,…,e n .

Let us consider a point AV 0, which in the basis e 0,e 1,…,e n of the space L has homogeneous coordinates (α 0:α 1:⋯:α n ), and repeating the arguments that we used in deriving formulas (9.6), let us find its coordinates with respect to the frame of reference (O;e 1,…,e n ) constructed in the manner outlined above. The point A corresponds to the line 〈e〉, where
$$ {\boldsymbol{e} }= \alpha _0 {\boldsymbol{e} }_0 + \alpha _1 {\boldsymbol{e} }_1 + \cdots+ \alpha _n {\boldsymbol{e} }_n, $$
(9.23)
and moreover, α 0≠0, since AV 0. By assumption, we must choose from both lines 〈e 0〉 and 〈e〉, vectors x and y with coordinate α 0=1 and examine the coordinates of the vector yx with respect to the basis e 1,…,e n . It is obvious that x=e 0, and in view of (9.23), we have
$${\boldsymbol{y} }= {\boldsymbol{e} }_0 + \alpha _1 \alpha _0^{-1} {\boldsymbol{e} }_1 + \cdots+ \alpha _n \alpha _0^{-1} {\boldsymbol{e} }_n. $$
Thus the vector yx has, in the basis e 1,…,e n , coordinates
$$ x_1 = \frac{\alpha _1}{\alpha _0},\qquad \ldots, \qquad x_n = \frac{\alpha _n}{\alpha _0}. $$
We shall now consider a nonsingular linear transformation Open image in new window and the associated projective transformation Open image in new window , given by formulas (9.22). It takes a point A with homogeneous coordinates α i to a point B with homogeneous coordinates β i . In order to obtain in both cases inhomogeneous coordinates in the subset V 0, it is necessary, by formula (9.6), to divide all the coordinates by the coordinate with index 0. Thus we obtain that a point with inhomogeneous coordinates \(x_{i} = \frac{\alpha _{i}}{\alpha _{0}}\) is mapped to the point with inhomogeneous coordinates \(y_{i} = \frac{\beta _{i}}{\beta _{0}}\), that is, taking into account (9.22), we obtain the expressions
$$ y_i = \frac{a_{i0} + a_{i1} x_1 + \cdots+ a_{in} x_n}{a_{00} + a_{01} x_1 + \cdots+ a_{0n} x_n},\quad i=1,\ldots, n. $$
(9.24)
In other words, in inhomogeneous coordinates, a projective transformation can be written in terms of the linear fractional formulas (9.24) with a common denominator for all y i . It is not defined at points where this denominator becomes zero, and these are the “points at infinity,” that is, points of the projective hyperplane ℙ(L 0) with equation β 0=0.
Let us consider projective transformations mapping “points at infinity” to “points at infinity” and consequently, “finite points” to “finite points.” This means that the equality β 0=0 is possible only for α 0=0, that is, taking into account formula (9.22), the equality
$$a_{00} \alpha _0 + a_{01} \alpha _1 + a_{02} \alpha _2 + \cdots+ a_{0n} \alpha _n = 0 $$
is possible only for α 0=0. Obviously, this latter condition is equivalent to the conditions a 0i =0 for all i=1,…,n. In this case, the common denominator of the linear fractional formulas (9.24) reduces to the constant a 00. From the nonsingularity of the transformation Open image in new window , it follows that a 00≠0, and we can divide the numerators in equalities (9.24) by a 00. We then obtain precisely the formulas for affine transformations ( 8.17). Thus affine transformations are special cases of projective transformations, namely, those that take the set of “points at infinity” to itself.

Example 9.15

In the case dimℙ(L)=1, the projective line ℙ(L) has a single inhomogeneous coordinate, and formula (9.24) assumes the form
$$y = \frac{a + bx}{c + dx},\quad ad - bc \neq0. $$
Transformations of the “finite part” of the projective line (x≠∞) are affine and have the form y=α+βx, where β≠0.

9.3 The Cross Ratio

Let us recall that in Sect.  8.2, we defined the affine ratio (A,B,C) among three collinear points of an affine space, and then, in Sect.  8.3, it was proved (Theorem 8.28) that the affine ratio (A,B,C) among three collinear points does not change under a nonsingular affine transformation. In projective spaces, the notion of a relationship among three collinear points cannot be given a natural analogue. This is the result of the following assertion.

Theorem 9.16

Let A 1,B 1,C 1 and A 2,B 2,C 2 be two triples of points in a projective space satisfying the following conditions:
  1. (a)

    The three points in each triple are distinct.

     
  2. (b)

    The points in each triple are collinear (one line for each triple).

     
Then there exists a projective transformation taking one triple into the other.

Proof

Let us denote the line on which the three points A i ,B i ,C i lie by l i , where i=1,2. Points A 1,B 1,C 1 are independent on l 1, and the points A 2,B 2,C 2 are independent on l 2. Let the point A i be determined by the line 〈e i 〉, point B i by the line 〈f i 〉, point C i by the line 〈g i 〉, and line l i by the two-dimensional space L i , i=1,2. They are all contained in the space L that determines our projective space. Repeating the proof of Theorem 9.14 verbatim, we shall construct an isomorphism Open image in new window taking the lines 〈e 1〉,〈f 1〉,〈g 1〉 to the lines 〈e 2〉,〈f 2〉,〈g 2〉 respectively. Let us represent the space L in the form of two decompositions:
$${\mathsf{L}}= {\mathsf{L}}_1 \oplus {\mathsf{L}}_1',\qquad {\mathsf{L}}= {\mathsf{L}}_2 \oplus {\mathsf{L}}_2'. $$
It is obvious that \(\dim {\mathsf{L}}_{1}' = \dim {\mathsf{L}}_{2}' = \dim {\mathsf{L}}- 2\), and therefore, the spaces \({\mathsf{L}}_{1}'\) and \({\mathsf{L}}_{2}'\) are isomorphic. We shall choose some isomorphism Open image in new window and define a transformation Open image in new window as Open image in new window on L 1 and as Open image in new window on \({\mathsf{L}}_{1}'\), while for arbitrary vectors xL, we shall use the decomposition \({\boldsymbol{x} }= {\boldsymbol{x} }_{1} + {\boldsymbol{x} }_{1}'\), x 1L 1, \({\boldsymbol{x} }_{1}' \in {\mathsf{L}}_{1}'\), to define Open image in new window . It is easy to see that Open image in new window is a nonsingular linear transformation, and the projective transformation Open image in new window takes the triple of points A 1,B 1,C 1 to A 2,B 2,C 2. □

Analogously to the fact that for a triple of collinear points A,B,C of an affine space, there is an associated number (A,B,C) that is unchanged under every nonsingular affine transformation, in a projective space we can associate with a quadruple of collinear points A 1,A 2,A 3,A 4 a number that does not change under projective transformations. This number is denoted by (A 1,A 2,A 3,A 4) and is called the cross or anharmonic ratio of these four points. We now turn to its definition.

Let us consider first the projective line l=ℙ(L), where dimL=2. Four arbitrary points A 1,A 2,A 3,A 4 on l correspond to four lines 〈a 1〉, 〈a 2〉, 〈a 3〉, 〈a 4〉 lying in the plane L. In the plane L, let us choose a basis e 1,e 2 and consider the decomposition of the vectors a i in this basis: a i =x i e 1+y i e 2, i=1,…,4. The coordinates of the vectors a 1,…,a 4 can be written as the columns of the matrix Consider the following question: how do the minors of order 2 of the matrix M change under a transition to another basis \({\boldsymbol{e} }_{1}', {\boldsymbol{e} }_{2}'\) of the plane L? Let us denote by [α i ] and \([{\boldsymbol{\alpha}}_{i}']\) the columns of the coordinates of the vector a i in the bases (e 1,e 2) and \(({\boldsymbol{e} }'_{1}, {\boldsymbol{e} }'_{2})\) respectively:
By formula ( 3.36) for changing coordinates, they are related by [α]=C[α′], where C is the transition matrix from the basis \({\boldsymbol{e} }_{1}',{\boldsymbol{e} }_{2}'\) to the basis e 1,e 2. From this it follows that for any choice of indices i and j, and by the theorem on multiplication of determinants, we obtain where |C|≠0. This means that for any three indices i,j,k, the relation is unaltered under a change of basis (we assume now that both determinants, in the numerator and denominator, are nonzero). Thus relationship (9.25) determines a number (a i ,a j ,a k ) depending on the three vectors a i ,a j ,a k but not on the choice of basis in L.

However, this is not yet what we promised: the points A i indeed determine the lines 〈a i 〉, but not the vectors a i . We know that the vector \({\boldsymbol{a} }_{i}'\) determines the same line as the vector a i if and only if \({\boldsymbol{a} }_{i}' = \lambda _{i} {\boldsymbol{a} }_{i}\), λ i ≠0. Therefore, if in expression (9.25) we replace the coordinates of the vectors a i ,a j ,a k with the coordinates of the proportional vectors \({\boldsymbol{a} }_{i}', {\boldsymbol{a} }_{j}', {\boldsymbol{a} }_{k}'\), then its numerator will be multiplied by λ i λ j , while its denominator will be multiplied by λ i λ k , with the result that the entire expression (9.25) will be multiplied by the number \(\lambda _{j} \lambda _{k}^{-1}\), which means that it will change.

However, if we now consider the expression then as our previous reasoning demonstrates, it will depend neither on the choice of basis of the plane L nor on the choice of vectors a i on the lines 〈a i 〉, but will be determined only by the four points A 1,A 2,A 3,A 4 on the projective line l. It is expression (9.26) that is called the cross ratio of these four points.
Let us write the expression for DV(A 1,A 2,A 3,A 4) assuming that homogeneous coordinates have been introduced on the projective line l. Let us begin with the formula written in the homogeneous coordinates (x:y). We shall now consider the points A i “finite” points of l, that is, we assume that y i ≠0 for all i=1,…,4, and we set t i =x i /y i ; these will be the coordinates of the point A i in the “affine part” of the projective line l. Then we obtain Substituting these expressions into formula (9.26), we see that all the y i cancel, and as a result, we obtain the expression
$$ \mathrm {DV}(A_1, A_2, A_3, A_4) = \frac{(t_1-t_3) (t_2-t_4)}{(t_1-t_4) (t_2-t_3)}. $$
(9.27)
If we assume that all four points A 1,A 2,A 3,A 4 lie in the “finite part” of the plane, then this means in particular that they belong to the affine part of the projective line l and have finite coordinates t 1,t 2,t 3,t 4 on the projective line l. Taking into account formula ( 8.8) for the affine ratio of three points, we observe that then the expression for the cross ratio takes the form
$$ \mathrm {DV}(A_1, A_2, A_3, A_4) = \frac{(A_3, A_2, A_1)}{(A_4, A_2, A_1)}. $$
(9.28)
Equality (9.28) shows the connection between the cross ratio and the affine ratio introduced in Sect.  8.2.

We have determined the cross ratio for four distinct points. In the case in which two of these points coincide, it is possible to define this ratio under some natural conventions (as we did for the affine ratio), setting the cross ratio in some cases equal to ∞. However, the cross ratio remains undefined if three of the four points coincide.

The above reasoning almost contains the proof of the following fundamental property of the cross ratio.

Theorem 9.17

The cross ratio of four collinear points in a projective space does not change under a projective transformation of the space.

Proof

Let A 1,A 2,A 3,A 4 be four points lying on the line l′ in some projective space ℙ(L). They correspond to the four lines 〈a 1〉,〈a 2〉,〈a 3〉,〈a 4〉 of the space L, and the line l′ corresponds to the two-dimensional subspace L′⊂L. Let Open image in new window be a nonsingular transformation of the space L, and Open image in new window the corresponding projective transformation of the space ℙ(L). Then by Theorem 9.12, φ(l′)=l″ is another line in the projective space ℙ(L); it corresponds to the subspace Open image in new window and contains the four points φ(A 1),φ(A 2),φ(A 3),φ(A 4). Let the vectors e 1,e 2 form a basis of L′ and write the vectors a i as a i =x i e 1+y i e 2, i=1,…,4. Then the cross ratio DV(A 1,A 2,A 3,A 4) is defined by the formula (9.26).

On the other hand, Open image in new window , and if we use the bases Open image in new window and Open image in new window of the subspace Open image in new window , then the cross ratio
$$\mathrm {DV}\bigl(\varphi (A_1), \varphi (A_2), \varphi (A_3), \varphi (A_4)\bigr) $$
is defined by the same formula (9.26), since the coordinates of the vectors Open image in new window in the basis f 1,f 2 coincide with the coordinates of the vectors a i in the basis e 1,e 2. But as we have already verified, the cross ratio depends neither on the choice of basis nor on the choice of vectors a i that determine the lines 〈a i 〉. Therefore, it follows that
$$\mathrm {DV}(A_1, A_2, A_3, A_4) = \mathrm {DV}\bigl(\varphi (A_1), \varphi (A_2), \varphi (A_3), \varphi (A_4)\bigr). $$
 □

Example 9.18

In a projective space Π, let us consider two lines l 1 and l 2 and a point O lying on neither of the lines. Let us connect an arbitrary point Al 1 to the point O of the line l A ; see Fig. 9.3. We shall denote the point of intersection of the lines l A and l 2 by A′. The mapping of the line l 1 into l 2 that to each point Al 1 assigns the point A′∈l 2 is called a perspective mapping.
Fig. 9.3

Perspective mapping

Let us prove that there exists a projective transformation of the plane Π defining a perspective correspondence between the lines l 1 and l 2. To this end, let us denote by l 0 the line joining the point O and the point P=l 1l 2, and let us consider the set V=Πl 0. In other words, we shall consider l 0 a “line at infinity” and the points of V will be considered “finite points” of the projective plane. Then on V, the perspective correspondence will be given by a bundle of parallel lines, since these lines in the “finite part” do not intersect; see Fig. 9.4.
Fig. 9.4

A bundle of parallel lines

More precisely, this bundle defines a mapping of the “finite parts” \(l_{1}'\) and \(l_{2}'\) of the lines l 1 and l 2. From this it follows that in the affine plane V, the lines \(l_{1}'\) and \(l_{2}'\) are parallel, and the perspective correspondence between them is defined by an arbitrary translation Open image in new window by the vector \({\boldsymbol{a} }= \overrightarrow {AA'}\), where A is an arbitrary point on the line \(l_{1}'\), and A′ is the point on the line \(l_{2}'\) corresponding to it under the perspective correspondence. As we saw above, every nonsingular affine transformation of an affine plane V is a projective mapping for Π, and this is even more obviously the case for a translation. This means that a perspective correspondence is defined by some projective transformation of the plane Π. Therefore, from Theorem 9.17, we deduce the following result.

Theorem 9.19

The cross ratio of four collinear points is preserved under a perspective correspondence.

9.4 Topological Properties of Projective Spaces*

The previous discussion in this chapter was related to a projective space ℙ(L), where L was a finite-dimensional vector space over an arbitrary field \({\mathbb{K}}\). If our interest is in a particular field (for example, ℝ or ℂ), then all the assertions we have proved remain valid, since we used only general algebraic notions (which derive from the definition of a field), and nowhere did we use, for example, properties of inequality or absolute value. Now let us say a few words about properties related to the notion of convergence, or as they are called, topological properties, of projective spaces. It makes sense to talk about them if, for example, L is a real or complex vector space, that is, the field in question is \({\mathbb{K}}= {\mathbb{R}}\) or ℂ.

Let us begin by formulating the notion of convergence of a sequence of vectors x 1,x 2,…,x k ,… in a space L to a vector x of the same space. Let us choose in L an arbitrary basis e 0,e 1,…,e n and let us write the vectors x k and x in this basis:
$${\boldsymbol{x} }_k = \alpha _{k0} {\boldsymbol{e} }_0 + \alpha _{k1} {\boldsymbol{e} }_1 + \cdots+ \alpha _{kn} {\boldsymbol{e} }_n,\qquad {\boldsymbol{x} }= \beta _{0} {\boldsymbol{e} }_0 + \beta _{1} {\boldsymbol{e} }_1 + \cdots+ \beta _{n} {\boldsymbol{e} }_n. $$
We shall say that the sequence of vectors x 1,x 2,…,x k ,… converges to the vector x if the sequence of numbers
$$ \alpha _{1i}, \alpha _{2i},\ldots, \alpha _{ki},\ldots $$
(9.29)
for fixed i converges to the number β i as k→∞ for each index i=0,1,…,n (in speaking about complex vector spaces, we assume that the reader is familiar with the notion of convergence of a sequence of complex numbers). The vector x is called, in this case, the limit of the sequence. From the formulas for changing coordinates given in Sect.  3.4, it is easy to derive that the property of convergence does not depend on the basis in L. We shall write this convergence as x k x as k→∞.
Let us move now from vectors to points of a projective space. In both cases that we are considering (\({\mathbb{K}}= {\mathbb{R}}\) or ℂ), there is a useful method of normalizing the homogeneous coordinates (x 0:x 1:⋯:x n ) defined, generally speaking, only up to multiplication by a common factor λ≠0. Since by definition, the equality x i =0 for all i=0,1,…,n is impossible, we may choose a coordinate x r for which |x r | (the absolute value in ℝ or ℂ, respectively) assumes the greatest value, and setting λ=|x r |, make the substitution y i =λ −1 x i for all i=0,1,…,n. Then, obviously,
$$(x_0 : x_1 : \cdots: x_n) = (y_0 : y_1 : \cdots: y_n), $$
and moreover, |y r |=1 and |y i |≤1 for all i=0,1,…,n.

Definition 9.20

A sequence of points P 1,P 2,…,P k ,… converges to the point P if on every line 〈e k 〉 that determines the point P k , and on the line 〈e〉 determining the point P, it is possible to find nonnull vectors x k and x such that x k x as k→∞. This is written as P k P as k→∞. The point P is called the limit of the sequence P 1,P 2,…,P k ,… .

We note that by assumption, 〈e k 〉=〈x k 〉 and 〈e〉=〈x〉.

Theorem 9.21

It is possible to choose from an arbitrary infinite sequence of points of a projective space a subsequence that converges to a point of the space.

Proof

As we have seen, every point P of a projective space can be represented in the form P=〈y〉, where the vector y has coordinates (y 0,y 1,…,y n ), and moreover, max|y i |=1.

It is proved in a course in real analysis that every bounded sequence of real numbers satisfies the assertion of Theorem 9.21. It is also very easy to prove the statement for a sequence of complex numbers. To obtain from this the assertion of the theorem, let us consider an infinite sequence of points P 1,P 2,…,P k ,… of the projective space ℙ(L). Let us focus attention first on the sequence of zeroth (that is, having index 0) coordinates of the vectors x 1,x 2,…,x k ,… corresponding to these points. Suppose they are the numbers
$$ \alpha _{10}, \alpha _{20},\ldots, \alpha _{k0},\ldots. $$
(9.30)
As we noted above, we may assume that all |α k0| are less than or equal to 1. By the assertion from real analysis formulated above, from the sequence (9.30), we may choose a subsequence
$$ \alpha _{n_1 0}, \alpha _{n_2 0}, \ldots, \alpha _{n_k 0}, \ldots, $$
(9.31)
converging to some number β 0 that therefore also does not exceed 1 in absolute value. Let us now consider a subsequence of points \(P_{n_{1}}, P_{n_{2}},\ldots, P_{n_{k}},\ldots\) and of vectors \({\boldsymbol{x} }_{n_{1}}, {\boldsymbol{x} }_{n_{2}},\ldots, {\boldsymbol{x} }_{n_{k}},\ldots\) with the same indices as those in the subsequence (9.31). Let us focus attention on the first coordinate of these vectors. For them, clearly, it is also the case that \(|\alpha _{n_{k} 1}| \le1\). This means that from the sequence
$$\alpha _{n_1 1}, \alpha _{n_2 1}, \ldots, \alpha _{n_k 1}, \ldots $$
we may choose a subsequence converging to some number β 1, and moreover, clearly |β 1|≤1.
Repeating this argument n+1 times, we obtain as a result, from the original sequence of vectors x 1,x 2,…,x k ,… , a subsequence \({\boldsymbol{x} }_{m_{1}}, {\boldsymbol{x} }_{m_{2}},\ldots, {\boldsymbol{x} }_{m_{k}},\ldots\) converging to some vector \({\overline{{\boldsymbol{x} }}} \in {\mathsf{L}}\), which, like every vector of this space, can be decomposed in terms of the basis e 0,e 1,…,e n , that is,
$${\overline{{\boldsymbol{x} }}} = \beta _0 {\boldsymbol{e} }_0 + \beta _1 {\boldsymbol{e} }_1 + \cdots+ \beta _n {\boldsymbol{e} }_n. $$
This gives us the assertion of Theorem 9.21 if we ascertain that not all coordinates β 0,β 1,…,β n of the vector \({\overline{{\boldsymbol{x} }}}\) are equal to zero. But this follows from the fact that by construction, for each vector \({\boldsymbol{x} }_{m_{k}}\) of the subsequence \({\boldsymbol{x} }_{m_{1}}, {\boldsymbol{x} }_{m_{2}},\ldots, {\boldsymbol{x} }_{m_{k}},\ldots\) , a certain coordinate \(\alpha _{m_{k} i}\), i=0,…,n, has absolute value equal to 1. Since there exists only a finite number of coordinates, and the number of vectors \({\boldsymbol{x} }_{m_{k}}\) is infinite, there must be an index i such that among the coordinates \(\alpha _{{m_{k}} i}\), infinitely many will have absolute value 1. On the other hand, by construction, the sequence \(\alpha _{m_{1}i}, \alpha _{m_{2} i},\ldots, \alpha _{m_{k} i},\ldots\) converges to the number β i , which therefore must have absolute value equal to 1. □

The property established in Theorem 9.21 is called compactness. It holds as well for every projective algebraic variety of a projective space (whether real or complex). We may formulate it as follows.

Corollary 9.22

In the case of a real or complex space, the points of a projective algebraic variety form a compact set.

Proof

Let the projective algebraic variety X be given by a system of equations (9.5), and let P 1,P 2,…,P k ,… be a sequence of points in X. By Theorem 9.21, there exists a subsequence of this sequence that converges to some point P of this space. It remains to prove that the point P belongs to the variety X. For this, it suffices to show that it can be represented in the form P=〈u〉, where the coordinates of the vector u satisfy equations (9.5). But this follows at once from the fact that polynomials are continuous functions. Let F(x 0,x 1,…,x n ) be a polynomial (in this case, homogeneous; it is one of the polynomials F i appearing in the system of equations (9.5)). We shall write it in the form F=F(x), where xL. Then from the convergence of the vectors x k x as k→∞ such that F(x k )=0 for all k, it follows that F(x)=0. □

For subsets of a finite-dimensional vector or affine space (whether real or complex), the property of compactness is related to their boundedness—more precisely, the property of boundedness follows from compactness. Thus while real and complex vector or affine spaces can be visualized as “extending unboundedly in all directions,” for projective spaces, such is not the case. But what does it mean to say “can be visualized”? In order to formulate this intuitive idea precisely, we shall introduce for the real and complex projective lines some simple geometric representations to which they are homeomorphic (see the relevant definition on p. xviii). This will allow us to give a precise meaning to the words that a given set “can be visualized.” Let us observe that the property of compactness established in Theorem 9.21 is unchanged under a transition from one set to another that is homeomorphic to it.

Let us begin with the simplest situation: a one-dimensional real projective space, that is, the real projective line. It consists of pairs (x 0:x 1), where x 0 and x 1 are considered only up to a common factor λ≠0. Those pairs for which x 0≠0 form an affine subset U, whose points are given by the single coordinate t=x 1/x 0, so that we may identify the set U with ℝ. Pairs for which x 0=0 do not enter the set U, but they correspond to only one point (0:1) of the projective line, which we shall denote by (∞). Thus the real projective line can be represented in the form ℝ∪(∞).

The convergence of points P k Q as k→∞ is defined in this case as follows. If points P k ≠(∞) correspond to the numbers t k , and the point Q≠(∞) corresponds to the number t, then P k =(α k :β k ) and Q=(α:β), where β k /α k =t k , α k ≠0, and β/α=t, α≠0. The convergence P k Q as k→∞ in this case implies the convergence of the sequence of numbers t k t as k→∞. In the case that P k →(∞), the convergence (in the previous notation) means that α k →0, β k →1 as k→∞, from which it follows that \(t_{k}^{-1} \to0\), or equivalently, |t k |→∞ as k→∞.

We can graphically represent the real projective line by drawing a circle tangent to the horizontal line l at the point O; see Fig. 9.5. Connecting the highest point O′ of this circle with an arbitrary point A of the circle, we obtain a line that intersects l at some point B. We thereby obtain a bijection between points AO′ of the circle and all the points B of the line l. If we place the coordinate origin of the line l at the point O and associate with each point Bl a number t∈ℝ resulting from a choice of some unit measure on the line l (that is, an arbitrary point of the line l different from O is given the value 1), then we obtain a bijection between numbers t∈ℝ and points AO′ of the circle. Then |t k |→∞ if and only if for the corresponding points A k of the circle, we have the convergence A k O′. Consequently, we obtain a bijection between points of the real projective line ℝ∪(∞) and all points of the circle that preserves the notion of convergence. Thus we have proved that the real projective line is homeomorphic to the circle, which is usually denoted by S 1 (the one-dimensional sphere).
Fig. 9.5

The real projective line

An analogous argument can be applied to the complex projective line. It is represented in the form ℂ∪(∞). On it, the convergence of a sequence of points P k Q as k→∞ in the case Q≠(∞) corresponds to convergence of a sequence of complex numbers z k z, where z∈ℂ, while the convergence of the sequence of points P k →(∞) corresponds to the convergence |z k |→∞ (here |z| denotes the modulus of the complex number z).

For the graphical representation of the complex projective line, Riemann proposed the following method; see Fig. 9.6. The complex numbers are depicted in the usual way as points in a plane. Let us consider a sphere tangent to this plane at the origin O, which corresponds to the complex number z=0. Through the highest point O′ of the sphere and any other point A of the sphere there passes a line intersecting the complex plane at a point B, which represents some number z∈ℂ. This yields a bijection between numbers z∈ℂ and all the points of the sphere, with the exception of the point O′; see Fig. 9.6. This correspondence is often called the stereographic projection of the sphere onto the plane. By associating the point (∞) of the complex projective line with the point O′ of the sphere, we obtain a bijection between the points of the complex projective line ℂ∪(∞) and all the points of the sphere. It is easy to see that convergence is preserved under this assignment. Thus the complex projective line is homeomorphic to the two-dimensional sphere in three-dimensional space, which is denoted by S 2.
Fig. 9.6

Stereographic projection of the sphere onto the plane

In the sequel, we shall limit our consideration to projective spaces ℙ(L), where L is a real vector space of some finite dimension, and we shall consider for such spaces the property of orientability. It is related to the concept of continuous deformation of a linear transformation, which was introduced in Sect.  4.4.

By definition, every projective transformation of a projective space ℙ(L) has the form Open image in new window , where Open image in new window is a nonsingular linear transformation of the vector space L. Moreover, as we have seen, the linear transformation Open image in new window is determined by the projective transformation up to a replacement by Open image in new window , where α is any nonzero number.

Definition 9.23

A projective transformation is said to be continuously deformable into another if the first can be represented in the form Open image in new window and the second in the form Open image in new window , and the linear transformation Open image in new window is continuously deformable into Open image in new window .

Theorem 4.39 asserts that a linear transformation Open image in new window is continuously deformable into Open image in new window if and only if the determinants Open image in new window and Open image in new window have the same sign. What happens under a replacement of Open image in new window by Open image in new window ? Let the projective space ℙ(L) have dimension n. Then the vector space L has dimension n+1, and Open image in new window . If the number n+1 is even, then it is always the case that α n+1>0, and such a replacement does not change the sign of the determinant. In other words, in a projective space of odd dimension n, the sign of the determinant Open image in new window of a linear transformation Open image in new window is uniquely determined by the transformation Open image in new window . This clearly yields the following result.

Theorem 9.24

In a projective space of odd dimension, a projective transformation Open image in new window is continuously deformable into Open image in new window if and only if the determinants Open image in new window and Open image in new window have the same sign.

The same considerations can be applied to projective spaces of even dimension, but they lead to a different result.

Theorem 9.25

In a projective space of even dimension, every projective transformation is continuously deformable into every other projective transformation.

Proof

Let us show that every projective transformation Open image in new window is continuously deformable into the identity. If Open image in new window , then this follows at once from Theorem 4.39. And if Open image in new window , then the same theorem gives us that the transformation Open image in new window is continuously deformable into Open image in new window , which has matrix Open image in new window , where E n is the identity matrix of order n. But Open image in new window , and the transformation Open image in new window has matrix Open image in new window . Since in our case, the number n is even, it follows that |−E n |=(−1) n >0, and by Theorem 4.38, the matrix Open image in new window is continuously deformable into E n+1, and consequently, the transformation Open image in new window is continuously deformable into the identity. Thus the projective transformation Open image in new window is continuously deformable into Open image in new window , and this means by definition, that Open image in new window is also continuously deformable into Open image in new window . □

Expressing these facts in topological form, we may say that the set of projective transformations of the space ℙ n of a given dimension has a single path-connected component if n is even, and two path-connected components if n is odd.

Theorems 9.24 and 9.25 show that the properties of projective spaces of even and odd dimension are radically different. We encounter this for the first time in the case of the projective plane. It differs from the vector (or Euclidean) plane in that it has not two, but only one orientation. It is the same with projective spaces of arbitrary even dimension. We saw in Sect.  4.4 that the orientation of the affine plane can be interpreted as a choice of direction of motion around a circle. Theorem 9.25 shows that in the projective plane, this is already not the case—the continuous motion in a given direction around a circle in the projective plane can be transformed into motion in the opposite direction. This is possible only because our deformation at a certain moment “passes through infinity,” which is impossible in the affine plane.

This property can be presented graphically using the following construction, which is applicable to real projective spaces of arbitrary dimension.

Let us assume that the vector space L defining our projective space ℙ(L) is a Euclidean space, and let us consider in this space the sphere S, defined by the equality |x|=1. Every line 〈x〉 of the space L intersects the sphere S. Indeed, such a line consists of vectors of the form α x, where α∈ℝ, and the condition α xS means that |α x|=1. Since |α x|=|α|⋅|x| and x0, we may set |α|=|x|−1. With this choice, the number α is determined up to sign, or in other words, there exist two vectors, e and −e, belonging to the line 〈x〉 and to the sphere S. Thus associating with each vector eS the line 〈x〉 of the projective space, we obtain the mapping f:S→ℙ(L). The previous reasoning shows that the image of f is the entire space ℙ(L). However, this mapping f is not a bijection, since two points of the sphere S pass through one point P∈ℙ(L), corresponding to the line 〈x〉, namely, the vectors e and −e. This property is expressed by saying that the projective space is obtained from the sphere S via the identification of its antipodal points.

Let us apply this to the case of the projective plane, that is, we shall suppose that dimℙ(L)=2. Then dimL=3, and the sphere S contained in three-dimensional space is the sphere S 2. Let us decompose it into two equal parts by a horizontal plane; see Fig. 9.7.
Fig. 9.7

A model of the projective plane

Each point of the upper hemisphere is diametrically opposite some point on the lower hemisphere, and we can map the upper hemisphere onto the projective plane ℙ(L) by representing each point P∈ℙ(L) in the form 〈e〉, where e is a vector of the upper hemisphere.

However, this correspondence will not be a bijection, since antipodal points on the boundary of the hemisphere will be joined together, that is, they correspond to a single point; see Fig. 9.8. This is expressed by saying that the projective plane is obtained by identifying antipodal points of the boundary of the hemisphere.
Fig. 9.8

Identification of points

Let us now consider a moving circle with a given direction of rotation; see Fig. 9.9. In the figure is shown that when the moving circle intersects the boundary of the hemisphere, the direction of rotation changes to its opposite.
Fig. 9.9

Motion of a circle

This property is expressed by saying that the projective plane is a one-sided surface (while the sphere in three-dimensional space and other familiar surfaces are two-sided). This property of the projective plane was studied by Möbius. He presented an example of a one-sided surface that is now known as the Möbius strip. It can be constructed by cutting from a sheet of paper the rectangle ABDC (Fig. 9.10, left) and gluing together its opposite sides AB and CD, after rotating CD by 180. The one-sided surface thus obtained is shown in the right-hand picture of Fig. 9.10, where is also shown the continuous deformation of the circle (stages 1→2→3→4), changing the direction of rotation to it opposite.
Fig. 9.10

Möbius strip

The Möbius strip also has a direct relationship to the projective plane. Namely, let us visualize this plane as the sphere S 2, in which antipodal points are identified. Let us divide the sphere into three parts by intersecting it with two parallel planes that pass above and below the equator. As a result, the sphere is partitioned into a central part U and two “caps” above and below; see Fig. 9.11.
Fig. 9.11

Partition of the sphere

Let us begin by studying the central section U. For each point of U, its antipodal point is also contained in U. Let us divide U into two halves—front and back—by a vertical plane intersecting U in the arcs AB and CD; see Fig. 9.12.
Fig. 9.12

The central part of the sphere

We may combine the front half (U′) with the rectangle ABDC in Fig. 9.10. Every point of the central section U either itself belongs to the front half or else has an antipodal point that belongs to the front half, of which there is only one, except for the points of the segments AB and CD. In order to obtain only one of the two antipodal points of these segments, we must glue these segments together exactly as is done in Fig. 9.10. Thus the Möbius strip is homeomorphic to the part U′ of the projective plane. To obtain the remaining part V=ℙ(L)∖U′, we have to consider the “caps” on the sphere; see Fig. 9.11. For every point in a cap, its antipodal point lies in the other cap. This means that by identifying antipodal points, it suffices to consider only one cap, for example the upper one. This cap is homeomorphic to a disk: to see this, it suffices simply to project it onto the horizontal plane. Clearly, the boundary of the upper cap is identified with the boundary of the central part of the sphere. Thus the projective plane is homeomorphic to the surface obtained by gluing a circle to the Möbius strip in such a way that its boundary is identified with the boundary of the Möbius strip (it is easily verified that the boundary of the Möbius strip is a circle).

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Igor R. Shafarevich
    • 1
  • Alexey O. Remizov
    • 2
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.CMAPÉcole Polytechnique CNRSPalaiseau CedexFrance

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