In mathematics, real projective space, denoted
It is a compact, smooth manifold of dimension n, and is a special case
can also be formed by identifying antipodal points of the unit
and merely identify antipodal points on the bounding equator.
As mentioned above, the orbit space for this action is
, it also serves as the universal cover in these cases.
A generator for the fundamental group is the closed curve obtained by projecting any curve connecting antipodal points in
-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
matrices of trace 1 that are also idempotent linear transformations.
[citation needed] 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.
Real projective spaces are smooth manifolds.
On Sn, in homogeneous coordinates, (x1, ..., xn+1), consider the subset Ui with xi ≠ 0.
Each Ui 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.
Real projective space RPn admits the structure of a CW complex with 1 cell in every dimension.
That is, take a complete flag (say the standard flag) 0 = V0 < V1 <...< Vn; then the closed k-cell is lines that lie in Vk.
In homogeneous coordinates (with respect to the flag), the cells are
This is not a regular CW structure, as the attaching maps are 2-to-1.
However, its cover is a regular CW structure on the sphere, with 2 cells in every dimension; indeed, the minimal regular CW structure on the sphere.
In light of the smooth structure, the existence of a Morse function would show RPn is a CW complex.
On each neighborhood Ui, g has nondegenerate critical point (0,...,1,...,0) where 1 occurs in the i-th position with Morse index i.
This shows RPn is a CW complex with 1 cell in every dimension.
More precisely, this is called the tautological subbundle, and there is also a dual n-dimensional bundle called the tautological quotient bundle.
by analogy with complex projective space.
The cellular chain complex associated to the above CW structure has 1 cell in each dimension 0, ..., n. For each dimensional k, the boundary maps dk : δDk → RPk−1/RPk−2 is the map that collapses the equator on Sk−1 and then identifies antipodal points.
RPn is orientable if and only if n is odd, as the above homology calculation shows.
The infinite real projective space is constructed as the direct limit or union of the finite projective spaces:
The double cover of this space is the infinite sphere
For each nonnegative integer q, the modulo 2 homology group