Real projective space explained

In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in the real space It is a compact, smooth manifold of dimension, and is a special case of a Grassmannian space.

Basic properties

Construction

As with all projective spaces, is formed by taking the quotient of

\Rⁿ⁺¹\setminus\{0\}

under the equivalence relation for all real numbers . For all in

\Rⁿ⁺¹\setminus\{0\}

one can always find a such that has norm 1. There are precisely two such differing by sign. Thus can also be formed by identifying antipodal points of the unit -sphere,, in

\Rⁿ⁺¹

One can further restrict to the upper hemisphere of and merely identify antipodal points on the bounding equator. This shows that is also equivalent to the closed -dimensional disk,, with antipodal points on the boundary,

\partialD^n=S^n-1

, identified.

Low-dimensional examples

is called the real projective line, which is topologically equivalent to a circle.
is called the real projective plane. This space cannot be embedded in . It can however be embedded in and can be immersed in (see here). The questions of embeddability and immersibility for projective -space have been well-studied.^[1]
is diffeomorphic to SO(3), hence admits a group structure; the covering map is a map of groups Spin(3) → SO(3), where Spin(3) is a Lie group that is the universal cover of SO(3).

Topology

The antipodal map on the -sphere (the map sending to) generates a Z₂ group action on . As mentioned above, the orbit space for this action is . This action is actually a covering space action giving as a double cover of . Since is simply connected for, it also serves as the universal cover in these cases. It follows that the fundamental group of is when . (When

n=1

the fundamental group is due to the homeomorphism with). A generator for the fundamental group is the closed curve obtained by projecting any curve connecting antipodal points in down to .

The projective -space is compact, connected, and has a fundamental group isomorphic to the cyclic group of order 2: its universal covering space is given by the antipody quotient map from the -sphere, a simply connected space. It is a double cover. The antipode map on has sign

(-1)^p

, so it is orientation-preserving if and only if is even. The orientation character is thus: the non-trivial loop in

	n)
\pi
	1(RP

acts as

(-1)ⁿ⁺¹

on orientation, so is orientable if and only if is even, i.e., is odd.^[2]

The projective -space is in fact diffeomorphic to the submanifold of

	(n+1)²
\R

consisting of all symmetric matrices of trace 1 that are also idempotent linear transformations.

Geometry of real projective spaces

Real projective space admits a constant positive scalar curvature metric, coming from the double cover by the standard round sphere (the antipodal map is locally an isometry).

For the standard round metric, this has sectional curvature identically 1.

In the standard round metric, the measure of projective space is exactly half the measure of the sphere.

Smooth structure

Real projective spaces are smooth manifolds. On Sⁿ, in homogeneous coordinates, (x₁, ..., x_n+1), consider the subset U_i with x_i ≠ 0. Each U_i is homeomorphic to the disjoint union of two open unit balls in Rⁿ that map to the same subset of RPⁿ and the coordinate transition functions are smooth. This gives RPⁿ a smooth structure.

Structure as a CW complex

Real projective space RPⁿ admits the structure of a CW complex with 1 cell in every dimension.

In homogeneous coordinates (x₁ ... x_n+1) on Sⁿ, the coordinate neighborhood U₁ = can be identified with the interior of n-disk Dⁿ. When x_i = 0, one has RPⁿ⁻¹. Therefore the n−1 skeleton of RPⁿ is RPⁿ⁻¹, and the attaching map f : Sⁿ⁻¹ → RPⁿ⁻¹ is the 2-to-1 covering map. One can put $\mathbf^n = \mathbf^ \cup_f D^n.$

Induction shows that RPⁿ is a CW complex with 1 cell in every dimension up to n.

The cells are Schubert cells, as on the flag manifold. That is, take a complete flag (say the standard flag) 0 = V₀ < V₁ <...< V_n; then the closed k-cell is lines that lie in V_k. Also the open k-cell (the interior of the k-cell) is lines in (lines in V_k but not V_k−1).

In homogeneous coordinates (with respect to the flag), the cells are $\begin[*:0:0:\dots:0] \\$

Notes and References

See the table of Don Davis for a bibliography and list of results.
Book: J. T. Wloka. B. Rowley . B. Lawruk . Boundary Value Problems for Elliptic Systems. 1995 . Cambridge University Press. 978-0-521-43011-1. 197.