What is Projective space: Definition and 18 Discussions

In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines.
This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, point and lines are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the following definition, which is more often encountered in modern textbooks.
Using linear algebra, a projective space of dimension n is defined as the set of the vector lines (that is, vector subspaces of dimension one) in a vector space V of dimension n + 1. Equivalently, it is the quotient set of V \ {0} by the equivalence relation "being on the same vector line". As a vector line intersects the unit sphere of V in two antipodal points, projective spaces can be equivalently defined as spheres in which antipodal points are identified. A projective space of dimension 1 is a projective line, and a projective space of dimension 2 is a projective plane.
Projective spaces are widely used in geometry, as allowing simpler statements and simpler proofs. For example, in affine geometry, two distinct lines in a plane intersect in at most one point, while, in projective geometry, they intersect in exactly one point. Also, there is only one class of conic sections, which can be distinguished only by their intersections with the line at infinity: two intersection points for hyperbolas; one for the parabola, which is tangent to the line at infinity; and no real intersection point of ellipses.
In topology, and more specifically in manifold theory, projective spaces play a fundamental role, being typical examples of non-orientable manifolds.

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  1. Structure seeker

    I Multiplication in projective space

    Let #F# be a field and consider the projective space of dimension #n# over it with added the point #0#. It seems to me that there is a valid definition of multiplication by just entrywise multiplicating the elements. Of course both can be multiplied by #x \in F# but that goes for the product as...
  2. Euge

    POTW Steenrod Squares over an Infinite Projective Space

    Let ##u## be a generator of ##H^1(\mathbb{R} P^\infty; \mathbb{F}_2)##. Prove the relations $$\text{Sq}^i(u^n) =\binom{n}{i} u^{n+i}$$
  3. D

    I Is the projective space a smooth manifold?

    Suppose you have the map $$\pi : \mathbb{R}^{n+1}-\{0\} \longrightarrow \mathbb{P}^n$$. I need to prove that the map is differentiable. But this map is a chart of $$\mathbb{P}^n$$ so by definition is differentiable? MENTOR NOTE: fixed Latex mistakes double $ signs and backslashes needed for math
  4. K

    A Hausdorff property of projective space

    Hello! I am reading "Differential Geometry and Mathematical Physics" by Rudolph and Schmidt. And they have and example of manifold (projective space). I believe that there is a typo in the book, but perhaps I miss something deep. Definitions are the following $$\mathbb{K}^n_\ast=\{\mathbf{x}\in...
  5. V

    A The map from a complex torus to the projective algebraic curve

    I am following the proof to show that the complex torus is the same as the projective algebraic curve. First we consider the complex torus minus a point, punctured torus, and show there is a biholomorphic map or holomorphic isomorphism with the affine algebraic curve in ##\mathbb{C}^2##...
  6. Euler2718

    Can Three Non-Collinear Points Always Define a Projective Plane?

    Homework Statement Let P(W) be a projective space whose dimension is greater than or equal to 2 and let three non-colinear projective points, [v_{1}],[v_{2}],[v_{3}]\in P(W) . Prove that there is a projective plane in P(W) containing all three points. Homework EquationsThe Attempt at a...
  7. S

    A Concept of duality for projective spaces and manifolds

    I firstly learned about duality in context of differentiable manifolds. Here, we have tangent vectors populating the tangent space and differential forms in its co-tangent counterpart. Acting upon each other a vector and a form produce a scalar (contraction operation). Later, I run into the...
  8. S

    A Why are conics indistinguishable in projective geometry?

    It is said that curves of the second order which we usually refer to as ellipse, parabola and hyperbola, i. e. conics, are all represented on projective plane by closed curves (oval curve), which means there is no distinction between them. Why is it? Projective space can, in principle, be...
  9. Ssnow

    I Difference between the projective space and one its part

    Hi everybody, I want to ask if there are big differences consider algebraic objects in the projective or in the projective space with only positive coordinates? I know that the question is generic (so permit the discussion ... ). I am interested to know what happen to classical invariants if...
  10. M

    2D Projective Complex Space, Spin

    Just reviewing some QM again and I think I'm forgetting something basic. Just consider a qubit with basis {0, 1}. On the one hand I thought 0 and -0 are NOT the same state as demonstrated in interference experiments, but on the other hand the literature seems to say the state space is...
  11. I

    From a vector space to the projective space

    Dear all, I am not very experienced in this field, so, I have a rather simple question :smile: -Consider a linear vector space V of dimension 4. -Prescribe that, if two vectors in V differ by a nonvanishing constant, they belong to the same equivalence class. -Put together all these...
  12. A

    Interpretation of projective space

    Hi, I am am currently taking a second course in geometry, the first part of the course concerns projective geometry, and I feel I'm not getting the picture. I would like to know what the motivation for the development of projective geometry is. What picture you guys have in your head of...
  13. K

    Holomorphic Quotient Maps on Projective Space

    Homework Statement Let \mathbb{CP}^n be n-dimensional complex projective space, and let \pi: \mathbb C^{n+1}\setminus\{0\} \to \mathbb{CP}^n be the quotient map taking \pi(z_1,\ldots,z_{n+1}) = [z_1,\ldots,z_{n+1}] where the square brackets represent the equivalence class of lines through...
  14. K

    Is Smoothness the Key to Understanding Complex Projective Space?

    So I have been tasked with what is likely a very simple problem, but have forgotten so much complex analysis that I would like to very the problem. Let \mathbb{CP}^n denote the n-dimensional complex projective space. We want to show that the quotient map \pi: \mathbb C^{n+1}\setminus\{0\}...
  15. T

    What is the meaning of equivalence classes in complex projective space?

    I've been thinking...and am starting to think that I don't understand complex projective space...So, it's defined as ( Cn+1 \{0,0} / C\{0} ). Now, I think this is just the set of planes in 4 space that pass through the origin... and one can consider how they would all intersect a 3 sphere and...
  16. MTd2

    Could bosonic strings be a kind of projective space of m-theory?

    Just like you can create the ilusion of superluminal motion by projecting a flashlight into a wall, some strings would be projected into tachyons. Has anyone heard or thought about that?
  17. T

    Projective Space: CP1, Homeomorphism to 2-Sphere?

    I have a question about complex projective space... specifically CP1 which can be thought of as the action of C on C^2\{0} which gives rise to the equivalence classes of "lines" passing through the origin in C^2 (but not including the 0) Now, any vector in complex space, when multiplied by the...
  18. I

    RP^2 vs CP: Comparing Stenographic Representations

    What is the relationship betwenn RP^2 and CP? Espesicially, why are their stenographic representations different? As far as I understand the stenographic representation for RP^2 goes like that: a sphere with antipodal points identified is put above the R^2 plane, lines through the origin...