map on Rp has sign , so it is orientation-preserving if and only if p is even. The orientation character is thus: the non-trivial loop in acts as on orientation, so RPn is orientable if and only if is even, i.e., n is odd. The projective n-space is in fact diffeomorphic to the submanifold of R(n+1)2 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, ..., 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. 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 ... xn+1) on Sn, the coordinate neighborhood U1 = {(x1 ... xn+1) | x1 ≠ 0} can be identified with the interior of n-disk Dn. When xi = 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 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 <...< Vn; then the closed k-cell is lines that lie in Vk. Also the open k-cell (the interior of the k-cell) is lines in (lines in Vk but not Vk−1). 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