Editing Real projective space (section)

==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 manifold]]s. On ''S<sup>n</sup>'', in homogeneous coordinates, (''x''<sub>1</sub>, ..., ''x''<sub>''n''+1</sub>), consider the subset ''U<sub>i</sub>'' with ''x<sub>i</sub>'' ≠ 0. Each ''U<sub>i</sub>'' is homeomorphic to the disjoint union of two open unit balls in '''R'''<sup>''n''</sup> that map to the same subset of '''RP'''<sup>''n''</sup> and the coordinate transition functions are smooth. This gives '''RP'''<sup>''n''</sup> a [[smooth structure]].

===Structure as a CW complex ===
Real projective space '''RP'''<sup>''n''</sup> admits the structure of a [[CW complex]] with 1 cell in every dimension. 

In homogeneous coordinates (''x''<sub>1</sub> ... ''x''<sub>''n''+1</sub>) on ''S<sup>n</sup>'', the coordinate neighborhood ''U''<sub>1</sub> = {(''x''<sub>1</sub> ... ''x''<sub>''n''+1</sub>) | ''x''<sub>1</sub> ≠ 0} can be identified with the interior of ''n''-disk  ''D<sup>n</sup>''. When ''x<sub>i</sub>'' = 0, one has '''RP'''<sup>''n''−1</sup>. Therefore the ''n''−1 skeleton of '''RP'''<sup>''n''</sup> is '''RP'''<sup>''n''−1</sup>, and the attaching map ''f'' : ''S''<sup>''n''−1</sup> → '''RP'''<sup>''n''−1</sup> is the 2-to-1 covering map. One can put
<math display="block">\mathbf{RP}^n = \mathbf{RP}^{n-1} \cup_f D^n.</math>

Induction shows that '''RP'''<sup>''n''</sup> is a CW complex with 1 cell in every dimension up to ''n''. 

The cells are [[Schubert cell]]s, as on the [[flag manifold]]. That is, take a complete [[flag (linear algebra)|flag]] (say the standard flag) 0 = ''V''<sub>0</sub> < ''V''<sub>1</sub> <...< ''V<sub>n</sub>''; then the closed ''k''-cell is lines that lie in ''V<sub>k</sub>''. Also the open ''k''-cell (the interior of the ''k''-cell) is lines in {{math|''V<sub>k</sub>'' \ ''V''<sub>''k''−1</sub>}} (lines in ''V<sub>k</sub>'' but not ''V''<sub>''k''−1</sub>). 

In homogeneous coordinates (with respect to the flag), the cells are
<math display="block">
\begin{array}{c}
[*:0:0:\dots:0] \\
{[}*:*:0:\dots:0] \\
\vdots \\
{[}*:*:*:\dots:*].
\end{array}</math>

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 '''RP'''<sup>''n''</sup> is a CW complex. One such function is given by, in homogeneous coordinates,
<math display="block">g(x_1, \ldots, x_{n+1}) = \sum_{i=1} ^{n+1} i \cdot |x_i|^2.</math>

On each neighborhood ''U<sub>i</sub>'', ''g'' has nondegenerate critical point (0,...,1,...,0) where 1 occurs in the ''i''-th position with Morse index ''i''. This shows '''RP'''<sup>''n''</sup> is a CW complex with 1 cell in every dimension.

===Tautological bundles===
Real projective space has a natural [[line bundle]] over it, called the [[tautological bundle]]. More precisely, this is called the tautological subbundle, and there is also a dual ''n''-dimensional bundle called the tautological quotient bundle.