In mathematics, a projective space is a fundamental construction, obtained from a vector space over an arbitrary division ring, in particular over a field. It generalises the notion of projective plane, which is constructed from a three-dimensional vector space. Projective spaces are essential to algebraic geometry through the rich field of projective geometry developed in the nineteenth century, but also in the constructions of the modern theory (based on graded algebras). Projective spaces and their generalisation to flag manifolds also play a big part in topology, the theory of Lie groups and algebraic groups, and their representation theory. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... :For other senses of this word, see dimension (disambiguation). ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Projective geometry is a non-metrical form of geometry that emerged in the early 19th century. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ... In mathematics, a flag manifold (or flag variety) is the set of all flags in a finite-dimensional vector space V. The flag variety on V is naturally a projective variety. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ...

The basic construction, given a vector space V over a division ring K, is to form the set of equivalence classes of non-zero vectors in V under the relation of scalar proportionality: we consider v to be proportional to w if v = cw with c in K non-zero. This idea goes back to mathematical descriptions of perspective. If K is the real numbers, and V has dimension n, then the projective space P(V)—which we can talk about as the space of lines through the zero element 0 of V—carries a natural structure of a compact smooth manifold of dimension n − 1. It is also highly symmetric, since any linear automorphism of V gives rise to a symmetry of P(V). These in the classical examples identify with 'perspectivity' and 'projectivity' transformations described geometrically, and account for the name. The group of these symmetries is the quotient of the general linear group of V by the subgroup of non-zero scalar multiples of the identity. In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ... A cube in two-point perspective. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. ...

The use of projective spaces makes quite rigorous the talk about a 'line at infinity' (where parallel lines meet), or a 'plane at infinity' for three dimensions: a translation of the latter can be made as part of the projective space associated to a four-dimensional real vector space. In that way geometrical ideas introduced by Poncelet and others become part of a theory founded on linear algebra. The part of a projective space not 'at infinity' is called affine space; but the symmetries of P(V) do not respect that division. Use of a basis of V allows, if required, the introduction of homogeneous co-ordinates for the handling of concrete calculations. In geometry and topology, the line at infinity is a line which is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. ... In projective geometry, the plane at infinity is a projective plane which is added to the affine 3-space in order to give it closure of incidence properties. ... Jean-Victor Poncelet (July 1, 1788 РDecember 22, 1867) was a mathematician and engineer who did much to revive projective geometry. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... In mathematics, homogeneous co-ordinates, introduced by August Ferdinand M̦bius, make calculations possible in projective space just as Cartesian co-ordinates do in Euclidean space. ...

Use of vector spaces over the field of complex numbers gives rise to different manifolds, also used by geometers. There are good reasons for using them, in order to get a theory about intersections of algebraic varieties with predictable properties. In the theory of Alexander Grothendieck there are reasons for applying the construction outlined above rather to the dual space V*. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... Alexander Grothendieck (Berlin, March 28, 1928) is one of the most important mathematicians of the 20th century. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...

Introduction

As outlined above, projective space is a geometric object formalizing statements like "Parallel lines intersect at infinity". For concreteness and simplicity, we will give the construction of the real projective plane Rℙ² in some detail. There are three equivalent definitions: First, the set of all lines in (real 3-)space R³ passing through the origin (0, 0, 0). Every such line meets the sphere of radius one centered in the origin exactly twice, say in P = (x, y, z) and its antipodal point (-x, -y, -z). Thus Rℙ² can also be described to be the points on the sphere S², where every point P and its antipodal point are not distinguished. For example, the point (1, 0, 0) (red point in the image) is identified with (-1,0,0) (light red point) etc. Finally, yet another equivalent definition is the set of equivalence classes of R³(0,0,0), i.e. 3-space without the origin, where two points P=(x, y, z) and P' =(x', y', z' ) are equivalent iff there is a nonzero real number λ such that P = λ·P' , i.e. x = λx' , y = λy' , z = λz' . The usual way to write an element of the projective plane, i.e. the equivalence class corresponding to an honest point (x, y, z) in R³ is Image File history File links Size of this preview: 225 × 225 pixelsFull resolution (225 × 225 pixel, file size: 2 KB, MIME type: image/png)Jakob. ... The fundamental polygon of the projective plane. ... A sphere is a perfectly symmetrical geometrical object. ... In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...

[x : y : z]

This goes under the name of homogenous coordinates. In mathematics, homogeneous co-ordinates, introduced by August Ferdinand Möbius, make calculations possible in projective space just as Cartesian co-ordinates do in Euclidean space. ...

Notice that any point [x : y : z] with z≠0 is equivalent to [x/z: y/z: 1]. So there are two disjoint subsets of projective plane. First the points [x : y : z] = [x/z: y/z: 1] if z≠0 and [x : y : 0]. The latter set can be subdivided similarly in two disjoint subsets [x/y : 1 : 0] and [x : 0 : 0]. In the latter case, x is necessarily nonzero, because the origin was not part of Rℙ². Thus the point is equivalent to [1 : 0 : 0]. Geometrically, the first subset, which is isomorphic (at this level, as a set, but later as a manifold, for example) to R² is in the image the yellow upper hemisphere (without the equator), equivalently the lower hemisphere. The second subset, isomorphic to R¹, corresponds to the green line (without the two marked points), or, again, equivalently the light green line. Finally we have the red point or the equivalent light red point. (Of course, the chosen order of the coordinates is by no means distinguished). We thus have a disjoint decomposition

Rℙ² = R² ⊔ R¹ ⊔ point.

Intuitively already clear, and made precise below, R¹ ⊔ point is itself the real projective line Rℙ¹. Considered as a subset of Rℙ², it is called line at infinity, whereas R² ⊂ Rℙ² is called affine plane, i.e. just the usual plane. In mathematics, the projective line is a fundamental example of an algebraic curve. ...

The next objective is make precise the saying: "parallel lines meet at infinity". A natural bijection between the the plane z=1 (which meets the sphere at the north pole N = (0, 0, 1)) and the affine plane (i.e. the upper hemisphere) inside projective plane is accomplished by the stereographic projection, i.e. any point P on this plane is mapped to the intersection point of the line through the origin and P and the sphere. Therefore two lines L₁ and L₂ (blue) in the plane are mapped to what looks like great circles (antipodal points are identified, though). Great circles intersect precisely in two antipodal points, which are identified in the projective plane, i.e. any two lines have exactly one intersection point inside Rℙ². This phenomenon is axiomatized and studied in projective geometry. Image File history File links Size of this preview: 800 × 480 pixelsFull resolution (865 × 519 pixel, file size: 9 KB, MIME type: image/png) Cirlce3D[p_,v1_,v2_,r_]= Polygon[Table[p+r Sin[i 2[Pi]/n]v1+r Cos[i 2[Pi]/n]v2,{i, 0, n}]]; c... North Pole Scenery When not otherwise qualified, the term North Pole usually refers to the Geographic North Pole – the northernmost point on the surface of the Earth, where the Earths axis of rotation intersects the Earths surface. ... Stereographic projection of a circle of radius R onto the x axis. ... A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the sphere into two equal hemispheres. ... Projective geometry is a non-metrical form of geometry that emerged in the early 19th century. ...

Definition of projective space

Real projective space is defined by In mathematics, real projective space, or RPn is the projective space of lines in Rn+1. ...

Rℙⁿ := (Rⁿ⁺¹)/~,

with the equivalence relation (x₀, ..., x_n) ~ (λx₀, ..., λx_n), where λ is an arbitrary non-zero real number. Equivalently, it is the set of all lines in Rⁿ⁺¹ passing through the origin (0, ..., 0).

Taking complex numbers or quaternions instead, one obtains the complex projective space Cℙⁿ and quaternionic projective space Hℙⁿ. In algebraic geometry, one uses arbitrary fields k, as well, the usual notation is then ℙⁿ_k. The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, complex projective space, or CPn, is the projective space of (complex) lines in Cn+1. ... In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H. Quaternionic projective space of dimension n is usually denoted by HPn and is a closed manifold of (real) dimension... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

If n is one or two, it is also called projective line or projective plane, respectively. In mathematics, a projective line is a one-dimensional projective space. ... Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ...

As in the above special case, the notation (so-called homogenous coordinates) for a point in projective space is

[x₀: ...: x_n].

Slightly more general, for a vector space V (over some field k), ℙ(V) is defined to be (V0)/~, where two non-zero vectors v₁, v₂ in V are equivalent if they differ by a non-zero scalar λ, i.e. v₁ = λ·v₂. The vector space need not be finite-dimensional, which is used, for example, in the theory of projective Hilbert spaces. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics and the foundations of quantum mechanics, the projective Hilbert space P(H) of a complex Hilbert space is the set of equivalence classes of vectors v in H, with v ≠ 0, for the relation given by v ~ w when v = λw with λ a scalar, that...

Projective space as a manifold

The above definition of projective space gives a set. For purposes of differential geometry which deals with manifolds, it is useful to endow this set with a (real or complex) manifold structure. This article is about sets in mathematics. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...

Namely consider the following subsets: U_i = {[x₀: ...: x_n,xi≠0]}, i=0, ..., n. By the definition of projective space, their union is the whole projective space. Further, U_i is in bijection to Rⁿ (or Cⁿ) via

(the hat means, that the i-th entry is missing).

We first define a topology on projective space by declaring that these maps shall be homeomorphisms, that is, an subset of U_i is open iff its image under the above isomorphism is an open subset (in the usual sense) of Rⁿ. An arbitrary subset A of Rℙⁿ is open if all intersections A ∩ U_i are open. This defines a topological space. A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... This word should not be confused with homomorphism. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

The manifold structure is given by the above maps, too.

The above decomposition in disjoint subsets reads in this generality:

Rℙⁿ = Rⁿ ⊔ R^n-1 ⊔ ... ⊔ R¹ ⊔ R⁰,

this so-called cell-decomposition can be used to calculate the singular cohomology of projective space. In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ...

All of the above for complex projective space, too. The complex projective line Cℙ¹ is an example of a Riemann surface. In mathematics, a projective line is a one-dimensional projective space. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ...

The covering by the above open subsets also shows, that projective space is a scheme, it is covered by n+1 affine n-spaces. The construction of projective scheme is an instance of the Proj construction. In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... Proj is a certain construction in mathematics, more precisely in the field of algebraic geometry. ...

Why the projective space is "better" than the affine space

There are a number of mathematically deeply meaningful advantages of the projective space against affine space (e.g. Rℙⁿ vs. Rⁿ).

• Projective space is a compact topological space, affine space is not. Thus, for example integrating functions or differential forms on ℙⁿ does not need cause convergence issues. In the parlance of algebraic geometry, projective space is proper.

• Probably surprisingly, for complex projective space, every complex submanifold X ⊂ Cℙⁿ, i.e. a manifold cut out by holomorphic equations is necessarily an algebraic variety, i.e. given by polynomial equations. This is Chow's theorem, it allows the direct use of algebraic-geometric methods for these ad-hoc analytically defined objects.

• As outlined above, lines in ℙ² or more generally hyperplanes in ℙⁿ always do intersect. This extends to non-linear objects, as well: appropriately defining the degree of an algebraic curve, which is roughly the degree of the polynomials needed to define the curve (see Hilbert polynomial), it is true (over an algebraically closed field k) that an two projective curves C₁ and C₂ ⊂ ℙⁿ_k of degree e and f intersect in exactly e·f points, counting them with multiplicities - see Bézout's theorem.

In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... Integration may be any of the following: Usually integration is the construction of an object, a theory, etc. ... Look up Function in Wiktionary, the free dictionary. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, a continuous function between topological spaces is called proper if inverse images of compact subsets are compact. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, algebraic geometry and analytic geometry are closely related subjects, where analytic geometry is the theory of complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... In mathematics, the Hilbert polynomial of a graded commutative algebra A = ⊕An over a field k that is generated by the finite dimensional space A1 is the unique polynomial f(x) with rational coefficients such that f(n) = dimk An for all but finitely many positive integers n. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ... This article refers to Bézouts theorem in algebraic geometry. ...

Axiomatic characterization of projective space

There is a alternative axiomatic approach to projective spaces defining them as an incidence structure with certain properties. This approach does not rely on the construction over vector spaces (ℙ(V)) and is in particular popular in the fields of finite geometry and combinatorics. For a projective space of dimension ≧3 it can be shown that it is isomorphic to ℙ(V) for some V. However for projective spaces of dimension 2 (projective planes) this is not true, i.e. there exist projective planes which are not isomorphic to ℙ(V) for any V, so the ℙ(V) construction does not describe all projective planes. A projective plane that is constructed over a Moulton plane is an example for such a projective plane, that cannot be described through ℙ(V) for some V. In mathematics, an axiomatic projective space S is a set P (the set of points), together with a set of subsets of P (the set of lines), all of which have at least three elements, satisfying these axioms : Each two distinct points p and q are in exactly one line. ... The Moulton plane. ...

Morphisms

Projective linear maps between two projective spaces over the same field, say, P(V) and P(W), have the form

where T is an element of L(V,W), the space of linear maps between V and W, v is an element of V, and we consider the equivalence classes under the defining identification of the respective projective spaces. Since members of the equivalence class differ by a scalar factor, and linear maps preserve scalar factors, this induced map is well-defined. On the other hand if T is not injective, it will have a null space larger than {0}; in this case the meaning of the class of T(v) is problematic if v is non-zero and in the null space. What to do in that case falls under birational geometry. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ... It has been suggested that this article or section be merged into kernel (mathematics). ... In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. ...

Two linear maps S and T in L(V,W) induce the same map between P(V) and P(W) iff they differ by a scalar multiple of the identity, that is if T=kS for some k ≠ 0. Thus if one identifies the scalar multiples of the identity map with the underlying field, the set of morphisms from P(V) to P(W) is simply P(L(V,W)). IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...

The automorphisms, the invertible projective linear maps from a projective space to itself, can be described more concretely. Consider the invertible linear maps from the underlying vector space to itself; these form a group, and the projective linear maps are an image of the group, under the map . Aut(P(V)) is the quotient group Aut(V)/Z(V), where Z(V) is again the group of nonzero scalar multiples of the identity, which is the kernel of the mapping. Z(V) is the center of Aut(V). This is why such quotient groups are known in general as projective linear groups. This picture illustrates how the hours in a clock form a group. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... In abstract algebra, the center (or centre) of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z ∈ G | gz = zg for all g ∈ G} Note that Z(G) is a subgroup of... The projective linear group of a vector space V over a field F is the quotient group PGL(V) = GL(V)/Z(V) where GL(V) is the general linear group on V and Z(V) is the group of all nonzero scalar transformations of V. The projective special linear...

Generalization

The projective space, being the "space" of all one-dimensional linear subspaces of a given vector space V is generalized to Grassmannian manifold, which is parametrizing higher-dimensional subspaces (of some fixed dimension) of V. More generally flag manifold is the space of flags, i.e. chains of linear subspaces of V. In mathematics, a Grassmannian is the space of all k-dimensional subspaces of a finite dimensional vector space V, often denoted Grk(V) or simply Grk(n) or G(k,n) when V is a standard n-dimensional vector space over a given field[1]. The Grassmannian is named after... In mathematics, a flag manifold (or flag variety) is the set of all flags in a finite-dimensional vector space V. The flag variety on V is naturally a projective variety. ...

Patching projective spaces together yields projective bundles.

See also

• projective transformation
• projective representation

A projective transformation is a transformation used in projective geometry: it is the composition of a pair of perspective projections. ... In mathematics, in particular in group theory, if G is a group and ρ is a vector space over a field K, then a projective representation is a homomorphism from G to Aut(ρ)/Kx where Kx here is the normal subgroup of Aut(ρ) consisting of multiplications of vectors...

External links

• http://mathworld.wolfram.com/ProjectiveSpace.html
• http://eom.springer.de/P/p075350.htm
• http://planetmath.org/encyclopedia/ProjectiveSpace.html

References

• Beutelspacher A./Rosenbaum U. : Projective Geometry. From Foundations to Applications: Cambridge University Press (1998)
• Coxeter, H.S.M. : Projective Geometry: University of Toronto Press (1974)
• Dembowski, P. : Finite Geometries. Springer (1968)